5.

The Hydrogen Atom

Based on the discussions in Chap. 4 we will now ap- it explains the physical interpretation of quantum
ply the quantum mechanical treatment to the simplest numbers, the description of the Zeeman effect and
atom, the H atom, which consists of one proton and one the fine structure by the model of angular momentum
electron moving in the spherical symmetric Coulomb vector couplings and gives a better understanding of
potential of the proton. These one-electron systems, the more complex many-electron systems that will be
such as the hydrogen atom and the ions He+ , Li++ , discussed in the next chapter.
Be+++ , etc., are the only real systems for which the
Schrdinger equation can be exactly (i.e., analytically)
solved. For all other atoms or molecules approxima- 5.1 Schrdinger Equation
tions have to be made. Either the Schrdinger equation for One-electron Systems
for these systems can be solved numerically (which of-
fers a mathematical solution within the accuracy of the The Schrdinger equation for a system consisting of
computer program, but generally gives little insight in- one electron (mass m 1 , charge q = e and radius vec-
to the physical nature of the approximation), or the real tor r 1 ) and a nucleus (mass m 2  m 1 , charge q =
atoms are described by approximate models that can be +Z e and radius vector r 2 ) is:
calculated analytically. In any case, for all multielec-
tron systems, one has to live with approximations, ei- h 2 h 2 Z e2
1 2 = E(r 1 , r 2 ) ,
ther in the numerical solution of the exact atomic model 2m 1 2m 2 4 0 r
or for the exact solution of the approximate model. (5.1)
A closer inspection of the spectrum of the hydrogen
where i is the Laplace operator with respect to ri . The
atom and other atoms reveals, however, that at high-
first term describes the kinetic energy of the electron,
er spectral resolution the lines show a substructure that
the second describes that of the nucleus and the third
cannot be described by the Schrdinger theory, but is
one the potential energy of the Coulomb interaction
due to new effects, such as fine structure, hyperfine
between the two particles, where r = |r 1 r 2 | is the
structure or the anomalous Zeeman effect. Therefore,
distance between the two particles. The wave function
even for the simple hydrogen atom the quantum me-
(r 1 , r 2 ) depends on the location of the electron and
chanical model of an electron in the Coulomb field of
nucleus, which means it depends on six coordinates.
the proton has to be modified by introducing new prop-
erties of electron and proton, such as electron spin or
5.1.1 Separation of the Center of Mass
proton spin and their mutual interactions. These effects,
and Relative Motion
which are small compared to the Coulomb energy, are
included in a relativistic theory, based on the Dirac In classical mechanics it is shown that the movement
equation, which is called quantum electrodynamics. of a closed system of particles can always be separated
The Schrdinger equation can be regarded as the fun- into the motion of the center of mass and the relative
damental equation of nonrelativistic quantum theory. motion of the particles in the center- of-mass system.
The treatment of the hydrogen atom illustrates in This is also possible in quantum mechanics as can be
a very clear way the basic ideas of quantum mechanics, seen by the following derivation.

W. Demtrder, Atoms, Molecules and Photons, 2nd ed., Graduate Texts in Physics,
DOI 10.1007/978-3-642-10298-1_5,  c Springer-Verlag Berlin Heidelberg 2010
160 5. The Hydrogen Atom

m2 the mixed terms in (5.3) cancel and the Schrdinger

equation becomes
r = | r | = | r1 r2 |
%  2 
m1 S h 2 2 2
r + +
M 2M X 2 Y 2 Z2
&
r2

z m 2 h 2 2 2 2
+ r + + 2+ 2 + E pot (r ) = E,
R M m1 2 x 2 y z
(5.4)

r1 where = (m 1 m 2 )/(m 1 + m 2 ) is the reduced mass of
y
the system.
For the solution of (5.4) we try the ansatz:
(R, r) = f (r)g(R).
x Inserting this into (5.4) yields after division by
Fig. 5.1. Transformation of laboratory frame into the center- h 2 R g h 2 r f
of-mass coordinate system = E pot (r ) + E , (5.5)
2M g 2 f
We regard a system of two particles with coordi- where R is the Laplace operator for (X , Y , Z ) and r
nates r 1 = {x1 , y1 , z 1 } and r 2 = {x2 , y2 , z 2 } written in that for (x, y, z). We can abreviate (5.5) by T1 = T2 +
lower case letters and the coordinates of the center of T3 + E.
mass (written in capital letters) (Fig. 5.1) The term T1 on the left side of (5.5) depends sole-
ly on the center-of-mass coordinates X , Y , Z . The two
m1 r 1 + m2 r 2
R= with M = m 1 + m 2 , other terms T2 and T3 , on the right side, depend sole-
M ly on the coordinates x, y, z. The total energy E of the
R = {X , Y , Z } ri = {xi , yi , z i } system is constant.
With the relative distance r = {x, y, z} = |r 1 r 2 | = Now we draw the same conclusion as in Sect. 4.3.2:
|{x1 x2 , y1 y2 , z 1 z 2 }| we obtain from Fig. 5.1 Since equation (5.5) has to be valid for arbitrary values
the relations: of the coordinates X , Y , Z and x, y, z, both sides of the
m2 m1 equation have to be constant. This means T1 as well as
r1 = R + r , r2 = R r. (5.2) T2 + T3 have to be constant, otherwise (5.5) can not be
M M
fulfilled for arbitrary choices of the coordinates. This
In order to properly write the Schrdinger equa-
gives the two conditions:
tion (5.1) in the coordinates r and R we have to
consider that the differentiation of the function #(r, R) h 2 R g
with respect to the variable xi (i = 1, 2) follows the = const = E g
2M g
chain rule:
h 2 r f
X x m 1 E pot (r ) = const = E f , (5.6)
= + = + 2 f
x1 X x1 x x1 M X x
  with E g + E f = E. We then obtain the two separate
2 m 1 X
= + equations
x12 X M X x x1
  h 2
m 1 x R g(R) = E g g(R) (5.7a)
+ + 2M
x M X x x1
h 2
m 21 2 2m 1 2 2 r f (r ) + E pot (r ) f (r ) = E f f (r ) . (5.7b)
= 2 + + . (5.3) 2
M X 2 M Xx x2
Analogous expressions are obtained for x2 . When (5.1) The first equation describes the kinetic energy E g =
(CM)
is written in the new coordinates (X , Y , Z ) and (x, y, z), E kin of the center-of-mass motion, which means the
 5.1. Schrdinger Equation for One-electron Systems 161

movement of the whole atom. Its solution is, as outlined Note:

in Chap. 4, the spatial part of the plane wave
The function R(r ) has nothing to do with the coordi-
i(k R(E g /h)t) nate R of the center of mass!
g(X , Y , Z ) = Ae .

With the de Broglie wavelength

5.1.2 Solution of the Radial Equation
2 h
CM = =$ , With the product-ansatz
k 2M E g
(r , , ) = R(r )Ylm (, )
which depends on the translational energy E g of the
center-of-mass motion. The correct description of this for the wave function #(r , , ) in Sect. 4.3.2 we had
motion has to use wave packets (see Sect. 3.3.13.3.4). already obtained (4.65) for the radial part R(r ), which
The relative motion of electron and nucleus is de- converts for m and C2 = l(l + 1) into
scribed by (5.7b). Renaming f (r ) as (r ), E f as E  
1 d 2 dR 2 " #
and r as , we obtain the Schrdinger equation 2
r + 2 E E pot (r ) R(r )
r dr dr h
l(l + 1)
h 2 = R(r ) . (5.9)
+ E pot (r ) = E, (5.8) r2
2 The integer l describes, according to (4.89), the integer
quantum number of the orbital angular momentum of
which is identical to the Schrdinger equation (4.40) the particle with respect to the origin r = 0 in our rel-
for a particle in a spherically symmetric potential if ative coordinate system, where the nucleus is at rest at
the mass m of the particle is replaced by the reduced r = 0.
mass = m 1 m 2 /(m 1 + m 2 ). Differentiation of the first term and introducing the
Coulomb-potential for E pot (r ) yields
The Schrdinger equation of a one-electron atom
d2 R 2 dR
can be separated into a term describing the center
2
+
of mass and a second term that describes the po- dr ( r dr  )
sition r of a particle with reduced mass relative 2 Z e2 l(l + 1)
+ E+ R=0.
to that of the nucleus at r = 0 in a potential with h 2 4 0 r r2
potential energy E pot . (5.10)

In the limit r all terms with 1/r and 1/r 2 in

In Sect. 4.3.2 we have already discussed the separa- the bracket approach zero and (5.10) becomes for this
tion of this equation in spherical coordinates (r , , ). limiting case:
It was shown there that the wave function
d2 R(r ) 2 d R 2
(r , , ) = R(r )Ylm (, ) 2
+ = 2 E R(r ) . (5.11)
dr r dr h
can be separated for arbitrary spherical potentials The solutions of this equation describe the asymp-
into a radial function R(r ) that depends on the r - totic behavior of the radial wave function R(r ). The
dependence of the potential and the angular part, which probability of finding the electron in a spherical shell
equals the spherical functions Ylm independent of the with volume 4r 2 dr around the nucleus between the
radial coordinate r . radii r and r + dr is given by 4|R|2r 2 dr . The ab-
In order to obtain the wave functions for the hy- solute square of the function r R therefore gives the
drogen atom we have to look for the radial wave probability of finding the electron within the unit vol-
function for the Coulomb potential. Inserting this func- ume of the spherical shell. We therefore introduce the
tion into the Schrdinger equation yields the energy W (r ) = r R(r ) into (5.11). This yields, with
function
eigenvalues E. k = 2E/h,
162 5. The Hydrogen Atom

Fig. 5.2. (a) In- This is an exponentially decreasing function which has
going and out- decayed to 1/e for r = 1/.
going spherical For the general solution, valid for all values of r , we
Electron wave waves as solutions
to the Schrdinger try the ansatz
Nucleus equation for an
electron with E > R(r ) = u(r )er . (5.12f)
= A eikr + B e ikr
0 in a spherical
potential. (b) Ex-
Inserting this into (5.10) we obtain for u(r ) the equation
E>0
perimentally de-   ( )
a) d2 u 1 du 2a 2 l(l + 1)
creasing wave am- +2 + u
plitude for E < 0 dr 2 r dr r r2
=0. (5.13)

= A e r Where the abbreviation a is

Z e2
b) E<0 a= .
r 4 0 h 2
The reciprocal value r1 = 1/a = 4 0 h 2 /(Z e2 )
gives, according to (3.85), the Bohr radius of the lowest
d2W energy level.
= k 2 W (5.12a)
dr 2 We write u(r ) as the power series
with the asymptotic solution .
u(r ) = bjr j . (5.14)
j
W (r ) = Aeikr + Beikr . (5.12b)
Inserting this into (5.13) the comparison of the co-
This gives for R(r ) = W (r )/r efficients of equal powers in r yields the recursion
A ikr B formula
R(r ) = e + eikr . (5.12c)
r r j a
b j = 2b j1 . (5.15)
For E > 0 k is real and the first term in (5.12b) j( j + 1) l(l + 1)
represents the spatial part of an outgoing spherical
Since R(r ) must be finite for all values of r , the pow-
wave
A er series can only have a finite number of summands.
(r , t) = ei(kr t) , (5.12d) If the last nonvanishing coefficient in the power se-
r ries (5.14) is bn1 , then b j becomes zero for j = n.
which describes an electron that can, with a positive This immediately gives, from (5.15), the condition that
total energy, leave the atom and can reach r only the coefficients b j with j < n contribute to the
(Fig. 5.2a). The second term corresponds to an incom- series (5.14).We therefore have the condition
ing spherical wave that represents an electron coming
from R = and approaching the nucleus (this is j <n. (5.16)
called a collision process). Since for j = n b j = 0 we obtain from (5.15)
For E < 0 we substitute = 2E/h = ik and
obtain the real asymptotic solutions a = n . (5.17)

R(r ) = Aer + Be+r . (5.12e) With = + 2E/h this yields the condition for the
energy values
Since R(r ) must be finite for all values of r (otherwise
the function R(r ) could not be normalized) it follows
that B = 0. We then obtain the asymptotic solution a 2 h 2 Z 2 e4 Z
2
En = = = Ry (5.18)
2n 2 802 h 2 n 2 n2
R(r ) = Aer .
 5.1. Schrdinger Equation for One-electron Systems 163

with the Rydberg constant Table 5.1. Normalized radial wave functions R(r ) (Laguerre-
Polynomials) of an electron in the Coulomb potential of
e4 the nucleus with charge Z e (N = (Z /na0 )3/2 ; x = Zr/na0 ;
Ry = . (5.18a)
802 h 2 a0 = 4 0 h 2 /(Z e2 ))

Note that (5.18) is identical to eq. (3.88) of Bohrs n l Rn,l (r )

model. 1 0 2N ex
2 0 2N ex (1 x)
The quantum mechanical calculation of one- 2 1 2 N ex x
electron systems gives the same energy values as 3  2

Bohrs atomic model. 3 0 2N ex 1 2x + 2x3

2 2N ex x(2 x)
3 1 3
Note: 3 2 4 N ex x 2
3 10  
3
4 0 2N ex 1 3x + 2x 2 x3
1. From the derivation of (5.18) it can be recognized   
2
that the discrete eigenvalues E n of possible energies 4 1 2 53 N ex x 1 x + x5
stem from the restraint (r ) 0, which im-  " #
plies that the electron is confined within a finite 4 2 2 15 N ex x 2 1 x3
spatial volume (see also Sect. 4.2.4). 4 3 2 N ex x 3
3 35
2. Besides the condition (5.18) for the energies there
is also a restraint for the angular momentum quan-
tum numbers l following from (5.15). According The energies E n can be calculated from (5.18)
to (4.59) l must be an integer. For the values j < n without the knowledge of the functions R(r ). These
that are allowed according to (5.16) the denomina- functions give, however, the radial electron distribution
tor in (5.15) would become zero for l = j, which and therefore the electric structure of the atom around
would result in an infinite coefficient b j . We there- the nucleus. This will be discussed in more detail in the
fore have to demand that in (5.15) all terms with following section.
l j must be zero in order to keep the function u(r )
finite.
We have then the condition 5.1.3 Quantum Numbers and Wave Functions
of the H Atom
l < j < n,
In the preceding section we have seen, that the energy
which gives the restraint for the angular momentum E(n, l) of an atomic state depends on certain quan-
quantum number l tum numbers: The principal quantum number n and the
quantum number l of the orbital angular momentum.
l n1. (5.19a) The stationary wavefunctions, which describe the
time-averaged spatial distribution of the electron, are
also characterized by these quantum numbers and
The projection of the angular momentum l onto the
in addition by the quantum number m of the z-
quantization axis (z-direction) is l z = m h with m =
component l z = m h of the orbital angular momentum
l, l + 1, . . . . . l 1, l. There are 2l + 1 possible
(see Sect. 4.4.2).
m-values.
The normalized wave functions
With the recursion formula (5.15) the functions u(r )
and with (5.12f) also the radial wave functions R(r ) (r , , ) = Rn,l (r )Ylm (, )
can be calculated successively. Table 5.1 lists, for the
lowest values of n and l, the radial functions R(r ). They discussed in Sects. 4.3.2 and 5.1.2 are also called
depend on n because of the condition (5.16) and on l atomic orbitals, because in the old Bohr-Sommerfeld
because of (5.15). theory the electron was assumed to move on certain
164 5. The Hydrogen Atom

Table 5.2. Normalized total wave functions of an electron in Table 5.3. Labeling of atomic states (l, m) using Latin and
the Coulomb potential E pot = Z e2 /(4 0 r ) Greek letters

n l m Eigenfunction n,l,m (r , , ) l state label |m| state label

  3 Zr 0 s 0
1 0 0 1 Z 2 e a0
a0
 3   Zr 1 p 1
2 0 0 1 Z 2 2 Zr e 2a0
4 2 a0 a0 2 d 2
 3 Zr
2 1 0 1 Z 2 Zr e 2a0 cos
3 f 3
4 2 a0 a0
 3 Zr
2 1 1 1 Z 2 Zr e 2a0 sin e 4 g 4
8 a 0 a 0
  3  Zr
Note: Often the nomenclature , , . . . is reserved for
3 0 0 1 Z 2 27 18 Zr + 2 Z 2 r 2 e 3a0
81 3 0 a 0 a a0 2 molecules (see Table 9.1). Since for atoms as well as for di-
  3  Zr atomic molecules this greek nomenclature gives the values of
3 1 0 2 Z 2 6 Zr Zr e 3a0 cos
81 a0 a0 a0 the projection m h, it is justified to use it for atoms as well
   3  Zr
1 1 2 Z 2
6 Zr Zr e 3a0 sin e
3 1 81 a0 0 0 a a
 3Zr
1 2 Z 2 r 2 3a0
Z
3 2 0
a0 e (3 cos2 1) index gives the direction of the lobe of the p-function
81 6 a02
  3 2 2 Zr (Fig. 4.27).
3 2 1 1
Z 2 Z r e 3a0 sin cos e
81 a0 a02 In Fig. 5.3 the radial wave functions of some atom-
  3 2 2 Zr ic states are illustrated. Together with the angular
3 2 2 1 Z 2 Z r e 3a0 sin2 e2
162 a0 a02 part Ylm (, ), drawn in Fig. 4.24, the total wave func-
tions can be visualized, as shown for two examples in
orbitals around the nucleus. This expression is, howev- Fig. 5.4, which represents the three-dimensional elec-
er, misleading, because we know from the discussion in tron distribution in the Coulomb potential for the 2s
Sect. 3.5 that we can not attribute to the electron a def- and the 2 p state.
inite path, but only a probability of finding it within Since, according to (5.18), the energy E n of an
a volume dV, given by ||2 dV . atomic state depends in this model solely on n and not
The normalized total wave functions for the low- on l or m, all states with possible combinations of l
est energy states of the hydrogen atom are compiled and m for the same n have the same energy. For each
in Table 5.2. They depend on the Quantum num- quantum number l there are 2l + 1 possible m values,
bers n, l and m. This also means that the probability because l m +l. The total number of different
of finding the electron at the position (r , , ), i.e., the
spatial electron distribution depends on these quantum
numbers.
R(r) n=1 2.0 R(r) n = 2 R(r) n=3
6 1.0
Each atomic state, described by its energy and its
spatial electron distribution is defined uniquely 4
l=0 1.0 l=0 0.5 l=0
by the three quantum numbers n, l and m.
0 0
2 8 4 8 12
In Fig. 4.28 the spatial density distributions
* m * 0.4 l=1 0.2 l=1
*Y (, )*2 are illustrated for l = 0, 1, 2. The different
l 0 0 0
(l, m) states are labeled according to an international 4 8 4 8 12
10 10
convention with lower case Latin letters, as compiled in r / 10 m r / 10 m 0.1 l=2
Table 5.3. For example, a state with quantum numbers 0
4 8 12
n = 2, l = 1 is a 2 p state, one with n = 4, l = 3 is a 4 f r / 1010 m
state. The magnetic quantum number m is indicated by Fig. 5.3. The radial wave function Rn,l (r ) for the principle
a coordinate index, e.g. the 2 p(m = 0) state is designat- quantum numbers n = 1, 2, 3. The ordinate is scaled in units
ed as 2 pz , the 2 p(m = 1) by 2 px on 2 p y , where the of 108 m3/2
 5.1. Schrdinger Equation for One-electron Systems 165

a) | ( x, z) |2 b) | ( x, z) |2

6
15 4
10 2
5 0
0 2
5 10 15 4 4 6
10 0 5 0 2
x / a0 15 10 5 66 4 2
z / a0 x / a0 z / a0
Fig. 5.4a,b. Illustration of the three-dimensional electron cases the two-dimensional projection of ||2 onto the x z-
charge distribution (a) for the spherical symmetric function plane is shown (calculated by H. von Busch, Kaiserslautern,
of the 2s state and (b) for the 2 p (m = 0) state. In both Germany)

atomic states with the same energy is then, because of EXAMPLES

l < n,
The state with the lowest energy (ground state) with
.
n1
n = 1, l = 0, m = 0 is a nondegenerate 1s-state.
k= (2l + 1) = n . 2
(5.19b)
States with n = 2 may have angular momentum
l=0
quantum numbers l = 0, m = 0 (2s) or l = 1 and m =
Different states with the same energy are called ener- 0(2 pz ) and m = 1 (2 px,y ). Such states are therefore
getically degenerate. The number of degenerate states fourfold degenerate.
is called the degeneracy order. The states in the
Coulomb potential (e.g., for the H atom or the He+
ion) are n 2 -fold degenerate, which means that n 2 states Note, that several effects (such as electron spin,
with different quantum numbers l and m and therefore nuclear spins, external fields or the relativistic mass de-
different wave functions, but the same quantum num- pendence), which are not included in the Schrdinger
ber n all have the same energy (Fig. 5.5). theory, may lift the degeneracy and split degenerate

E / eV
n= r n l=0 l=1 l=2 l=3 l=4 Ionization limit
0 5
0.84 n = 4 4 5s 5p 5d 5f 5g
4s 4p 4d 4f
1.5 3
n=3 3s 3p 3d

3.37 n = 2 2 2s 2p

e2
Ep =
4 0r

13.6 n=1 1 1s

Fig. 5.5. Level scheme of the H atom, drawn on a correct scale according the Schrdinger equation
166 5. The Hydrogen Atom

levels into components with different energies (see The function P(r ) for n = 1 is maximum for rm =
Sects. 5.35.6). a0 /Z , as can be seen immediately by differentiation of
(5.21). For Z = 1 one obtains the Bohr-radius rm = a0
(the maximum probability of finding the electron is at
5.1.4 Spatial Distributions and Expectation the Bohr-radius!). However, one should keep in mind
Values of the Electron in Different that the angular momentum obtained from the quantum
Quantum States mechanical treatment is l = 0, while the Bohr mod-
el gives l = 1. All experiments performed so far have
The spatial distribution of the electron in s states is
confirmed the quantum mechanical result.
spherically symmetric. The electron has the angular
If one would like to use a classical model for the
momentum
$ movement of the electron in the 1s state, one has to re-
|l| = l(l + 1)h = 0 place the circular path of the Bohr model by periodic
linear motions of the electron through the nucleus. The
in contrast to the Bohr-model, where the electron
direction of this oscillation is, however, randomly dis-
moves on a circular path around the nucleus with an an- tributed, causing an average electron distribution that
gular momentum |l| = h. We can see from Table 5.2, is spherically symmetric (Fig. 5.6). Arnold Sommerfeld
that the spatial probability density ||(r , , )|2 in the (18681951) showed that the electron motion can be
1s state has its maximum at r = 0, i.e, at the location described to proceed on very eccentric elliptical orbits
of the nucleus. passing close to the nucleus, which causes a fast pre-
When we want to calculate the probability P(r )dr cession of the large axis and brings about that the outer
to find the electron within a spherical shell in a distance turning points are uniformly distributed on a spherical
between r and r + dr from the nucleus, independent of surface.
the angles and , we have to solve the integral The expectation value r  for the mean distance


2 between electron and nucleus is given by
P(r ) dr = |(r , , )|2r 2 dr sin d d .



2
=0 =0
r  = r |(r , , )|2r 2 sin d d dr .
(5.20)
r =0 =0 =0
Inserting the wave function for n = 1, l = 0 and (5.22a)
m = 0 (i.e., for the ground state of the hydrogen atom),
we obtain For the 1s state this yields, after inserting the 1s wave
4Z 3 function from Table 5.2,
P(r ) dr = r 2 e2Zr/a0 dr . (5.21)
a03
| (r ) |2 = maximum for r = 0
Comparing this with the wave function for the 1s state
4 r 2 | (r ) |2 dr = maximum
we get the result: for r = a0 | 2|
r 2 | 2 |
The probability to find the electron within the
distance r to r + dr from the nucleus is in the
r( t)
rm
1s state given by
P(r ) dr = 4r 2 |(r , , )|2 dr . a) b) 1 r 2 r / a0
Fig. 5.6. (a) Classical model of electron paths as oscilla-
tions on straight lines through the nucleus of the 1s state.
Note: The orientation of the lines is statistically distributed to give
an isotropic average. (b) Comparison between the probabil-
Spherical symmetric electron distributions are obtained ity density |(1s)|2 and the probability 4r 2 |(1s)|2 dr of
for all s states with arbitrary quantum number n. finding the electron within the spherical shell 4r 2 dr
 5.1. Schrdinger Equation for One-electron Systems 167

r 2R10
2
a0

0.5

0.4

0.3 r 2R30
2
a0
0.1

0.2 0.08
1s 3s
0.06
0.1 0.04
rm r
r 0.02 r
0 1 2 3 4 5 a0 0 5 10 15 a0

2 2 0.1 r 2R31
2
a0
0.2 r R20a0
0.08
2s 3p
0.06
0.1
0.04

0.02
r r
0 2 4 6 8 10 12 a0 0 5 10 15 a0

r 2R32
2
a0
r 2R21
2
a0 0.12
0.1
0.2 2p 3d
0.08
0.06
0.1 0.04
0.02
r r
0 2 4 6 8 10 12 a0 0 5 10 15 a0

Fig. 5.7. Radial charge distribution of the electron in different states of the H atom. Note the different ordinate and abzissa
scales

The plotted curves are directly proportional to the prob-
r 3
r  = 4r 2 e2r/a0 dr = a0 , (5.22b) ability 4r 2 |Rnl (r )|2 dr of finding the electron within
a0 3 2 the spherical shell between r and r + dr .
r =0
The probability P(r < a0 ) of finding the electron
which differs from the Bohr radius a0 ! within the Bohr radius a0 is, for s-functions (l = 0),
In Fig. 5.7 the functions r 2 a0 |Rnl (r )|2 are plotted
for some states against the abscissa r/a0 , i.e., in units
a0
of the Bohr radius. They are normalized in such a way Pn,l (r a0 ) = 4 r 2 |n,0 (r )|2 dr , (5.23)
that the shaded area under the curve becomes r =0

dr and can be calculated with the functions in Tables 5.1

r 2 |Rn,l (r )|2 a0 =1. and 4.2.
a0
r =0
168 5. The Hydrogen Atom

For n = 1 and l = 0, for example, we obtain: circular orbit, while the angular momentum is given by
the quantum mechanical expression

a0 $
4 |l| = l(l + 1)h .
P1,0 (r a0 ) = r 2 e2r/a0 dr = 0.32 .
a03
r =0 An electron with charge e moving with the velocity v
and the circular frequency = v/(2r ) on a circle with
For n = 2 and l = 0 one can verify the result: radius r represents an electric current
P2,0 (r a0 ) ev
I = e = , (5.24)

a0   2r
1 4r 3 r4
= 3 4r
2
+ 2 er/a0 dr = 0.034 . which causes a magnetic moment
8a0 a0 a0
r =0 = I A = I r 2 n , (5.25)

While for n = 2 and l = 1 one obtains the smaller where A = r 2 n is the area vector perpendicular to the
probability plane of the motion (Fig. 5.8).
The angular momentum of the circulating electron

a0 is
1
P2,1 (r a0 ) = r 4 er/a0 dr = 0.0037 . l = r p = m er v n . (5.26)
24a05
r =0
The comparison of (5.25) and (5.26) gives the relation
These results are illustrated by the curves in Fig. 5.7.
In the simplified classical model this means that e
= l (5.27)
the orbits with l = 0 correspond to very eccentric el- 2m e
liptical paths where the electron is often close to the
nucleus, while orbits with maximum possible l are between magnetic moment and angular momentum l
close to circular orbits. With increasing principal quan- of the electron. Since is proportional to l, the orbital
tum number n the maximum possible values of l< n magnetic moment is often labeled l .
become larger and the quantum mechanical spatial In an external magnetic field the potential energy of
probabilities approach more and more closely the clas- a magnetic dipole with magnetic moment is
sical circular orbits.
E pot = B . (5.28)
An interesting result arises:
Using the relation (5.27) this can be expressed by the
Summing the spatial probability |(r , , )|2
at angular momentum l as
a given n over all allowed values of l and m e
E pot = + lB. (5.29)
gives the total probability in the state n, which 2m e
is always spherically symmetric! Therefore the When the magnetic field points into the z-direction
sum over the electron distributions in all possi- (B = {0, 0, Bz = B}), we obtain from (5.29), because
ble states (l, m) for a given value of n is called an of l z = m h,
electron shell.

5.2 The Normal Zeeman Effect

We will now discuss the behavior of the H atom in
an external magnetic field. In the beginning we will Fig. 5.8. Classical model of or-
use a semiclassical model (called the vector model), bital angular momentum l and
where the electron motion is described by a classical magnetic moment
 5.2. The Normal Zeeman Effect 169

eh Using the Bohr magneton (5.31) we can write the

E pot = mB , (5.30) orbital magnetic moment of the electron as
2m e
where m (which had been introduced before as the pro- l = ( B /h)l . (5.35a)
jection of l onto the z-axis) is called the magnetic quan-
tum number, that can take the values l m +l. Since the external magnetic field with cylindrical sym-
The constant factor in (5.30) metry breaks the spherical symmetry of the Coulomb
eh potential the orbital angular momentum l of the elec-
B = = 9.274015 1024 J/T (5.31) tron is no longer constant, because the torque
2m e
D = l B (5.35b)
is called the Bohr magneton.
We can now write the additional energy caused by acts on the electron. In the case of a magnetic field B =
the magnetic field as {0, 0, Bz = B} in the z direction the z component of l
stays constant. The vector l precesses around the z-axis
E m = B m B , (5.32) on a cone with the apex angle 2 (Fig. 5.10), where
which gives for the energies of the hydrogen atomic lz m
cos = = . (5.35c)
states in an external magnetic field: |l| l(l + 1)
The component l z has the values
E n,l,m = E Coul (n) + B m B . (5.33)
l z = m h with l m +l . (5.35d)

The 2l + 1 m-sublevels that are degenerate without Also, the absolute value of l
$
magnetic field split into 2l + 1 equidistant Zeeman |l| = l(l + 1)h (5.36)
components with energies between E nmin = E Coul (n)
Bl B and E nmax = E Coul (n) + B l B, with an energetic is well defined, while the two other components l x
distance (Fig. 5.9) and l y are not defined (see Sect. 4.4.2). Their quantum
mechanical expectation value is zero, as is the classical
E = E n,l,m E n,l,m1 = B B , (5.34) time averaged value. Since the magnetic moment  l is
 it also precesses around the magnetic
anti-parallel to l,
between adjacent levels which is determined by the field axis with a well defined component z .
product of Bohr magneton B and magnetic field
strength B.

The splitting of the 2l + 1 degenerate m compo-

nents in an external magnetic field B due to the
orbital magnetic moment related to the angular
momentum |l| = l(l + 1)h is called the normal
Zeeman effect.

m
+2
+1
I= 2 E = B B
0
1 Fig. 5.9. Zeeman splitting of Fig. 5.10a,b. Vector model of the normal Zeeman effect.
a level with l = 2 in a homo- (a) Classical model of orbital angular momentum precess-
2
geneous magnetic field (nor- ing around the field axis. (b) Possible orientations of l and
B=0 B0 mal Zeeman effect) projections m h in the quantum mechanical description
170 5. The Hydrogen Atom

For the absorption or emission of light by atoms B=0 B0 m B=0 B0 m

E
in a magnetic field, our model makes the following +2 +2
predictions. +1 +1
l=2 l=2
When a circularly polarized + -light wave propa- 0 0
1 1
gates into the z direction all photons have the spin +h k. 2 2
If they are absorbed by atoms in the magnetic field
B = {0, 0, Bz } they transfer their spin +h k to the atoms m = +1 0 1 m = 1 0 +1
(because of conservation of angular momentum) and
therefore cause transitions with l z = +h, where the
quantum number m changes by +1. l=1
+1 +1
l=1
For , polarization of the light wave transitions 0 0
1 1
with m = 1 and l z = h are induced.

A similar consideration is valid for the emission of
0
light by atoms in a magnetic field. For the light emit- Absorption Emission
B
ted into the direction of the magnetic field (i.e., in the = B
h
z direction) the two circularly polarized + and
components are observed, while for light emitted in- Fig. 5.12. Level scheme and transitions m = 0, 1 between
Zeeman level in absorption and emission for the normal
to the direction perpendicular to the field three linearly Zeeman effect
polarized components are observed. One component
with the E-vector parallel to B, which is not shifted Next we will discuss how this prediction and all the
against the field-free transition, and two components other conclusions drawn from the Schrdinger model
with EB, which are shifted to opposite sides of the of the atoms match the experimental results?
unshifted line (Fig. 5.11).
According to (5.34), the Zeeman splitting E =
B B is independent of the quantum numbers n and l.
This implies that all atomic states should have the same 5.3 Comparison of Schrdinger
separation of the Zeeman components. Therefore ev- Theory with Experimental Results
ery spectral line corresponding to a transition (n 1l1 )
(n 2 , l2 ) should always split in a magnetic field into Although the hydrogen atom is, from a theoretical point
three Zeeman components (Fig. 5.12) with + , and of view, the simplest atomic system, and can be cal-
-polarization and a frequency separation of culated analytically (at least within the Schrdinger
model), its experimental investigation is not as simple.
= B B/ h . There are several reasons for this:
1. First of all hydrogen atoms cannot be bought in
a bottle. They have to be produced by dissociation

EB E || B EB of H2 molecules. This can be achieved in several
Transverse observation ways.
linearly polarised The easiest method is dissociation by electron
m = + 1 m = 0 m = 1 impact in gas discharges, where electrons with

EB Longitudinal observation sufficient kinetic energy collide with hydrogen
0 circularly polarised molecules H2 to form H atoms, according to the
scheme
Without H2 + e H + H + e .
With
magenetic field The excited atoms H in the state |n k  release
Fig. 5.11. Normal Zeeman effect. Zeeman splitting and po- their excitation energy partly or completely by
larizations of a spectral line with frequency 0 observed in emitting photons with energy h = E(n k ) E(n i )
emission. The splitting is = B B/ h with E i < E k :
 5.3. Comparison of Schrdinger Theory with Experimental Results 171

h
H (n k ) H(n i ) .
S1 M1
Another technique is the thermal dissocia-
tion of H2 molecules at high temperatures (T = S
1500 2000K ) in the presence of catalysts (e.g.,
tungsten surfaces).
Nowadays the preferred method is the dissocia-
tion by microwave discharges, which has proved to
be the most efficient way of forming H atoms. M2
2. The absorption spectrum of ground state hydrogen G
atoms lies in the vacuum ultraviolet (VUV) spectral
region. It therefore can be measured only in the vac-
S2
uum, i.e, in evacuated spectrographs (Fig. 5.13b).
In most experiments the emission rather than the CCD-detector
absorption is measured. The hydrogen discharge is a) Grating spectrometer
placed in front of the entrance slit of the evacuated
spectrograph (Fig. 5.13) and the dispersed emis-
sion spectrum of the hydrogen atoms is detected Hydrogen
discharge Vacuum chamber
on a photoplate. Since most materials absorb in the
VUV, no lenses are used and the curved grating +
Rowland-
(Rowland arrangement) images the entrance slit on- grating
to the photoplate. For wavelengths below 120 nm
no entrance window can be used and the air coming
from the outside through the open entrance slit has Pump
to be pumped away by differential pumping in or-
der to maintain the vacuum inside the spectrograph.
Since the emission of excited H atoms has
Curved
a spectrum covering the whole range from the in- Photoplate
frared to the VUV region, that part of the emission
spectrum with wavelengths above 200nm can be
measured with spectrographs in air. Here, the in-
tensity can be enlarged by imaging the discharge

source S by a lens onto the entrance slit and two
curved mirrors M1 and M2 image the entrance b) Photoplate
slit S1 onto the CCD camera at the exit (Fig. 5.13a). Fig. 5.13. (a) Experimental setup for measuring the emission
As has already been discussed in Sect. 3.4.1 the spectrum of atomic hydrogen for lines with > 200 nm. (b)
lines in the spectrum of the H Atom can be arranged Vacuum UV spectrograph for measuring the Lyman-series
with < 200 nm
in series (Fig. 3.40) with wavenumbers that can be
fit by the simple relations
  Accurate measurements with higher spectral reso-
1 1 lution showed, however, significant deviations of the
ik = Ry 2
n i2 nk measured line positions and line structures from the
mp predictions of the Schrdinger theory discussed so
with Ry = Ry far.
me + mp
= 109,677.583 cm1 (5.37) a) The wavenumbers ik of the different transitions
between levels |i and |k depend not only on the
in accordance with the formula (3.79) and (5.18). principal quantum number n but also slightly on the
172 5. The Hydrogen Atom

Table 5.4. Comparison of the measured wavenumbers vac

of the Balmer series and the calculated values Ry obtained
from the Rydberg formula
n air / vac /cm1 Ry /cm1
H 3 6,562.79 15,233.21 15,233.00
H 4 4,861.33 20,564.77 20,564.55
H 5 4,340.46 23,032.54 23,032.29
H 6 4,101.73 24,373.07 24,372.80
H 7 3,970.07 25,181.33 25,181.08
H 8 3,889.06 25,705.84 25,705.68
H 9 3,835.40 26,065.53 26,065.35
H 10 3,797.91 26,322.80 26,322.62
H 11 3,770.63 26,513.21 26,512.97
H 12 3,750.15 26,658.01 26,657.75
H 13 3,734.37 26,770.65 26,770.42
H 14 3,721.95 26,860.01 26,859.82
H 15 3,711.98 26,932.14 26,931.94

c) The experimentally observed splittings of the

Zeeman components agrees only for a few atomic
species with the prediction of the normal Zeeman
pattern. For most atoms it is more complicated.
For the H atom, for instance, it looks completely
different from the regular triplet pattern in Fig. 5.12.
d) The ground state of the H atom (n = 1, l = 0)
shows a very narrow splitting into two compo-
nents (hyperfine structure), which differs for the
two isotopes 1 H and 2 H = 2 D.
In order to explain these deviations, the Schrdinger
theory has to be extended and new atomic parameters
have to be included. This will be discussed in the next
sections.

Fig. 5.14ac. Balmer series of the hydrogen atom. (a) Fine 5.4 Relativistic Correction
structure of H measured with conventional Doppler-limited of Energy Terms
spectroscopy. (b) High-resolution Doppler-free spectrum of
H showing the Lamb shift of the 22 S1/2 level. (c) Level The last section has shown, that the coarse structure
scheme of the hydrogen spectrum is well described by the
Schrdinger-theory for an electron in the Coulomb-
field of the atomic nucleus. The small discrepancies
angular momentum quantum number l. The abso-
found by comparison of experimental results with the-
lute wavenumbers for the H atom deviate from the
oretical predictions are due to relativistic effects. There
predictions by up to 0.2 cm1 .
are essentially three contributions to these effects:
b) All spectral lines starting from s levels with l = 0
consist of two narrowly spaced components (dou- 1. The relativistic dependence of the electron mass on
blets). Those starting from levels with l > 0 contain its velocity in the Coulomb-field which results in a
even more components (Fig. 5.14). small decrease of its kinetic energy
 5.4. Relativistic Correction of Energy Terms 173

2. A closer inspection of the relativistic treatment Inserting for the wave functions of the hydrogen
shows that the electron charge is smeared out atom n,l,m gives (see Problem 5.10)
over a volume c 3 = (h/m e c)3 where c is the  
Compton wavelength of the electron. This leads to Z 22 3 1
E r = E nr . (5.42)
a change of the potential energy of the electron n 4n l + 1/2
(Darwin-term)
3. The interaction between the magnetic moment of The constant
the electron due to its orbital angular momentum
e2 1
and the magnetic moment due to the electron spin = = 7.297353 103 =
results in a shift and a splitting of the energy levels 4 0 hc 137
(fine-structure). (5.43)
We will now briefly discuss these three contribu- is called Sommerfelds fine structure constant.
tions and their magnitude.
1) Relativistic mass increase The total energy of an eigen-state for the H atom is
then
Instead of the nonrelativistic energy relation (  )
Z2 2 Z 2 3 1
E = p 2 /2m + E pot (5.38) E n,l = Ry 2 1 ,
n n 4n l + 1/2
anticipated by the Schrdinger theory, we have to use (5.44)
the relativistic energy relation which now depends not only on n but also on l! The
 relativistic correction is maximum for n = 1 and l = 0.
E = c m 20 c2 + p 2 m 0 c2 + E pot . (5.39)

For the electron in the hydrogen atom, the velocity v EXAMPLES

of the electron is still small compared to the velocity of
1. For n = 1, l = 0 the magnitude of the relativistic
light c, which means that E kin  m 0 c2 or p 2/m 20 c2  1.
correction is
We can therefore expand the square root in (5.39) into
the power series E r = Z 2 E 1 5 2 /4 .
!
p2 1 p2 1 p4 2. For Z = 1 this becomes
1+ 2 =1+ + ... ,
m 0 c2 2
2 m 0 c2 8 m 40 c4 E r = 9 104 eV .
which gives for the energy expression (5.38) 3. For n = 2, l = 0 we obtain
 2  13
p p4 E r (n = 2, l = 0) = Z 2 E 2 2
E= + E pot + . . . = E nr E r . 16
2m 0 8m 30 c2
= 1.5 104 eV. for Z = 1
(5.40)
4. For n = 2, l = 1 the correction is only
For E kin  m 0 c2 we can neglect the higher order
terms. In this approximation the last term in (5.40) 7
E r (n = 2, l = 1) = E 2 Z 2 2
represents the relativistic correction E r to the non- 48
relativistic energy (5.38). = Z 2 2.6 105 eV .
We can obtain the quantum mechanical expectation
value of this correction by substituting p i h,
which leads to the expression This illustrates that
4

h
E r =
n,l,m 4 n,l,m d . (5.41) a) The relativistic energy shift is maximum for the
3
8m 0 c 2 ground state of atoms (n = 1, l = 0).
174 5. The Hydrogen Atom

b) The correction depends on both quantum numbers n For the Coulomb potential E pot (r ) = (Z e2/40r )
and l. The n-fold degeneracy of states (n, l), de- we obtain
duced from the Schrdinger theory is lifted by the
relativistic correction. E pot = (Z e2 /0 )(r )
c) At a given value of n, the electron comes closest (where (r) is the delta function (r ) = 1 for r = 0 and
to the nucleus (and therefore acquires the largest (r  = 0) = 0) and the relativistic correction becomes
velocity) for small values of l (the Sommerfeld or- 2
bits are then ellipses with large eccentricity). The Wrel = +(Z e2 h 2 /0 m 2e c ) (r) (5.47)
relativistic mass increase is then maximum, which
The quantum mechanical expectation value, ob-
decreases the energy term value. For the maximum
tained with the hydrogen wavefunction gives the
allowed l = n 1 the orbit is circular and the ve-
Darwin term
locity of the electron has a constant medium value.
2
The relativistic mass correction is then minimum. W D = +(Z e2 h 2 /0 m 2e c ) |(0)|2 (5.48)
where (0) is the wavefunction at the origin r = 0.
Note:
Since only the s-functions have nonvanishing values at
As the numerical examples show, the relativistic mass r = 0 we can insert the hydrogen 1s-function
correction only amounts to less than 104 of the
1s = 1/ (Z /a 0 )3/2 eZr/a0
Coulomb energy.
and obtain with the definition of a0 and the fine-
2) Darwin-Term structure constant
The second relativistic contribution comes from the W D = 4Z 4 m e c2 4 . (5.49)
fact, that, even in a model which describes the electron
With the nonrelativistic energy
as a point charge, the momentary position of the elec-
tron cannot be defined more precisely than within the 2
W0 = E nr = (1/2)m e c2 (Z 2 /n ) 2 (5.50)
volume 3c = (h/m e c)3 given by the Compton wave-
length c of the electron. The potential energy of the the ratio WD /W 0 of Darwin term to the nonrelativistic
electron in the Coulomb field of the nucleus is then the energy becomes
weighted average of all values of the electric field with- W D /W 0 = 2Z 2 2 8(Z /137)2 . (5.51)
in the volume 3c around the point r. This means: the
potential energy is no longer E pot (r ) but is determined For Z = 1 the Darwin correction is about 104 E nr .
by the integral We will now discuss the third relativistic correction,
namely the interaction between the orbital magnetic
E pot (r) = f()E pot (r + ) d3 (5.45) moment and the spin moment of the electron, causing
the fine structure of spectral lines.
over the volume 3c around the point r. Expanding
E pot (r + ) into a Taylor series of powers of around
the point = 0 gives 5.5 The Electron Spin
E pot (r + ) = E pot (r ) + (dE pot /d)= 0 Several experimental results such as the Stern-Gerlach
+ 1/2(d2E pot /d 2 ) 2 +. . . . . (5.46) experiment, the fine-structure of spectral lines or the
anomalous Zeeman effect indicated that the electron
the first term gives the unperturbed potential used in must have, besides its charge e and mass m e an
the Schrdinger theory. The second term vanishes be- additional characteristic property, which was called
cause of the spherical symmetry of the Coulomb field. electron spin and which must cause an additional mag-
The third term, inserted into (5.45) gives the relativistic netic moment s in addition to the orbital magnetic
Darwin correction which is of the order of magnitude moment l . This had been already postulated by Fermi,
(h/m e c)2 E pot (r ), where  is the Laplace operator. before it could be experimentally confirmed.
 5.5. The Electron Spin 175

5.5.1 The SternGerlach Experiment The silver atoms were evaporated in a hot furnace
and emerged through a narrow hole A in the furnace
The space quantization of the angular momentum in-
into the vacuum chamber (Fig. 5.15). They were then
troduced by Arnold Sommerfeld in 1916 was consid-
collimated by the slit S before they entered, traveling in
ered by many physicists as a purely theoretical model
the x direction, the inhomogeneous magnetic field Bz ,
without any real correspondence in nature. It was there-
which pointed in the z direction. In the plane x = x0
fore in doubt if the explanation of the Zeeman effect
the atoms were condensed on a cold glass plate. With
by the corresponding space quantization of the mag-
a densitometer the density N (z) of silver atoms on the
netic moment could ever be proved experimentally.
glass plate could be measured.
Nevertheless this issue was intensely discussed among
Without a magnetic field, the symmetric blue den-
experimental physicists. Otto Stern (18881969), who
sity profile corresponding to the central curve in
had a lot of experience with atomic beams, proposed
Fig. 5.15c was obtained. It represents the density pro-
to test the idea of space quantization by measuring
file of the non-deflected atomic beam due to the spread
the deflection of atoms in a collimated atomic beam
of the transverse velocity components of the silver
passing through a transverse inhomogeneous magnetic
atoms behind the collimating slit S.
field. He found in Walther Gerlach (18891979) an en-
In the inhomogeneous magnetic field the force
thusiastic and experimentally skillful assistant whom
he convinced to try this experiment. They started in F = grad B
1919 and after many unsuccessful efforts and follow-
ing improvements to their beam apparatus, the two on the atoms depends on the spatial orientation of the
researchers were finally rewarded with an unexpected magnetic moment relative to B. The intensity distri-
result in 1921 [5.1]. bution I (z) is therefore expected to split into as many
They chose as test objects silver atoms because peaks as the possible values of the scalar product B.
these atoms could be detected on a glass plate, where At that time the quantum number l of the orbital an-
they condensate and form a thin layer with an optical gular momentum of silver was assumed to be l = 1.
transparency depending on the thickness of the layer Therefore three possible values of B were expected,
and therefore the number of incident atoms. corresponding to the three magnetic quantum numbers

z x = x0

N
x
A

S Beam of
silver atoms
Oven Collimation Inhomogeneous Glas plate
a) slit magnetic field

I
Without magnetic field Fig. 5.15. (a) SternGerlach
N Weak field apparatus for measuring the
Strong field
space quantization of angu-
Atomic beam lar momentum l z = m h. (b)
Cut in the yz-plane of the in-
homogeneous magnetic field.
(c) Observed density pattern
S
z N (z) of silver atoms in the
b) c) detection plane x = x0
176 5. The Hydrogen Atom

m l = 0, 1. The experimental result clearly showed electron. The astronomical analogy is the earth revolv-
only two peaks and a minimum intensity in between ing around the sun in one year but simultaneously
the peaks where atoms with m = 0 should arrive. Bohr, turning around its own axis within one day. The total
who was asked for advice, explained this result by angular momentum of the earth is the orbital angu-
a model that assumed that atoms with their magnetic lar momentum plus the spin of the earth. Similarly,
moment perpendicular to the direction of the magnetic the electron of the hydrogen atom has for l > 0 an
field (m = 0) were unstable and would flip into one of orbital angular momentum l = r p and in addition
the other quantum states with m = 1 [5.1]. its spin s. However, as will be discussed in Sect. 5.8
Stern and Gerlach had proved with their pioneer- this interpretation of the electron spin as a mechan-
ing experiment that space quantization is a real effect ical angular momentum runs into serious difficulties.
and does not only exist in the brain of theoreticians. Nevertheless the spin can be treated as a vector obey-
O. Stern later on received the Nobel Price 1944 for his ing the same mathematical rules as the orbital angular
contribution to the development of the molecular beam momentum.
technique and the discovery of the magnetic moment of In analogy to the orbital angular momentum the
the proton. absolute value of the spin s is written as
Although their experiment was a very ingenious
demonstration of space quantization, it turned out $
that the interpretation of their results was not correct, |s| = s(s + 1)h , (5.52)
because the splitting was not due to the orbital an-
gular momentum, but to a new quantity called the
electron spin, which was postulated as a new charac- where s is the spin quantum number, which had already
teristic property of the electron after further convincing been introduced by Pauli as an additional quantum
experimental discoveries. number in order to explain the different components in
the fine structure of observed spectra (see below). Pauli,
however, regarded this as a pure mathematical quanti-
ty and resisted giving it a physical meaning as a real
5.5.2 Experimental Confirmation of Electron Spin angular momentum.
When spectroscopic measurements showed that the The magnetic spin moment s is related to the
ground state of silver atoms is in fact an s state with spin by
l = 0, Bohrs explanation of the results of the Stern
Gerlach experiment could no longer be regarded as s = s s , (5.53)
correct.
Samuel A. Goudsmit (19021978) and George where is the gyromagnetic ratio of magnetic moment
E. Uhlenbeck (19001988) proposed a new model to spin.
where the electron possesses an intrinsic angular mo- From the experimental result of the SternGerlach
mentum, called the electron spin in addition to a pos- experiment, that the beam of silver atoms was split
sible orbital angular momentum. This model attributes in the inhomogeneous magnetic field into two compo-
a new property to the electron, which is then charac- nents, it can be concluded that the angular momentum
terized by its mass m, its charge q = e, its orbital should have two possible orientations. Since spectro-
angular momentum l and its spin s which is connected scopic investigations of the silver atom proved that the
with a magnetic moment s that can interact with mag- orbital angular momentum in the ground state is zero,
netic fields. Many further experimental findings such as the splitting must be due to electron spin. The mag-
the fine structure in atomic spectra and deviations from netic spin moment s must have two orientations and
the normal Zeeman splittings (called the anomalous therefore the electron spin must also have two compo-
Zeeman effect) corroborated this hypothesis. nents sz = m s h. If the electron spin should be treated
This spin can be mathematically treated like an as an angular momentum then the quantum number m s
angular momentum and is therefore often regarded must obey the relation s m s +s. Since m s can
as a mechanical eigen-angular momentum of the only change by an integer value, the explanation of two
 5.5. The Electron Spin 177

z Fig. 5.16. Space quantization Light source Fig. 5.17. Ein-

of electron spin steinde Haas ex-
1
Mirror Light beam periment
+ h S
2 Torsion wire

1
h |S | = 1
2 3 h
2
Magnetic
U field coil

components in the SternGerlach experiment gives the

condition (Fig. 5.16):
reason. We will see that it is caused by the spins of the
s = 1/2 m s = 1/2 . (5.54) free electrons.

Note. that for a one-electron atom the quantum Note:

number s is always +1/2, but m s can be 1/2.
In iron, the magnetic moment is not exclusively caused
by the free electrons in the conduction band. A mi-
The absolute value of the electron spin is then
nor contribution also comes from the electrons bound
$ 1 in the iron atoms. They can have orbital angular mo-
|s| = s(s + 1)h = 3h . (5.54a)
2 mentum and spin. However, this contribution is small
And the two components in z direction are and we will neglect it for the following discussion. (For
a more detailed discussion see textbooks on solid state
1
sz = h . (5.54b) physics).
2
If the magnetic field is reversed, all magnetic mo-
We will at first introduce further experiments lead-
ments flip into the opposite direction. This causes
ing to the introduction of the electron spin and will then
a change in magnetization
discuss some basic theoretical considerations.
M = 2M = 2N . (5.55)
5.5.3 Einsteinde Haas Effect
From the measurement of M the product N can be
Einstein proposed the following experiment in order to determined.
gain insight into the causes of magnetic properties of The reversal of the magnetic moments also results
solids. It was actually performed some years later by in a flip of the corresponding angular momenta s of all
the Dutch physicist Wander Johannes de Haas (1878 electrons. The resulting change
1960).
An iron cylinder with mass m and radius R hangs S = 2N s z = L = I (5.56)
on a thin wire (Fig. 5.17) in a vertical magnetic field
B = {0, 0, Bz }, produced by an electric current through has to be compensated by the opposite change L
a coil. The magnetic field was chosen sufficiently of the mechanical angular momentum L of the cylin-
strong to saturate the magnetization M = N of the der with the moment of inertia I = (1/2)M R 2 . With
cylinder with N free electrons, each possessing the a reversal of the magnetic field, the cylinder that was
magnetic moment . In cases of saturation, all mag- initially at rest acquires the angular momentum L =
netic moments point in one direction opposite to that (1/2)M R 2 and a rotational energy
of the magnetic field. Since the free electrons in the
conduction band of iron have no orbital angular mo- E rot = L 2 /2I = L 2 /(M R 2 ) , (5.57a)
mentum and therefore also no orbital magnetic moment
(there is no force center to cause a circular motion of which results in a torsion of the suspension wire
the electrons) the magnetic moment must have another with a restoring force Fr = Dr . The maximum
178 5. The Hydrogen Atom

torsion angle max is reached when the potential a theory has been developed by Paul Dirac (1902
energy 1984) who replaced the Schrdinger equation with the
1 Dirac equation. Its representation exceeds, however,
E pot = Dr max
2
= E rot = L 2 /2I (5.57b) the level of this textbook.
2
of the twisted suspension wire equals the kinetic ro- 5.5.4 Spin-Orbit Coupling and Fine Structure
tational energy. The measurement of max therefore
allows the determination of L. The experiment gave the We will now discuss why the energy levels of the H
surprising result atom with l > 0, split into two components, which
could not be explained by the Schrdinger theory.
L = N h = 2N h/2 . (5.57c) Since this splitting is very small and can be only re-
This means that the z component of the angular mo- solved with high resolution spectrographs, where the
mentum of each electron must have the amount sz = hydrogen lines appear as a fine substructure, it was
h/2. From the measured magnetization change M it named fine structure.
was, however, clear that the magnetic moment relat- We start with a semiclassical model, treating the
ed to this angular momentum must be = B , i.e, the angular momenta as vectors with quantized absolute
same as for the orbital angular momentum with l = 1h. values and quantized z components. In Sect. 5.2 it was
The gyromagnetic ratio shown that an electron with charge e, moving with
s | | B the orbital angular momentum l on a circle around the
M/S = = s = 1 = s (5.58) nucleus, produces a magnetic moment
sz |s| /2h
e
of magnetic moment |s | and angular momentum |s| l = l = (B /h) l .
2m e
is therefore twice as large as for the orbital angular
momentum, where it is That is proportional to l.
In a coordinate system where the electron rests at
|l |/|l| = B /h = l . (5.59) the origin, the nucleus with positive charge Z e moves
This means that s is twice as large as l ! with the frequency on a circle around the electron.
This causes a circular current Z e that produces a mag-
netic field B at the location of the electron (Fig. 5.18).
For the electron spin, the ratio of magnetic mo-
According to BiotSavarts law (see textbooks on mag-
ment to mechanical angular momentum is twice
netic fields) this magnetic field is
as large as for the orbital angular momentum of
the electron. 0 Z e 0 Z e
Bl = (v (r)) = (v r)
4r 3 4r 3
The magnetic spin moment is written analogously 0 Z e
to the orbital moment l = (B /h)l as =+ l (5.61)
4r 3 m e
s = gs (B /h)s . (5.60a)
The factor gs 2 is called the Land factor.
The absolute value of the magnetic spin moment is
$
|s | = gs B s(s + 1) . (5.60b)

Remark
Here the electron spin has been introduced phenomeno-
logically. The exact value of the Land factor gs = Fig. 5.18a,b. Vector model of spin-orbit interaction. (a) vec-
tor model with the electron circling around the at. (b) trans-
2.0023 can only be explained by a theory that already formation to a coordinate system, where the electron rests at
includes the electron spin in the basic equations. Such r =0
 5.5. The Electron Spin 179

because the angular momentum of the electron in a co- Fig. 5.19. Vector coupling of orbital
ordinate system where the electron moves around the angular momentum l and electron

proton at rest is l = m e (
v r) and the transformation
j
s spin s to form the total angular mo-
mentum j of the electron
to the rest frame of the electron changes the sign of the
vector product.
The magnetic spin moment of the electron has two l
spatial orientations in this field according to the two
j=l+s
spin directions sz = h/2. This causes an additional
energy (in addition to the Coulomb energy) h 2
l sn, l, s, j, m l = [ j ( j + 1) l(l + 1)
0 Z e 2
E = s Bl = gs B (s l) s(s + 1)]n, l, s, j, m l .
4r 3 m e h
0 Z e2 With this relation we can write (5.63) as
(s l) . (5.62)
4 m 2er 3 a
E n,l, j = E n + [ j( j + 1) l(l + 1) s(s + 1)] .
because gs 2 and B = eh/2m e . In this coordinate 2
system the electron is not in an inertial frame of refer- (5.66a)
ence. Transforming the coordinate system back to the With the spin-orbit coupling constant
rest-frame of the nucleus by a Lorentz transformation
gives a factor 1/2 (Thomas factor [5.2]), which is due
0 Z e2 h 2
to the fact that the electron spin in the rest-frame of a= . (5.66b)
the nucleus precesses when moving around the nucleus 8 m 2er 3
(Thomas precession).
The energy levels E n of (5.18), which had been ob- For s = 12 the energy levels split, depending on the
tained without taking into account the electron spin, orientation of the spin, into the two components with
now split, due to the spin-orbit coupling, into the fine j = l + 1/2 and j = l 1/2 (see Fig. 5.20).
structure components with energies Note: Finestructure splittings are observed only for
levels with l 1, i.e. for p, d, f , . . . levels, not for s-
0 Z e2 levels with l = 0.
E n,l,s = E n s Bl = E n + (s l) .
8m 2er 3
(5.63) The fine structure may be regarded as Zeeman
The scalar product (s l) may be positive or negative splitting due to the interaction of the magnet-
depending on the orientation of the spin relative to the ic spin moment with the internal magnetic field
orbital angular momentum. generated by the orbital motion of the electron.
When we introduce the total angular momentum
$ In the quantum mechanical model the distance r
j = l + s with | j | = j( j + 1)h (5.64a) of the electron from the nucleus cannot be given ex-
as a vector sum of orbital angular momentum l and actly. Only the time-averaged value of r related to the
electron spin s (Fig. 5.19), we can square this sum and probability of finding the electron at the location r is
obtain a measurable quantity

j 2 = l 2 + s2 + 2l s . (5.64b)
3
j=
This gives for the scalar product of the vectors l and s 2
P3 / 2
P a/2
1
l s = h 2 [ j( j + 1) l(l + 1) s(s + 1)] . l=1
a
2
(5.65) P1/ 2 Fig. 5.20. Energy level
1 scheme of fine structure
j=
In operator notation this is written as 2 splitting of a 2 P(l = 1) state
180 5. The Hydrogen Atom

As can be seen from (5.70), the fine structure split-

r  = n,l,m r n,l,m d ting decreases with increasing quantum numbers n
0 1
and l, but it increases proportionally to the product
1 1
3
= n,l,m 3 n,l,m d . (5.67a) Z 2 E n . Since the energies E n of the levels with prin-
r r
cipal quantum number n follow the relation E n
The quantum mechanical average a of the spin-orbit Z 2 /n 2 , we can write the fine structure splitting as
coupling constant is then

Z4
0 Z e2 h 2 1
a = n,l,m n,l,m d . (5.67b) E n,l,s . (5.71)
2
8 m e r3 n 3l(l + 1)

Inserting the hydrogen wave functions n,l,m (r , , ),

the integral can be solved and one obtains
EXAMPLE
Z 22
a = E n , (5.68) For the 2 p level of the H atom, we have Z = 1, n = 2,
nl(l + 1/2)(l + 1) l = 1 and E n = 3.4 eV. From (5.70) we therefore
where the constant obtain for the fine structure splitting El,s = 4.6
105 eV El,s / hc = = 0.37 cm1 .
0 ce2 e2 1
= = (5.69)
4 h 4 0 hc 137
If both effects, the relativistic increase of the elec-
is Sommerfelds fine structure constant, which was
tron mass and the spin-orbit coupling, are taken into
already introduced in Sect. 5.4 for the relativistic cor-
account we have to add (5.42) and (5.66a) and obtain
rection of the level energies.
for the energy of a fine structure component (n, l, j)
The energy separation of the two fine structure
(see Problem 5.11)
components (n, l, j = l + 1/2) and (n, l, j = l 1/2)
is then, according to (5.66) and (5.68),
  (  )
1 Z 22 Z 22 1 3
E n,l,s = a l + = E n E n, j = E n 1+ ,
2 nl(l + 1) n j + 1/2 4n
Z 2
5.3 105 E n . (5.70) (5.72)
nl(l + 1)
This shows that the splitting is very small compared to which turns out to be independent of l (Fig. 5.21).
the energy E n,l of the levels (n, l) and justifies the name Note: Since E n < 0, levels with larger j-values
fine structure. Note, that E n < 0! have a higher energy (see Fig. 5.22).

2p
3/2

2
m = 2, l = 0,1 s1/2
2
2
s1/2;2p1/2; 2p3/2 p3/2

Schrdinger 2
p1/2

spin-orbit 2s 2 2s Fig. 5.21. Energies of the levels

1/2; p1/2 1/2

2p
n = 1, l = 0, 1, s = 1/2 of the H-
spin-orbit 1/2 atom within the Schrdinger theory
+relativistic Lamb- and including the different rela-
correction Shift tivistic effects
 5.5. The Electron Spin 181

E/eV

0.108 cm1 0.036 cm1

1.5 n=3 3d 5 / 2
EFS = 0.2 104 eV 3p 3 / 2
3d 3 / 2
3s1/ 2 3p1/ 2

3.37 n=2
EFS = 0.56 104 eV 2p 3 / 2 . 10 5 eV
EFS = 113

= 0.09 cm1
2s1/ 2 2p1/ 2

0.365 cm1

n=1 1eV = 8065.54 cm1

13.6

. 104 eV
EFS = 18 = 1,4517cm1

1s
Fig. 5.22. Energy level scheme of the hydrogen atom, taking into account the relativistic mass increase and the spin-orbit
coupling. The dashed lines mark the energies obtained from the Schrdinger equation

no longer a Coulomb potential because of the mutual

In the Coulomb field with E pot 1/r the energy interaction between the electrons. Here levels with dif-
of a fine structure component (n, l, j) does not ferent values of the quantum number l have different
depend on the quantum number l. All levels with energies even for equal quantum numbers j.
equal quantum numbers n and j have the same
energy (Fig. 5.22).
5.5.5 Anomalous Zeeman Effect
When the electron spin s and the magnetic spin mo-
EXAMPLE ment s are taken into account, the total magnetic
The two levels 2s1/2 and 2 p1/2 or 3 p3/2 and 3d3/2 have moment depends on the coupling of the two vectors
the same energy (Fig. 5.22). l + s . The Zeeman splittings of levels in a magnetic
field now become more complicated than those for the
normal Zeeman effect shown in Figs. 5.11 and 5.12,
Note: This degeneracy applies only to one-electron which are only observed,
- if the total spin of the atom-
systems such as the hydrogen atom or the ions He+ , ic electrons is S = si = 0. This is, for instance,
Li++ , etc., where the electron moves in a Coulomb the case for the helium atom with two electrons with
potential. This is because the assumption E pot 1/r antiparallel spins in its ground state (see Sect. 6.1).
enters into the relativistic mass correction, as well as Without a magnetic field, the total angular mo-
in (5.67), for the calculation of the fine structure con- mentum j = l + s of the electron in a Coulomb field
stant. For atoms with more than one electron there is (central force field) is constant, which means that its
182 5. The Hydrogen Atom

z
3
h Because of the Land factor gs 2 for the spin

2 moment s , the total magnetic moment
jz j 1
h
j
s 2 j = l + s = (B /h)(l + gs s) (5.73)
0
s

l 1
h is for l > 0 no longer parallel to the total angular
2
l momentum j = l + s!
3
h
2
Without an external field the absolute value and the
a) b) c)
direction of j is constant in time. Since the vector s
Fig. 5.23. (a) Coupling of l and s and their precession around precesses around the axis of the internal magnetic field
the space-fixed vector j = l + s without external field. (b) produced by the orbital movement of the electron and
Precession of j in an external magnetic field Bz . (c) Possible
orientations of j with components jz = m h j is not parallel to j , j has to precess around the
direction
  of the space-fixed vector j . The time aver-
age j of j is then the projection of j onto j
orientation in space and its absolute value are con- (Fig. 5.24a). This gives
stant, independent of time (Fig. 5.23a). In an external
homogeneous magnetic field B = {0, 0, Bz }, the mag-
 
netic moment j , and therefore j , precess around   j j l j s j
j = j = e + gs 2 j .
the field axis with constant components z and jz | j |2 2m e | j| 2 | j|
(Fig. 5.23b). (5.74)
If the external magnetic field is weaker than the
magnetic field generated by the orbital movement of From j = l + s follows
the electron, the Zeeman splitting is smaller than the
fine structure splittings. In other words, the coupling 1 2
l j= [ j + l 2 s2 ]
between orbital angular momentum and spin is stronger 2
than the coupling of l and s to the external field. 1
= [ j( j + 1) + l(l + 1) s(s + 1)]h 2
The spin-orbit coupling is still valid and the absolute 2
value | j | of the total angular momentum (5.74a)
$
j = l + s with | j | = j( j + 1)h (5.72a)

is conserved in the external magnetic field. Its direction

is, however, no longer space-fixed because the mag-
netic moment j = l + s , which is related to j ,
experiences a torque

D = j B . (5.72b)

For one-electron systems the component jz can take

the values jz = m j h with the half-integer values jz
m j + jz (Fig. 5.23c).
The magnetic moments of orbital motion and spin
of the electron are:

l = (B /h)l and s = gs (B /h)s .

Fig. 5.24. (a) Projection of j onto the direction of j . (b)
(5.72c) Precession of the
 angular
 momentum j and the average mag-
netic moment j around the z-axis in an external magnetic
There is an important point to mention: field B = {0, 0, Bz }
 5.5. The Electron Spin 183

and similar from l = j s: the value of the Lande factor g can be calculated from
1 eq. (5.76).
s j = [ j( j + 1) + s(s + 1) l(l + 1)]h 2 . In an external magnetic field B = {0, 0, Bz }, the
2 spatial orientation of the total angular momentum is no
We can therefore write (5.74) with gs 2 as longer constant. The vector j precesses around the field
  3 j( j + 1) + s(s + 1) l(l + 1) direction. The projection of j can take the values
j = B
2 j( j + 1)
jz = m j h with j mj + j .
= g j B . (5.75)
The Land factor g j is defined here as The precession of j around j is faster than that of j
around B as long as the finestructure splitting is larger
than
  the Zeeman splitting. Therefore the z component
j( j + 1) + s(s + 1) l(l + 1)
gj = 1 + . j z of the average magnetic moment j is
2 j( j + 1)
 
(5.76) j z
= m j g j B (5.76a)

For s = 0 (pure orbital magnetism) it follows j = l and and the additional energy of the Zeeman component m j
we obtain g j = 1. For l = 0 (pure spin magnetism) is is
j = s and therefore g j 2. If orbital angular momen-  
tum and spin both contribute to the magnetic moment, E m j = j z B = m j g j B B . (5.77a)

4
B
3 B mj mj gj

+3 / 2 +2
2
3 P3 / 2 +1 / 2 +2 / 3
2 gj = 4 / 3 1 / 2 2 / 3
B mj mj gj
3 B
3 / 2 2
3 2 P1/ 2 +1 / 2 +1 / 3
gj = 2 / 3 1 / 2 1 / 3

1 6
3 4
1 2 3 4 5 2

+1 / 2 +1 +1 / 2 +1
3 2 S1/ 2 3 2 S1/ 2
2 B B 2 B B
gj = 2 gj = 2
1 / 2 1 1 / 2 1
1 2 3 4 1 23 4 56

a) D1 b) D2

Fig. 5.25a,b. Anomalous Zeeman effect of the transitions (a) 2 P1/2 2 S1/2 and (b) 2 P3/2 2 S1/2 neglecting hyperfine
structure
184 5. The Hydrogen Atom

The energy of a specific Zeeman component m j of 5.6 Hyperfine Structure

a fine structure level |n, l, j > is them obtained from
(5.76) and (5.77) as In the previous sections we have described the atomic
nucleus as a point-like charge Z e that interacts with the
(  ) electron merely through the electric Coulomb-potential
Z 2 1 3
E n, j,m j = E n 1 +
n j + 1/2 4n Ze
(r ) = .
+ m j g j B B (5.77b) 4 0 r
With this potential the Schrdinger equation allowed
The energy separation between two adjacent Zeeman the calculation of the term values of all levels in the
components comes out as H atom and the wavenumbers of all transitions between
these levels. The fine structure of the spectral lines
E m j E m j1 = g j B B . (5.78) was explained by the magnetic interaction between the
magnetic moments of the orbital angular momentum
and the electron spin. This magnetic interaction was
Since the Land factor g j depends on the quan- just added to the Coulomb interaction. It cannot be
tum numbers l and j,according to (5.76), the calculated from the Schrdinger equation, which does
Zeeman splitting for the anomalous Zeeman ef- not include the electron spin.
fect differs for the different levels (n, l, j), con-
trary to the situation for the normal Zeeman 5.6.1 Basic Considerations
effect. Therefore, the Zeeman pattern of spec- If the hydrogen spectrum is observed with very high
tral lines is more complicated here. There are spectral resolution, one finds that even the fine struc-
generally more than three Zeeman components. ture components are split into two subcomponents.
The separation of these sub-components is, for the
The following examples shall illustrate the situa- H atom, smaller than the Doppler width of the spec-
tion for the anomalous Zeeman effect. In Fig. 5.25 tral lines and therefore these components cannot be
the Zeeman pattern of the two D-lines in the sodi- recognized with Doppler-limited resolution. This very
um spectrum are shown, corresponding to the transi- small splitting, which for many atoms can only be
tions Na 32 S1/2 3 2P1/2 (D1 line) and Na 3 2S1/2 resolved with special Doppler-free spectroscopic tech-
3 2P3/2 (D2 line). For the H atom a completely sim- niques (see Chap. 12), is called hyperfine structure. It
ilar pattern is obtained. Only the spin-orbit coupling is explained as follows.
constant a is smaller and therefore the fine structure Atomic nuclei have a small but finite volume
splitting smaller. The Lande factors of the different and possess, besides their electric charge Z e another
levels are characteristic quantity I, called the nuclear spin ana-
    logue to the electron spin s. Its absolute value
gj 2
S1/2 = 2 , g j 2 P1/2 = 2/3 ,
  $
|I| = I (I + 1)h (5.79)
g j 2 P3/2 = 4/3 .
is described by the nuclear spin quantum number I .
The projection of I onto the z-axis is
The spectrum shows four Zeeman components for the
transition 3 2S1/2 3 2P1/2 and six components for the Iz = m I h with I m I +I . (5.80)
3 2S1/2 3 2P3/2 , which are not equidistant.
The nuclear spin I can be treated as an angular momen-
As for the normal Zeeman effect, transitions with
tum in complete analogy to the electron spin.
m j = 1 are circularly polarized and those with
A magnetic moment is connected with the nuclear
m j = 0 are linearly polarized with the electric field
spin
vector E in the direction of the external magnetic
field. N = K I . (5.81)
 5.6. Hyperfine Structure 185

The unit of the nuclear magnetic moment is the nuclear E pot (I , j) = N Bint = |N |B j cos( j , I) .
magneton
(5.84)
e me B
K = h = B = Introducing the total angular momentum F = j + I
2m p mp 1836
of the atom as the vector sum of the total electronic
= 5.05 1027 J T1 (5.82) angular momentum j = l + s and the nuclear spin I
analogue to the Bohr magneton B . However, the (Fig. 5.27), we obtain, because of j I = 1/2(F 2
nuclear magneton is smaller by a factor m e /m p j 2 I 2 ) = | j ||I| cos( j , I) ,
1/1836. The magnetic moment of the proton is jI
I (p) = 2.79K and is determined by the movements cos  ( j , I) =
| j ||I|
and charges of the three quarks (u,u,d) inside the
proton. 1 F(F + 1) j( j + 1) I (I + 1)
= .
The magnetic moment of any atomic nucleus can be 2 j( j + 1)I (I + 1)
written in units of the nuclear magneton as (5.84a)
K
N = K I = gN I , (5.83) The hyperfine energy of the H atom is then
h
A
where the dimensionless factor gN = K h/K is called E HFS = [F(F + 1) j( j + 1) I (I + 1)] ,
the nuclear g-factor. 2
(5.85)
The nuclear magnetic moment gives two contribu-
tions to the shift and splitting of energy levels of the where the hyperfine constant
atomic electrons:
gN K B j
a) The interaction of the nuclear magnetic moment N A= (5.86)
j( j + 1)
with the magnetic field produced by the electrons
at the nucleus (Zeeman effect of N in the internal depends on the internal magnetic field produced by the
magnetic field produced by the electrons). electron, and is therefore dependent on the electronic
b) The interaction of the electronic magnetic mo- angular momentum j.
ment j with the nuclear moment N (magnetic
dipole-dipole interaction). Each energy level E n,l, j splits into hyperfine
components, due to the interaction between nu-
The potential energy of the nuclear magnetic mo-
clear magnetic moment and electronic magnetic
ment N in the magnetic field produced by the electron
moments. The energy of these components is
at the location of the nucleus is (Fig. 5.26)
E HFS = E n,l, j
1
+ A[F(F + 1) j( j + 1) I (I + 1)] .
2
(5.87)

Fig. 5.27. Coupling of total

electronic angular momen-
Fig. 5.26. Interaction between nuclear magnetic moment I tum j = l + s and nuclear
and the magnetic field B j produced at the location of the spin I to the total atom-
nucleus by the orbital motion of the electron and the magnetic ic angular momentum F =
moment s due to the electron spin s j+I
186 5. The Hydrogen Atom

For the H atom, with a proton as the nucleus, the absorption 1S 2S (Fig. 5.29), where two photons
experiments give the values are simultaneously absorbed out of two antiparallel
laser beams. The splitting of the two lines in Fig. 5.29
I = 1/2, g I = +5.58 (N )z = 2.79K . reflects the difference = (E(1S) E(2S))/ h
of the hyperfine splittings of the lower and the upper
For the ground state 2 S1/2 is j = 1/2, I = 1/2 F =
state of the transition with F = F(1S) F(2S) = 0.
0 or F = 1. This gives the two hyperfine components
The splitting E(2S) is small compared to that of the
(Fig. 5.28).
ground state.
3 The total angular momentum F has to be conserved
E HFS (F = 0) = E 1,0,1/2 A for the two photon transition because the two absorbed
4
1 photons have opposite spins. They therefore do not
E HFS (F = 1) = E 1,0,1/2 + A , (5.88) transfer angular momentum to the atom.
4
The hyperfine splitting of the ground state 1S can
with the separation E = A = 5.8 106 ev. be directly measured by a magnetic dipole transition
= E/ hc = 0.047 cm1 = (see Sect. 7.2.6) between the two HFS components.
1.42 GHz, = 21 cm. This transition lies in the microwave range with a wave-
length of = 21 cm. It plays an important role in radio
astronomy, because H atoms are the most abundant
species in the universe and H atoms in interstellar
Fig. 5.28. Hyperfine clouds can be excited by star radiation into the upper
structure of the HFS level and can emit this transition as radio signals
12 S1/2 state of the received on earth by large parabolic radio antennas.
H atom. The The measurements of the signal amplitude with spatial
hyperfine cou-
pling constant is
resolution gives information about the density distri-
A = 0.047cm1 bution, velocities and temperatures of H atoms in the
universe.

5.6.2 Fermi-contact Interaction

The internal magnetic field at the location r = 0 of the
nucleus depends on j and on the spatial probability dis-
tribution |n,l |2 of the electron. The hyperfine splitting
is particularly large for 1S states where has a max-
imum for r = 0. The magnetic interaction of the 1s
electron with the nucleus is called Fermi-contact inter-
action, because there is close contact between the elec-
tron and the nucleus. A more detailed calculation shows
that for S states the hyperfine constant is given by
2
A= 0 gl B gN K |n (r = 0)|2 . (5.89)
3
This is the dominant contribution for the HFS
of the H atom. The absolute value of the hyperfine
splitting in the 2 S1/2 ground state of the H atom is Fig. 5.29. The two resolved hyperfine components F =
0, 1 of the two-photon transition 2S 1S in the H atom
= 0.0474 cm1 . In the optical spectral region it (with kind permission of Th.W. Hnsch from G.F. Bassani,
can only be resolved with special Doppler-free tech- M. Inguscio, T.W. Hnsch (eds) The Hydrogen Atom
niques. One example is the Doppler-free two-photon (Springer, Berlin Heidelberg New York, 1989))
 5.6. Hyperfine Structure 187

5.6.3 Magnetic Dipole-Dipole Interaction 5.6.4 Zeeman Effect of Hyperfine Structure Levels
The second contribution to the hyperfine splitting is the In a weak external magnetic field B the hyperfine
dipole-dipole interaction between the magnetic dipoles component with F = 1 splits into three Zeeman
of the electron and the nucleus. This contribution is sublevels with m F = 0, 1, while the component with
zero for S states with a spherically symmetric charge F = 0 does not split. This is, however, only observed
distribution, because the average of the electronic mag- for weak fields as long as the interaction energy E HFS
netic moment is zero. It therefore plays a role only for between nuclear magnetic moment and electron mo-
states with l 1 (Fig. 5.30), where the first contribu- ments is larger than the Zeeman coupling energy s B
tion is small because the electron density at r = 0 is between the electronic spin moment and the exter-
zero (Fig. 5.3). nal magnetic field. This gives rise to the anomalous
For larger atoms there are also electrostatic contri- Zeeman effect of the hyperfine levels.
butions to the hyperfine structure if the nucleus has an For stronger fields, when s B becomes larger
electric quadrupole moment. For the H atom, howev- than E HFS the electron spin s and the nuclear spin I
er, this electrostatic contribution is absent because the become uncoupled and the energy E F of the levels
proton has no electric quadrupole moment. is governed by the interaction energy s B between
electronic magnetic moment and external field. In this
case there are only two Zeeman components with sz =
1/2. Each of these components shows a hyperfine
splitting into two HFS components with m I = 1/2
Fig. 5.30. Fine and
hyperfine split- (Fig. 5.31).
tings of the 2P This uncoupling of angular momentum by the
state of the H atom magnetic field is called the PaschenBack effect
with n = 2, l = 1, (Fig. 5.32). It appears for hyperfine structure at rather
j = 1/2, 3/2, small magnetic fields. It is also observed for the fine
Schrdinger
Electron Nuclear F = 0, 1, 2 (not to
spin spin scale) structure levels, but only at higher fields, because the

Fig. 5.31a,b. Zeeman effect of the 12 S1/2 ground state of the hydrogen atom. (a) Weak magnetic field. (b) Energy
dependence E F,I ,S (B) of hyperfine components
188 5. The Hydrogen Atom

Fig. 5.32. Vector model The introduction of the electron spin with its two
of the PaschenBack ef- possible orientations sz = 1/2h against the z-axis
fect (which is chosen as preferential axis by general agree-
ment and is called the quantization axis) adds a new
quantum number m s = 1/2, which defines the pro-
jection of the electron spin onto the quantization axis.
Each of the spatial electron distributions n,l,m (x, y, z)
can be realized with two spin orientations. This is
described by multiplying the spatial wave function
n,l,m l (x, y, z) with a spin function m s (sz ) that defines
the projection sz = m s h of the electron spin s onto the
quantization axis. We label the spin function as + for
m s = +1/2 and as for m s = 1/2. The total wave
function, including the electron spin, is then

n,l,m l ,m s (x, y, z, sz ) = n,l,m l (x, y, z)m s .

(5.90b)

interaction energy between s and the internal mag-

netic field produced by l is generally much larger than
Each electronic state of a one-electron atom
the Zeeman energy j B ext , while the Zeeman energy
is unambiguously defined by the four quantum
N Bext can exceed the hyperfine energy E HFS .
numbers n, l, m l and m s . It is described by a sin-
gle wave function (5.90).

5.7 Complete Description

of the Hydrogen Atom 5.7.2 Term Assignment and Level Scheme
The preceding sections have shown that all the effects For the complete assignment of an atomic state by
discussed so far make the spectrum of the simplest its quantum numbers (n, l, m l , s, m s ) the short hand
atom more complicated than was assumed in Bohrs notation
model of the H atom. In this section we will summa-
n 2s+1X j (5.91a)
rize all phenomena discussed in this chapter and some
new ones for a complete description of the hydrogen is used. The letter n denotes the principal quantum
spectrum. number. The capital letter X stands for S (l = 0),
P (l = 1), D (l = 2), F (l = 3), . . .. The upper left in-
dex 2s + 1 is the multiplicity, which gives the number
5.7.1 Total Wave Function and Quantum Numbers
of fine structure components for s > 0, provided the or-
The solutions of the Schrdinger equation for the H bital angular momentum quantum number is l > 0. For
atom gave (without taking into account the electron systems with only one electron outside closed shells is
spin) n 2 different wave functions for each value of the s = 1/2 and the multiplicity is 2s + 1 = 2. Atoms with
principal quantum number n. They represent n 2 differ- a single electron always have doublet states, which split
ent atomic states with the same energy (they are n 2 -fold into two fine structure components for l > 0. The low-
degenerate), but with different spatial distributions of er right index in (5.91a) gives the quantum number j
the electron density. Each of these n 2 wave functions of the total electronic angular momentum j = l + s
(Fig. 5.19).
n,l,m l (x, y, z) = Rn,l (r )Ylm (, ) (5.90a)
The hyperfine components are labeled by the quan-
is unambiguously defined by the quantum numbers n, l, tum number F of the total angular momentum F =
and m l . j + I, including the nuclear spin I (Fig. 5.27).
 5.7. Complete Description of the Hydrogen Atom 189

Fig. 5.33. Complete level

scheme of the H atom includ-
ing all interactions known
so far. Note: The fine, HFS
structure, and the Lamb shift
are not drawn to scale. They
are exaggerated in order to
illustrate the splittings and
shifts

2s, 2p

Schrdinger Dirac Lamb shift Fig. 5.34. Fine structure and Lamb shift of the n =
theory theory quantum electrodynamics 2 level of the H atom. Note the different energy
without spin fine structure scale compared to Fig. 5.33.
190 5. The Hydrogen Atom

EXAMPLE For the ground state 12 S1/2 of the H atom no

Zeeman splitting is observed for the HFS component
The first excited state 2 2P of the H atom that can F = 0, while the other HFS component with F = 1
be reached by one-photon excitation from the 1 2S1/2 splits into three Zeeman sublevels (Fig. 5.31).
ground state is defined by the quantum numbers n = 2, For higher magnetic fields ( j B > E HFS ) the
s = 1/2, l = 1, and j = 1/2 or j = 3/2. The two fine coupling between j and I breaks down. The quantum
structure components are therefore labeled as 2 2P1/2 number F is no longer defined and the Zeeman shift of
and 2 2P3/2 . Both of them split into two hyperfine levels the levels depends on j B. For still higher magnet-
2 2P1/2 (F = 0 or 1) and 2 2P3/2 (F = 1 or 2). ic fields ( j B > E FS ) even the coupling between s
and l breaks down. In this case there is no longer a de-
Without nuclear spin interaction and without Lamb fined total electronic angular momentum j but l and s
shift (see Sect. 5.7.3) all levels of the H atom with equal precess separately around the field axis (PaschenBack
quantum numbers (n, j) have the same energy, because effect Fig. 5.32).
the energy shift due to the relativistic increase of the The complete level scheme of the H atom is shown
electron mass m e and that due to spin-orbit coupling in Fig. 5.33 where, on the left side, the energy levels
just cancel. This (2 j + 1) fold degeneracy is lifted by without effects of the electron spin are plotted, which
the hyperfine interaction, because the magnitude of are the energies obtained from Bohrs atomic model
this interaction depends on the spatial distribution of and also from the Schrdinger equation. The level en-
the electron density and is therefore different for dif- ergies plotted in the second column take into account
ferent values of the quantum number l. Levels with the relativistic mass increase and the fine structure due
equal quantum numbers n and j but different val- to spin-orbit coupling. The next column adds the Lamb
ues of l do experience different hyperfine shifts and shift (see next section) and the last column includes the
splittings. hyperfine interaction.
In an external magnetic field, each atomic state Note that the energy scales for fine and hyperfine
(n, l, s, j) splits without hyperfine interaction into interactions are widely spread, in order to show these
2 j + 1 Zeeman components. The energy separation small splittings in the same diagram. The absolute val-
of these components depends on the Lande fac- ues for the splitting of the 2P1/2 level are illustrated in
tor g j (5.76), which might be different for the different Fig. 5.34.
levels. Generally the Zeeman splittings of different
Remark
states are therefore different (anomalous Zeeman ef-
fect). For states with total electron spin S = 0 (which In this chapter the electron spin was introduced in
can be only realized for atoms with an even number a phenomenological way, based on the results of ex-
of electrons) the normal Zeeman effect applies and the periments, such as the Einsteinde Haas effect, the fine
total Zeeman splittings are equal for all states with structure in the atomic spectra and the SternGerlach
the same orbital angular momentum quantum number experiment. Mathematically, the total wave function
l, while the energy difference between neighbouring was written as the product of spatial wave function
Zeeman levels is independent of l. (solution of the Schrdinger equation) and spin func-
If the Zeeman splittings are small compared to tion. This heuristic introduction of the electron spin is
the hyperfine splittings ( j B  A j I), the exter- able to explain all experimental results discussed so far,
nal magnetic field can not break the coupling between although it does not meet the requirements of a strict
electronic and nuclear magnetic moments. The to- mathematical derivation.
tal angular momentum including nuclear spin is then A complete theory, including the electron spin ab
the vector sum F = j + I, which has 2F + 1 pos- initio, was developed by Paul A.M. Dirac (1902
sible orientations against the external magnetic field 1984) who used as a master equation, instead of the
with different energies. Therefore the hyperfine levels Schrdinger equation, an equation that includes all rel-
with the quantum number F split into 2F + 1 Zeeman ativistic effects (Dirac equation). This equation can be
components. solved analytically for all one-electron systems as long
 5.7. Complete Description of the Hydrogen Atom 191

as they can be reduced to real one body systems (for Random absorption and emission
instance the treatment of the H atom can be reduced to of virtual photons
a one-body system, where one particle with the reduced
mass moves in the spherical symmetric Coulomb po-
tential). This treatment is no longer possible for the
two particle system e+ e (positronium consisting of a)
positron and electron) because the interaction between
the two spins of e+ and e represents a strong pertur-
bation of the Coulomb potential (see Sect. 6.7.4).

5.7.3 Lamb Shift

An atom can absorb or emit electromagnetic radiation. b)
The correct description has to take into account the
interaction of this atom with the radiation field. This
interaction is not only present during the absorption or
emission of photons, but also for so-called virtual in-
teractions, where the atomic electron in the Coulomb
field of the nucleus can absorb and then emit a pho-
ton of energy h during a time interval t < h/E =
1/. The uncertainty relation Et h allows such
processes without violating the energy conservation
law.
This interaction leads to a small shift of the energy
levels (Lamb-shift), which depends on the spatial prob-
ability distribution of the electron in the Coulomb field
c)
of the nucleus and therefore on the quantum numbers n
and l. Fig. 5.35ac. Illustration of the random shaky motion of the
The Lamb shift can be understood at least qual- electron due to absorption and emission of virtual photons.
itatively by an illustrative simple model. Because of Motion of a free electron in a radiation field without taking
the photon recoil, the statistical virtual absorption and into account the photon recoil (a), with recoil (b), and shaky
motion of an electron in a Coulomb field on the first Bohr
emission of photons results in a shaky movement of the orbit including the photon recoils (c)
electron in the Coulomb field of the nucleus (Fig. 5.35),
where its distance from the nucleus r varies in a random
way by r . Its average potential energy is then
0 1 The effects of these interactions are generally very
  Z e2 1 small. Therefore, in most cases the Schrdinger theory,
E pot = . (5.91b)
4 0 r + r including the electron spin, is sufficiently accurate to
match the experimental results. Only in special cases,
For
 a random
  distribution
 of r is r  = 0 but and in particular for high precision measurements, does
(r + r )1  = r 1 . Therefore an energy shift occurs. the Lamb shift have to be taken into account.
Its quantitative calculation is not possible within the The complete term diagram of the levels with n =
framework of the Schrdinger theory but can be per- 2 in the H atom is drawn in Fig. 5.34. The Lamb
formed in an extended theory called quantum electro- shift E L is maximum for the S states, because the
dynamics, which contains the complete description of wave function has a maximum at the position of the
atoms and their electron shells including the interaction nucleus and the effect of the random variations r are
with the radiation field [5.3]. largest for small r values.
192 5. The Hydrogen Atom

The numerical values for the Lamb shifts are collected by a detector. The rate of emitted electrons
represents a small electric current that can be measured.
E La (12 S1/2 ) = +3.55 105 eV During their flight to the detector the electrons
La = +8.176 GHz pass a radio frequency field with a tunable fre-
quency. If the frequency matches the energy separa-
E La (22 S1/2 ) = +4.31 106 eV tion E = E(22 S1/2 ) E(2 P1/2 ) = 4.37 106 eV
La = +1.056 GHz ( res = 1.05 109 Hz or = 30 cm) between the
22 S1/2 state and the 22 P1/2 state, transitions 22 S1/2
E La (22 P1/2 ) = 5.98 108 eV 22 P1/2 are induced. The lifetime of the 22 P1/2 state is
La = 14 MHz . only 2 109 s, because it decays spontaneous-
ly into the 1S state by emitting Lyman- radiation.
The first measurement of the Lamb shift was per- Therefore atoms in the 2P state cannot reach the de-
formed in 1947 by Willis Lamb (*1912) [5.4] and tector. Hydrogen atoms in the 1S ground state cannot
Robert Retherford (*1912) using the experimental set- release electrons from the tungsten target. Therefore
up shown in Fig. 5.36. the measured electron current decreases and I (rf )
In a heated tungsten oven, hydrogen is thermal- shows a sharp dip at the resonance radio frequency.
ly dissociated. (In modern devices, a higher degree of An alternative way for detecting the transitions
dissociation is achieved with a microwave discharge.) between the 22 S1/2 and the 22 P1/2 states is the mea-
The H atoms emerging from a hole in the oven in- surement of the Lyman -fluorescence emitted from
to the vacuum are collimated by the aperture B into the 2 P1/2 state. It can be detected with a solar blind
a nearly parallel atomic beam. The atoms are excit- photomultiplier viewing the rf field region.
ed into the metastable 22 S1/2 state by collisions with The numerical value res = 1.05 109 Hz =
electrons crossing the atomic beam. The lifetime of 1.05 GHz obtained from these experiments is in
the 2S state is about 1s and therefore longer than the good agreement with theory. However, recent, much
flight time of the atoms through the apparatus. After more accurate measurements, show that for a reliable
a pathlength L, the metastable atoms impinge onto comparison with theory the charge distribution in the
a tungsten target, where they transfer their excitation proton, which affects the Lamb shift, must be known
energy, which is higher than the energy necessary to more accurately than is presently possible from high
release electrons from the conduction band, which are energy scattering experiments.

rf
rf

rf

rf rf

Fig. 5.36. LambRetherford experiment

 5.7. Complete Description of the Hydrogen Atom 193

Note: While the LambRetherford experiment measured

only the Lamb shift of the 2S state, a modern ver-
In real experiments [5.4] very small electric stray fields,
sion of Lamb shift measurement can also determine the
which are difficult to eliminate completely, already
much larger Lamb shift of the 12 S1/2 ground state [5.5].
cause Stark shifts that are different for the 2S and the
It is based on the precise comparison of the frequen-
2P levels. These shifts not only add to the Lamb shift
cies of two different optical transitions in the H atom
but can also mix the 2S and 2P levels, causing Lyman-
(Fig. 5.38):
emission without applying the rf field. This effect can
Firstly the two-photon transition 12 S1/2 22 S1/2 ,
be avoided by applying a static magnetic field B, which
which is only possible if two photons are simultaneous-
causes a Zeeman splitting and an increase of the en-
ly absorbed (see Sects. 7.2.6 and 10.5). And secondly,
ergy separation between the 22 S1/2 and 22 P1/2 levels
the one-photon transition 22 S1/2 42 P1/2 .
(Fig. 5.37). Instead of tuning the rf field, the magnet-
According to the Schrdinger theory (and also the
ic field is now varied at a fixed radio frequency until
Dirac theory) the relation
the resonance is reached for transitions between the
Zeeman levels. This has the additional advantage that    
the radiofrequency can always stay in resonance with 10 12 S1/2 22 S1/2 = 420 22 S1/2 42 P1/2
the rf resonator and therefore the rf field amplitude in (5.91c)
the interaction zone is always constant at its maximum
value. holds. Taking into account the Lamb shift (which is
If the experiment is repeated at different radio fre- negligible for the 42 P1/2 level) we obtain for the actual
quencies, the resonance will occur at different magnetic frequencies
fields B. Plotting the measured values of rf as a func-
tion of B (Fig. 5.37b) allows the extrapolation towards 1 = 10 E La (1S) + E La (2S),
B = 0, which yields the field-free Lamb shift. 2 = 20 E La (2S) . (5.91d)

rf

rf

Fig. 5.37. (a) Zeeman splittings of the 22 P1/2 , 22 S1/2 and 22 P3/2 levels for measuring the Lamb shift. (b) Frequencies of the
rf transitions as a function of the magnetic field strength
194 5. The Hydrogen Atom

Fig. 5.38a,b. Optical measurement of the Lamb shift. (a) Level scheme. (b) Experimental arrangement

The difference 5.8 Correspondence Principle

= 1 42
For many qualitative results, estimates are sufficient
= 10 420 (E La (1S) 5E La (2S))/ h and can save much of the time necessary for more de-
= (E La (1S) + 5E La (2S))/ h (5.92a) tailed calculations. Here, a correspondence principle,
formulated by Niels Bohr, is very useful. It illustrates
is measured. Since the Lamb shift of the 2S state is
the relation between classical and quantum physical
known from the LambRetherford experiment, the shift
quantities [5.7]. Its statements are as follows.
of the 1S state can be determined from (5.92a).
The predictions of quantum mechanics have to con-
The two-photon transition 1S 2S is excited by
verge against classical results for the limit of large
two photons from the frequency-doubled output of
quantum numbers.
a dye laser, tuned to the optical frequency L =
Selection rules for transitions between atomic
(1/4)(1S 2S) = (1/4)1 . The Lamb shift of the 1S
states are valid for all quantum numbers. This means
ground state is then
that rules obtained from classical considerations for
E La (1S) = 5E La (2S) h (1 42 ) . large quantum numbers must also be valid for quan-
(5.92b) tum mechanical selection rules for small quantum
numbers.
The very precisely measured frequencies 1 This correspondence principle allows a quantitative
and 2 [5.6] furthermore yields the present most relation between classical and quantum physics and
accurate value of the Rydberg constant gives the validity area for a classical description and
its correspondence to a quantum mechanical model at
Ry = 10,973,731.568639(91) m1 . the borderline of the classical realm. We will illustrate
this using some examples.
 5.9. The Electron Model and its Problems 195

EXAMPLES is described by l = 1 in the Bohr model while the

quantum theory demands l = 0.
1. According to classical electrodynamics, the fre- For large values of l and n both models con-
quency of an electromagnetic wave emitted by an verge against l [l(l + 1)]1/2 [(n 1)n]1/2 n
electron on an orbit around the nucleus equals (because l n 1).
the revolution frequency of the electron. In Bohrs 3. For the limiting case of small frequencies (large
atomic model, this frequency is on the nth orbital wavelengths) Plancks radiation law converges
v m Z 2 e4 against the RayleighJeans law (see Sect. 3.1). The
cla = = 2 . (5.92c) mean energy of the black body radiation at the fre-
2r 40 n 3 h 3
quency is E = n h, where n is the mean
The quantum theory demands that h = E = population density of photons h in a mode of the
E i E k . This gives radiation field. From Plancks formula we can see
  that for 0, the energy converges as E kT .
m e Z 2 e4 1 1 This gives
QM = 2
802 h 3 n i2 nk
n h kT n kT /(h) . (5.95c)
m e Z 2 e4 (n k + n i )(n k n i )
= . (5.93) For h  kT the mean photon density n becomes
80 h 3 nk 2 ni 2
very large, and n h  h. The quantum struc-
For large quantum numbers n and small quantum ture of the photon field becomes less prominent,
jumps n = n k n i  n i we can approximate because the energy E = n h is now a nearly con-
(n k + n i )(n h n i ) 2nn and we obtain: tinuous function of n and the classical model does
m Z 2 e4 not differ much from the quantum mechanical one.
QM n . (5.94) 4. For the harmonic oscillator, the probability
402 n 3 h 3 |n (R)|2 of finding the system in the nth vibra-
For n = 1 the quantum model gives the classi- tional level at a distance R is for small quantum
cal fundamental frequency (5.92b) and for n = numbers n very different for the classical and
2, 3, . . . the corresponding harmonics (Table 5.5). the quantum mechanical models. However, for
2. The angular momentum of the electron is, accord- large values of n the classical probability Pcl (R)
ing to Bohrs model, approaches more and more the average of |(R)|2
(see Fig. 4.21).
|l| = n h with n = 1, 2, 3, . . . , (5.95a)
while the Schrdinger theory yields The correspondence principle is particularly use-
$
|l| = l(l + 1)h . (5.95b) ful for the discussion of selection rules for radiative
transitions between atomic or molecular levels (see
For small values of l the differences between the Chap. 7).
two models are significant, because the lowest state

Table 5.5. Comparison of quantum mechanical and classical 5.9 The Electron Model
transition frequencies n = 1 for the H atom and its Problems
n QM cla Difference (%)
We have learned so far that the electron has a rest
5 5.26 1013 7.38 1013 29 mass m e = 9.1 1031 kg, a negative electric charge
10 6.57 1012 7.72 1012 14 e = 1.6 1019 Coulomb, a spin s with the absolute
100 6.578 109 6.677 109 1.5 value
1000 6.5779 106 6.5878 106 0.15
1
10,000 6.5779 103 6.5789 103 0.015 |s| = 3h , (5.95d)
2
196 5. The Hydrogen Atom
 
which can be mathematically treated like an angular re = e2 / 4 0 m e c2 = 2.8 1015 m . (5.96b)
momentum, and a magnetic moment
|s | = gs B 2B , In this model the magnetic moment s of the elec-
tron is produced by the rotating charge. The elementary
which is related to the spin by calculation gives the relation
s = s s with s = e/m e . 1
s = e re2 with s = |
s| (5.97)
Up to now we have neither discussed the size of the 3
electron, nor the spatial mass and charge distribution. between s and the angular rotation frequency .
In a simple classical model, one assumes that the Inserting the absolute value s = 2B = 1.85
electron can be described by a charged sphere where 1023 Am2 , obtained from the Einsteinde Haas ex-
the mass is uniformly distributed over the volume of periment and the classical electron radius re = 1.4
this sphere and, because of the electric repulsion be- 1015 m, yields the angular frequency
tween charges of equal sign, the charge is uniformly
distributed over its surface (Fig. 5.39). The radius re of 3s
= = 1.7 1026 s1 . (5.98)
this sphere (the classical electron radius) can then be e re2
calculated as follows.
The capacity of the charged surface is This would result in a velocity at the equator of the
sphere of
C = 4 0re . (5.95e)
v = re = 2.3 1011 m/s  c = 3 108 m/s!!
In order to bring a total charge Q = e onto this
capacitor, one needs the energy (5.99)

1 2 1 This is clearly a problematic result.

W = Q /C = e2 /C = e2 /(8 0re ) = E pot . A similar result is obtained, in contradiction to spe-
2 2
(5.95f) cial relativity, when the electron spin is interpreted as
mechanical angular momentum of a sphere with the
This potential energy corresponds to the energy W = classical electron radius.
2 0 |E| of the static electric field produced by the
1 2
The moment of inertia of the sphere is I = 25 m ere2
charged electron. If this energy equals the mass ener- and the angular momentum
gy m e c2 of the electron, the classical electron radius
becomes 1 2
|s| = 3 h = I = m ere2 . (5.100)
e2 2 5
re = = 1.4 1015 m . (5.96a)
8 0 m e c2 This gives an angular velocity
If the charge is not only on the surface of the sphere
5 3 h
but is uniformly distributed over the volume, an ana- = (5.101)
logue consideration yields twice the energy, i.e., W = 4m ere2
e2 /(4 0 re ) and a radius and a velocity of a point at the equator of

5 3 h
v= . (5.102)
4m ere
Inserting the numerical values yields

v = 9 108 m/s > c = 3 108 m/s .

Fig. 5.39. Classical model
of the electron as a sphere From high energy scattering experiments it can be con-
with mass m, uniform sur- cluded that the charge e of the electron is localized
face charge e, spin s and within a smaller volume with r < 1016 m. The re-
magnetic moment s sultant smaller value of r would, however, increase
 5.9. The Electron Model and its Problems 197

the discrepancies of this mechanical model even more, of the electron in the atomic electron shell gives the
because a smaller re in the denominator of the expres- probability to find the (probable point-like electron) in
sion (5.102) would further increase the equator velocity the volume element d around the location (r , , ).
v 1/re . These considerations illustrate a general problem
in the realm of microparticles. Is the distinction be-
tween particles with mass m and field energy E = mc2
Apparently the mechanical model of the elec-
still meaningful? What are the lower limits of volumes
tron as a charged sphere and the interpretation
V = x y z in space, where our geometrical
of its spin as mechanical angular momentum
concept of space is still valid? Do we have to go to
must be wrong. Up to now there does not exist
a higher dimensional space when we want to describe
a convincing realistic model of the electron.
elementary particles?
There have been several attempts to answer these
The high energy experiments and precision mea- questions, but a definite indisputable model has not yet
surements of the magnetic spin moment indicate that been developed. There are, however, mathematical the-
the electron can be treated as a point-like charge. Its ories which are consistent with all experimental results,
mass m e = E/c2 can be interpreted as the energy E of although they do not provide a clear and vivid picture
the electric field produced by its charge e. The spin of the electron.
is an additional characteristic of the electron. Although The Dirac theory starts from a relativistic equa-
it follows the same mathematical relations as other an- tion (the Dirac equation) that describes all properties
gular momenta, such as the commutation relations, and of the electron correctly (except its self-interaction
it has the properties of a vector, it apparently cannot with its radiation field resulting in the Lamb shift).
be regarded as a mechanical angular momentum in the Analogous to the situation for the Schrdinger equation
classical sense. the Dirac equation cannot be derived in a mathematical
The charge distribution way from first physical principles. The complete the-
ory that includes all aspects of atomic and molecular
dq(r , , ) =
el (r , , ) d physics is quantum electrodynamics (QED) [5.810].
Its introduction is, however, beyond the scope of this
= e|(r , , )|2 r 2 sin dr d d book.
198 5. The Hydrogen Atom

S U M M A R Y

The three-dimensional Schrdinger equation for The normal Zeeman effect results from the in-
the hydrogen atom can be separated in the center- teraction of the magnetic moment l (due to the
of-mass system into three one-dimensional equa- orbital motion of the electron) with an external
tions. This is possible because of the spher- magnetic field. This interaction splits the energy
ically symmetric potential. The solutions of states E n,l into (2l + 1) equidistant Zeeman com-
the Schrdinger equation are wave functions ponents with energies shifted by E = B m l B
(r , , ) = R(r )!()"(), which can be writ- against the field-free energies, where B is the
ten as the product of three functions of only one Bohr magneton.
variable. While the radial part R(r ) depends on the Several experimental results (anomalous Zeeman
special r -dependence of the potential, the angular effect, SternGerlach experiment, Einsteinde
part Ylm (, ) = !()"() represents spherical Haas experiment) force an extension of the Schr
surface harmonics Ylm for all spherical poten- dinger theory. This was achieved by the intro-
tials. These functions depend only on the quantum duction of the electron spin s with an additional
numbers l of the orbital angular momentum l spin magnetic moment s = gs (B / h)s with
and m l of its projection l z . the Lande factor gs 2. The total angular mo-
The constraints of normalization and unambigui- mentum of the electron is the vector sum j =
ty for the wave function lead to the quantization l + s. The total wave function is now written as
of bound energy states with E < 0 (only discrete a product of the spatial part and a spin function.
energy levels exist) while for states with E > 0 The fine structure, observed in the atomic spectra,
all energies are allowed (continuous states). One can be interpreted as Zeeman splitting, caused by
can also say that if the wave function is restricted the interaction of the spin magnetic moment s
to a finite volume in space, the energies are quan- with the internal magnetic field, produced by the
tized. If the particle can move all over the space, orbital motion of the electron. The energies of the
a continuous energy spectrum appears. fine structure components are
Each wave function = n,l,m (r , , ) of the a
H atom is unambiguously defined by the three E n,l, j = E n + [ j( j + 1) l(l + 1) s(s + 1)] ,
2
quantum numbers n (principal quantum number),
l (quantum number of orbital angular momen- where
tum l) and m l (projection quantum number of l z ). 0 Z e2 h 2
The absolute square |(r , , )|2 of the wave a=
8 m 2er 3
function describes the probability density func-
tion. This means, that ||2 dV gives the probabil- is the spin-orbit coupling constant.
ity to find the particle within the volume dV . In the Coulomb potential all energy terms with
The energy eigenvalues E n are obtained by insert- equal quantum number j are degenerate when the
ing the wave functions n,l,m into the Schrdinger Lamb-shift is neglected. This is due to the can-
equation. cellation of the energy shift due to the relativistic
Within the Schrdinger model the energies E n of increase of the electron mass and the shift caused
the discrete states of the hydrogen atom depend by the spin-orbit interaction. This degeneracy is
solely on n, not on l and m. All states with equal n lifted in non-Coulombic potentials, even if they
but different values of l or m have the same ener- are spherically symmetric, because here the two
gy (they are degenerate). For each possible value shifts are different. It is furthermore lifted by the
-n1 Lamb-shift.
of E n there are k = l=0 (2l + 1) = n 2 differ-
The anomalous Zeeman effect is observed for
ent wave functions n,l,m (r , , ) that describe n 2
all states with total spin S  = 0: and total orbital
different spatial charge distributions of the elec-
angular momentum L  = 0. The energy shift
tron. The energy states of the hydrogen atom are
of the Zeeman components is E = j B,
therefore n 2 -fold degenerate.

 Summary 199

with j = l + s . Each term E n, j splits into absorption of photons is taken into account, the
(2 j + 1) Zeeman components, which are gener- energy levels experience a small additional shift,
ally not equidistant as for the normal Zeeman called the Lamb shift. The shift is maximum for
effect. the 1S state, smaller for the 2S state and negligi-
Atoms with a nuclear spin I and a corresponding ble for the P or D states. The Lamb shift can only
(very small) nuclear magnetic moment N show be calculated within the framework of quantum
an additional small energy shift E = N B electrodynamics.
of the atomic states, caused by the interaction The Schrdinger theory describes the hydrogen
of the nuclear magnetic moment with the inter- atom correctly if relativistic effects (mass increase
nal magnetic field produced by the electrons at and electron spin) are neglected. The Dirac theory
the position of the nucleus (hyperfine structure). includes these effects, but does not take into ac-
The energy levels split into (2F + 1) hyperfine- count the Lamb shift. A complete description of
components, where F is the quantum number of all effects observed so far, is possible within the
the total angular momentum F = J + I = L + quantum electrodynamic theory.
S + I, including the nuclear spin I. Up to now no concrete model of the electron exists
If the interaction of the electron with the ra- that consistently describes all characteristics such
diation field produced by virtual emission and as mass, size, charge, spin and magnetic moment.
200 5. The Hydrogen Atom

P R O B L E M S
1. Calculate the expectation values r  and 1/r  for 7. Assume you want to measure the Zeeman split-
the two states 1s and 2s in the hydrogen atom. ting of the Balmer -line on the transition
2. Which spectral lines in the emission spectrum of 22 S1/2 32 P1/2 in a magnetic field of B = 1 T.
hydrogen atoms can be observed if the atoms are (a) What should the minimum spectral resolution
excited by electrons with kinetic energy E kin = of a grating spectrograph be in order to resolve
13.3 eV? all components? What is the minimum number of
3. By what factor does the radius of the Bohr or- grooves that must be illuminated if you observe
bit increases if the H atom in its ground state is in the second diffraction order? (b) What is the
excited by (a) 12.09 eV and (b) 13.387 eV? minimum magnetic field B needed to resolve the
4. Show that within the Bohr model the ratio l /l Zeeman components with a FabryPerot interfer-
of orbital magnetic moment and angular mo- ometer (plate separation d = 1 cm, reflectivity of
mentum is independent on the principal quantum each plate R = 95%)?
number n. 8. How large is the internal magnetic field produced
5. By how much does the mass of the hydrogen by the 1s electron in the H atom at the location
atom differ in the state with n = 2 from that in of the proton that causes the splitting of the two
the state n = 1 (a) because of the relativistic in- hyperfine components observed in the transition
crease of the electron mass and (b) because of the with = 21 cm between the two components?
higher potential energy? Assume circular motion 9. Compare the frequencies of the absorption lines
of the electron. 1S 2P for the three isotopes 1 H, 2 D, and
6. In the classical model, the electron is described as 3 T of the hydrogen atom (a) by taking into ac-

a rigid sphere with radius r , mass m, charge e count the different reduced masses and (b) by
and uniform charge distribution. (a) What is calculating the hyperfine shifts and splittings
the velocity of a point on the equator of this with the nuclear spin quantum numbers I (H) =
sphere when the angular momentum is 1/2 3h? 1/2, I (D) = 1 and I (T) = 3/2 and the nuclear
(b) What would the rotational energy of this magnetic moments N (H) = 2.79K , N (D) =
sphere be? Compare the result with the mass 0.857K ; N (T) = 2.98K .
energy m e c2 . Use both numerical values re = 10. Derive the expression (5.49) for the Darwin
1.4 1015 m (obtained from the classical mod- term.
el of the electron) and re = 1018 m (obtained 11. Derive eq. (5.72) by adding (5.42) and (5.66a)
from scattering experiments). using (5.68).