| BSC Sem5 : Introduc on to QM : GSS College Belgaum

1. What is an Atom:

The search for “what is matter made of ?” began more than 2500 years ago. Some of the early
philosophers proposed that matter was composed of tiny particles. They named this tiny
particle an “atom”, (it means “cannot be divided”).
The atom is smallest particle of an element which possesses all properties of that element. For
example, atom of carbon shows all properties of carbon, atom of oxygen shows all properties
of oxygen and so on.
After this a search for structure, shape, size and contents of an atom began. Many
philosophers and scientists proposed various models for an atom. Some of these models finally
led to proper atom model explaining all features of an atom. The prominent amongst them
are Dalton’s atom model, Thomson’s atom model, Rutherford’s atom model, Bohr’s atom
model, Sommerfeld atom model, vector atom model and quantum-mechanical model. These
attempts discovered that an atom has nucleus containing protons and neutrons with electrons
revolving around nucleus.

2. Thomson’s Atom Model:

The first model of atom was proposed by J. J. Thomson in 1898. According to this model, the
posi ve charge of the atom is uniformly distributed throughout the volume of the atom. And
the nega vely charged electrons are embedded in it like seeds in a watermelon. This model
was called plum pudding model of the atom. However subsequent studies on atoms, showed
that the distribu on of the electrons and posi ve charges are different from that proposed in
this model.

3. Alpha Par cle Sca ering Experiment (Rutherford’s Sca ering):

Ernst Rutherford, in 1911, proposed an experiment of alpha par cle sca ering. It was
performed by H. Geiger and E. Marsden.
The experimental arrangement is as shown in figure.

Alpha-par cles emi ed by a 83 B 214 radioac ve source are collimated into a narrow beam by
passing through lead bricks. The beam is allowed to fall on a thin foil of gold of thickness 2.1
× 10–7 m. The sca ered alpha-par cles are observed through a rotatable detector consis ng
of zinc sulphide screen and a microscope. The sca ered alpha-par cles strike the screen and
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 | BSC Sem5 : Introduc on to QM : GSS College Belgaum

produce brief light flashes or scin lla ons. These flashes are viewed through a microscope.
The distribu on of the number of sca ered par cles can be studied as a func on of angle of
sca ering. Following is the graph of total number of sca ered par cles against sca ering
angle.

Following are some of the important observa ons.

a. Many of the alpha par cles pass through the foil. It means that they do not suffer any
collisions on their way. Hence there is sufficient vacuum inside the atoms of the foil.
b. Only about 0.14% of the incident a-par cles sca er by more than 1°; and about 1 in 8000
deflect by more than 90°. Rutherford argued that, to deflect the a-par cle backwards, it
must experience a large repulsive force. This force can be provided only if almost en re
part of the mass of the atom and its posi ve charge were concentrated at its centre.
Rutherford termed this centre as nucleus.
c. The nega vely charged electrons are outside the nucleus. To avoid their falling into the
nucleus, Rutherford suggested that the a rac ve force exerted by nucleus should act as
centripetal force. And hence electrons revolve around the nucleus in circular orbit.
d. This experiment suggested the size of the nucleus to be about 10 -15 m to 10 -14 m.

4. Rutherford’s Sca ering Formula:

Following the Rutherford’s sca ering formula,

=
Ω ɛ

Where,
dσ is sca ering cross sec on.
ze is charge of projec le.
Ze is charge of target.
Ɵ is sca ering angle.
dΩ is solid angle.
E is incident par cle beam energy.
ɛ0 is absolute permi vity of free space.

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5. Postulates of Bohr’s Theory of Hydrogen atom:

The Bohr model of the atom was proposed by Neil Bohr in 1915. Following are postulates of
Bohr’s Hydrogen atom theory.
a. Almost the entire mass and entire positive charge of an atom are concentrated at
centre of atom called nucleus. and the electrons (negatively charged) revolve around
the positively charged nucleus in a definite circular path called stationary orbits. The
attractive electrostatic force between nucleus and electron acts as necessary
centripetal force required for circular motion.
b. Stationary orbits are those orbits in which orbital angular momentum of electron is
equal to integral multiple of .
i.e. 𝐿 = 𝑛 , where h is Planck’s constant. Where n=1, 2, 3, 4…
c. Electrons orbiting in stationary orbit doesn’t radiate energy.
d. Electrons orbiting in stationary orbit possesses fixed amount of energy. Inner orbit
electrons have less energy than outer orbit electrons.
e. When electron jumps from higher to lower orbit, the difference in energy is emitted in
the form of photon of energy. 𝐸 = ℎ 𝝂 where h is Planck’s constant and 𝝂 is frequency
of radiation.

6. Derivation of Expression of Radius of Stationary Orbit:

Consider an electron with mass m and charge e circulating a nucleus with atomic number z.
its mass is z mp where mp is mass of nucleon.

Ler r be the radius of the orbit. The electrosta c force

between electron and nucleus is,

𝐹 =
e
z The centripetal force required for circular mo on is

𝐹 =

Hence,
𝐹 =𝐹

i.e.

𝑚𝑣 𝑟= 𝑧 𝑒 ………………………. (1)

According to Bohr’s condi on, orbital angular momentum of electron is,

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𝑚𝑣 𝑟 = 𝑛 ………………. (2)
Squaring equa on (2) we get,
𝑚 𝑣 𝑟 =𝑛 …………………… (3)

Dividing equa on (3) by equa on (2) we get,

𝑚𝑟=

𝑟=

𝑟=𝑛 ………………… (4)

For hydrogen z = 1 and hence radius of first orbit of hydrogen atom is,

𝑟=𝑛

Equa on (4) is the expression for radius of nth orbit.

We understand that the radius of nth orbit is propor onal to square of orbit number. Hence
orbits are not equally spaced.
If radius of first orbit is r0 then radii of other orbits are,
r2 = 4 r 0 ,
r3 = 9 r 0
r4 = 16 r0 and so on…..

7. Total Energy of Electron in Hydrogen Atom:

Consider an electron with mass m and charge e circulating a nucleus with atomic number z.
its mass is z mp where mp is mass of nucleon. The electron is orbiting around nucleus in
electric field of nucleus. and hence it possesses potential energy. And due to velocity of
electron it also possesses kinetic energy.
Hence total energy of electron is,

T.E. = KE + PE

Ler r be the radius of the orbit. The electrosta c force

e between electron and nucleus is,
z

𝐹 =
The centripetal force required for circular mo on is

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𝐹 =

Hence,
𝐹 =𝐹

i.e.

𝑚𝑣 =

And hence, kine c energy of electron is,

𝐾𝐸 = 𝑚𝑣

𝐾𝐸 =

𝐾𝐸 = …………………. (1)

And,
PE of electron is,

𝑃𝐸 = 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑎𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑟 𝑑𝑢𝑒 𝑡𝑜 𝑛𝑢𝑐𝑙𝑒𝑎𝑟 𝑐ℎ𝑎𝑟𝑔𝑒 𝑥 𝑐ℎ𝑎𝑟𝑔𝑒 𝑜𝑛 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛

𝑃𝐸 = (−𝑒)

𝑃𝐸 = ………………. (2)

𝑃𝐸 =
Hence,
𝑇𝐸 = 𝐾𝐸 + 𝑃𝐸

𝑇𝐸 = +

𝑇𝐸 =

Subs tu ng value of r we get,

𝑇𝐸 =

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𝑇𝐸 =

𝑇𝐸 =

For hydrogen atom z = 1,

Hence total energy of electron in hydrogen atom in nth orbit is,

𝐸 =

Subs tu ng the values of all constants on RHS and conver ng energy in electron volt we get,

.
𝐸 = 𝑒𝑉

(1 electron volt is the energy gained by an electron when it is accelerated through a poten al
difference of 1 volt)
Therefore, energy of electron in
first orbit is -13.6 eV
second orbit is -13.6/4 = -3.4 eV
third orbit is -13.6/9 = -1.51 eV
and so on..
With increase in orbit number n, the numerical value of energy goes on decreasing but due to
nega ve sign, it becomes less and less nega ve. Hence energy of electron goes on increasing
as one moves from inner to outer orbit.

8. Excita on and Ioniza on Poten als:

When the electron exists in first orbit then the hydrogen atom is said to be in ground state or
normal state. The process of shi ing electron from the given lower energy orbit to another
higher energy orbit is known as excita on. The minimum energy required to excite an electron
from the ground state of an atom to any excited state is called excita on energy.
The excita on poten al is the poten al to be applied to an external electron so that it gains
sufficient energy and can raise the electron from one orbit to another higher orbit by impact
is known as excita on poten al. If an electron is raised from first orbit to second orbit, then
excita on poten al is known as first excita on poten al. If it is raised from first to third orbit,
then it is called second excita on poten al and so on.
Therefore,
First excita on poten al is –3.4 – (–13.6) = 10.2 Volt
Second excita on poten al is – 1.51– (–13.6) = 12.09 Volt and so on.
The process of shi ing electron from the given lower energy orbit to completely out of atom
is known as ioniza on.
The ioniza on poten al is the poten al to be applied to an external electron so that it gains
sufficient energy and can completely remove electron given orbit to completely out of atom
by impact is known as ioniza on poten al.

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The general name for all the excita on and ioniza on energies in the electron orbits of an
atom is cri cal poten al. The corresponding electronic states of the atom can be excited in
various ways, for example by inelas c collisions with electrons.

9. Hydrogen atom Spectrum:

Consider a sample of hydrogen. However small in amount, it contains huge number of atoms
in it (refer Avogadro number). Therefore, when different-different amount of energy is
supplied to sample then the atoms absorb the energy and different atoms go to different
energy states. i.e. electrons are raised to different orbits. When these electrons jump back to
lower energy states tending to achieve normal state, they may jump directly or in steps. Each
of such jump gives rise to a radia on in the form of photon having energy equal to difference
in energy of ini al orbit and final orbit.
E2 – E 1 = h 𝝂
Where h is Planck’s constant and 𝝂 is frequency of radia on emi ed.
And hence we can expect huge number of photons of different energies. The group of photons
of par cular energy gives rise to a line in hydrogen spectrum. And hence there are number of
spectral lines in hydrogen spectrum. These all lines are classified into various following series
depending upon the ini al orbit during jump.
a. Lyman Series:
Theodore Lyman found this series between
1906 and 1914. As a result, it is named for him.
According to Bohr’s concept, the Lyman series
appears when electrons shi from higher
energy levels (n2 = 2,3,4,5,6,…) to n1 = 1 energy
state. The wavelengths of the Lyman series are
all in the ultraviolet range.
b. Balmer Series:
This series was discovered in the year 1885 by
Johann Balmer. Thus, the series is named a er
him. Balmer series appears when electron transi on takes place from higher energy states
(n2 =3,4,5,6,7,…) to n1=2 energy state. All the wavelengths of Balmer series fall in visible
part of electromagne c spectrum (400nm to 740nm).
c. Paschen Series:
This series was first observed in 1908 by German Physicist Friedrich Paschen. Thus, the
series is named a er him. Paschen series appears when electron transi on takes place
from higher energy states n2 =( 4,5,6,7,8,…) to n1=3 energy state. All the wavelengths of the
Paschen series fall in the Infrared region of the electromagne c spectrum.
d. Bracke Series:
The series was first observed in 1922 by American Physicist Friedrich Sumner Bracke .
Thus, the series is named a er him. Bracke series appears when electron transi on takes
place from higher energy states (n2=5,6,7,8,9…) to n1=4 energy state. All the wavelengths
of the Bracke series fall in the Infrared region of the electromagne c spectrum.
e. Pfund Series:
The series was first observed during the year 1924 by August Harman Pfund. Thus, the
series is named a er him. Pfund series is appears when electron transi on takes place from
higher energy states (n2=6,7,8,9,10,…) to n1=5 energy state. All the wavelengths of Pfund
series fall in Infrared region of the electromagne c spectrum.

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10. Ritz Combina on Principle:

The Ritz combina on principle is an empirical rule proposed by Walther Ritz in 1908 to
describe the rela onship between the spectral lines for all atoms.
The principle states that the spectral lines of any element include frequencies that are either
the sum or the difference of the frequencies of two other lines.
By using this principle, the lines of the spectra of elements can be predicted from exis ng
lines.

According to Bohr’s theory of hydrogen atom, the energy of electron in the n th orbit is,

𝐸 =
Hence energy of electron in orbit n 1 is,

𝐸 =
And in orbit n2 is,

𝐸 =
Therefore, energy of photon emi ed when an electron jumps from n2 to n1 is,

𝐸 −𝐸 =ℎ𝝂
i.e.
1 𝑚 𝑒4
ℎ𝝂=
𝑛21
− 𝑛12 2 …………. (1)
2 8 ℎ 𝜀20

= −

And hence,

= −

All terms in the bracket on RHS are constant.

According to Rydberg’s formula for hydrogen spectrum,
=𝑅 −
The value of Rydberg’s constant as given by Rydberg is 1.097 x 107 m-1
By comparing above equa ons we get,

𝑅=

This value is also equal to 1.097 x 107 m-1. This is first confirma on of Bohr’s theory.

From equa on (1) we have,

1 𝑚 𝑒4
ℎ𝝂=
𝑛21
− 𝑛12 2
2 8 ℎ 𝜀20

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And hence,

𝜈= −

𝜈= − (𝑐 𝑅)

Where c is velocity of light and R is Rydberg’s constant.

Consider an electron jumping from n2 to n1. Let the frequency of photon emi ed be 𝜈 .
Hence,
𝜈 = − (𝑐 𝑅) ………… (2)

Similarly, when electron jumps from n3 to n2 let frequency emi ed be 𝜈 .

𝜈 = − (𝑐 𝑅) …………….. (3)

Let 𝜈 be the frequency of photon emi ed when electron jumps from n3 to n1 directly, then,

𝜈 = − (𝑐 𝑅) ……………….. (4)

Adding equa ons (1) and (2) we get,

𝜈 +𝜈 = = − (𝑐 𝑅) + − (𝑐 𝑅)

𝜈 +𝜈 = − + − (𝑐 𝑅)

𝜈 +𝜈 = − (𝑐 𝑅) …………. (5)

From equa ons (4) and (5) we get,

𝜈 = 𝜈 + 𝜈 ……………….. (6)

This proves Ritz Combina on Principle.

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11. Correspondence Principle:

In physics, the correspondence principle states that the behaviour of systems described by
the theory of quantum mechanics reproduces classical physics in the limit of large quantum
numbers. In other words, it says that for large orbits and for large energies, quantum
calcula ons agree with classical calcula ons.
The principle was formulated by Niels Bohr in 1920.
The correspondence principle establishes the connec on between classical and quantum
physics.
Einstein's Special Theory of Rela vity sa sfies the Correspondence Principle since it reduces
to classical mechanics for veloci es small compared with the velocity of light. Similarly, the
General Theory of Rela vity reduces to Newton's Law of Gravita on in the limit of weak
gravita onal fields.

12. Atomic Excita ons:

The hydrogen atomic model proposed by Niels Bohr (1915) was the first stable model of the
atom. It gave the concept of electrons revolving around the nucleus in fixed orbits.
Erwin Schrödinger’s quantum mechanical atomic model (1926) states electrons revolve
around the nucleus in shells, orbits, orbitals, and spinning orientations with discrete quantum
states.
Hence, the energy of the electrons is quantized. Atomic excitation means an electron in an
atom can be moved to a higher energy orbit when the atom absorbs an amount of energy
that exactly equals the energy difference between the orbits. Similarly, atomic de-excitation
means, an electron in an atom can be moved to a lower energy orbit when the atom emits an
amount of energy that exactly equals the energy difference between the orbits.
Generally atomic excitations are carried out by the absorption of photons with discrete
energy that is exactly equal to the energy gap between two orbits. Many larger atoms are
excited using electron- or ion-beams in which fast-moving electrons and ions collide with the
electrons in the atom and transfer their kinetic energy through perfect elastic collision.
Semiconductors have a very small energy gap, which can be excited simply by thermal
excitation. Thermal excitation is a process in which vibrations of crystal lattice provide
sufficient energy to transit a ground-state electron to a higher energy state.

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