Theoretical Physics KTH June 12, 2007

Quantum Mechanics Formula Collection

To an operator corresponds an adjoint operator

: |V V |

Hermitean operator:

= .
Unitary operator:

= 1,

= 1
Completeness relations:

i
|ii| = I,
_
||d = I
Fourier transforms:
f(k) =
1
(2)
1/2
_

e
ikx
f(x)dx, f(x) =
1
(2)
1/2
_

e
ikx
f(k)dk
Dirac delta function:
_

f(x

)(x x

)dx

= f(x), (x x

) = (x

x),
(ax) = (x)/|a|,
d
dx
(x x

) =
d
dx

(x x

) = (x x

)
d
dx

(x x

) =
d
dx
(x x

), step function: (x) = 0, x < 0; (x) = 1, x > 0

e
ik(x

x)
dk = (x

x)
2 Classical mechanics and electromagnetism
Lagranges equations:
L
q
i

d
dt
L
d q
i
= 0
Lagrangian: L = T V , Action: S =
_
t
1
t
0
Ldt
Canonical momentum conjugate to q
i
: p
i
=
L
q
i
Generalized force conjugate to q
i
: F
i
=
L
q
i
Hamiltonian: H =

i
p
i
q
i
L = T + V =
p
2
2m
+ V (r)
Hamiltons equations: q
i
=
H
p
i
, p
i
=
H
q
i
Force on charge q: F = q
_
E +
v
c
B
_
(no factor c in SI units)
Electric eld: E =
1
c
A
t
. Magnetic eld B = A
Electromagnetic Lagrangian: L =
1
2
mv
2
q +
q
c
v A
Electromagnetic Hamiltonian: H =
|pqA/c|
2
2m
+ q
3 Basic Quantum Mechanics
Schrodinger equation:
i
|(t)
t
= H|(t)
1
Expansion in discrete eigenfunction basis: | =

i
|
i

i
|.
Expectation value: =

i
|
i
||
2
. Probability for a system in state
| to be in state |
i
: P(
i
) = |
i
||
2
Expansion in continuous eigenfunction basis: | =
_
d||.
Expectation value: =
_
d|||
2
. Probability for a system in state
| to have a value for between and + d is P()d, where P() =
|||
2
Canonical commutation relation: [x, p] = i
Position basis: x| = (x), x|x| = x(x), x|p| = i
d(x)
dx
Momentum basis: p| = (p), p|p| = p(p), p|x| = i
d(p)
dp
Ehrenfests theorem: i
d
dt
= [, H]
Probability conservation:
P
t
= j
Probability current: j =

2mi
(

)
Density matrix: =

i
p
i
|ii|
Ensemble average: =

i
p
i
i||i = Tr ()
where Tr denotes summation of the diagonal matrix elements
4 Free particle
Plane waves:
p
(x) =
e
ipx/
(2)
1/2
Propagator U(x, x

, t) = x|U(t)|x

=
_
m
2it
_
1/2
e
im(xx

)
2
/2t
5 Harmonic oscillator
Hamiltonian H =
p
2
2m
+
m
2
x
2
2
Creation and annihilation operators
a

=
_
m
2
_
x
ip
m
_
, a =
_
m
2
_
x +
ip
m
_
, [a, a

] = 1
x =
_

2m
_
1/2
(a

+a), p = i
_
m
2
_
1/2
(a

a)
Number operator: N = a

a
Spectrum: N|n = n|n, n = 0, 1, 2, 3, ...
a|n = (n)
1/2
|n 1 , a

|n = (n + 1)
1/2
|n + 1
Energy eigenstates: H = (N +
1
2
)
H|n = E
n
|n, E
n
= (n +
1
2
)
Groundstate:
0
(x) = x|0 =
1

x
0

exp
1
2
_
x
x
0
_
2
, x
2
0
=

m
2
6 Path integrals
Propagator: U(x, x

, t) =
_
[Dx] exp
_
i

_
t
0
dtL(x, x)
_
where
_
[Dx] = lim
N
_
m
2i
_
N/2

N1
n=1
_
dx
n
, = t/N.
7 Uncertainty relations
Uncertainty: = [( )
2
]
1/2
General uncertainty relation: [, ] = i
||
2
Position-momentum uncertainty relation: xp /2
Energy-time uncertainty relation: Et /2
8 Many particles
Center of mass coordinates: x
CM
= (m
1
x
1
+ m
2
x
2
)/M, p
CM
= p
1
+ p
2
,
M = m
1
+ m
2
,
Relative coordinates: x = x
1
x
2
, p = x, = m
1
m
2
/M
9 Translations and momentum
Translation operator: T(a)|x = |x + a
Translated wave function: x|T(a)| = (x a)
Finite translations: T(a) = e
iap/
10 Rotations and angular momentum
Spatial rotation operator: R(k)| = | +
Rotated wave function: |R(k)| = ( )
Finite rotations: R(n) = e
inJ/
Commutation relations: [

J
x
,

J
y
] = i

J
z
[

J
z
,

J
2
] = 0,

J
2
=

J
2
x
+

J
2
y
+

J
2
z
Spectrum m = j, ..., j

J
2
|j, m = j(j + 1)
2
|j, m,

J
z
|j, m = m|j, m
Raising and lowering operators

J

=

J
x
i

J
y
:
j, m

|j, m =
_
j(j + 1) m(m1)
m

,m1
3
11 Rotation invariant problems
Schrodinger equation in spherical coordinates:
_

2
2
1
r
2

r
r
2
r
+
L
2
2mr
2
+ V (r)
_
R
El
(r)Y
m
l
(, ) = ER
El
(r)Y
m
l
(, )
Spherical harmonics:
L
2
Y
m
l
(, ) = l(l + 1)
2
Y
m
l
(, )
|L
z
| = i

() = m(), () = e
im
12 Hydrogen like atoms
V (r) = Ze
2
/r, Ze = nuclear charge. SI units: e
2
e
2
/4
0
Energy: E
n
=
mZ
2
e
4
2
2
n
2
, n = 1, 2, 3, . . .
Hydrogen (Z = 1): E
n
=
13.6
n
2
eV=
1
n
2
Ry
E
n
is n
2
-fold degenerate.
R
n,l=n1
peaks at r = n
2
a
0
/Z
Bohr radius: a
0
=
2
/me
2
= 0.529

A for H.
13 Spin
Spin 1/2 operators: S =

2
, = (
x
,
y
,
z
)
Pauli spin matrices:

x
=
_
0 1
1 0
_
,
y
=
_
0 i
i 0
_
,
z
=
_
1 0
0 1
_
Rotation matrix:
R(n) = exp (in /2) =
_
cos(/2) sin(/2)e
i
sin(/2)e
i
cos(/2)
_
Magnetic moment: H = B =
ge
2mc
S B
Spin 1 matrices:
S
x
=

2
_
_
0 1 0
1 0 1
0 1 0
_
_
, S
y
=

2
_
_
0 i 0
i 0 i
0 i 0
_
_
, S
z
=
_
_
1 0 0
0 0 0
0 0 1
_
_
14 Addition of angular momenta
Total angular momentum operator: J = J
1
+J
2
Addition of two spin-1/2. Basis states in total spin |s, m-basis:
4
singlet: |0, 0 =
1

2
| +
1

2
| +
triplet: |1, 1 = | + +, |1, 0 =
1

2
| + +
1

2
| +, |1, 1 = |
General case. Vector coupling model:
|j
1
j
2
jm =

m
1
m
2
|j
1
j
2
m
1
m
2
j
1
j
2
m
1
m
2
|j
1
j
2
jm
Clebsch-Gordan coecient is zero unless:
m = m
1
+m
2
and |j
1
j
2
| j j
1
+ j
2
15 Variational method
Variational trial wave function: (x, ).
Ground state estimate: E
0
min

E()
16 Time independent perturbations
H = H
0
+ H
1
, H
0
= unperturbed Hamiltonian, H
1
= small perturbation
E
0
n
, |n
0
unperturbed energies and eigenstates
E
1
n
, |n
1
rst order corrections to energies and eigenstates, etc
First and second order corrections to energy:
E
1
n
= n
0
|H
1
|n
0
, E
2
n
=

m=n
|n
0
|H
1
|m
0
|
2
E
0
n
E
0
m
First order correction to state:
|n
1
=

m=n
|m
0

m
0
|H
1
|n
0

E
0
n
E
0
m
17 Time dependent perturbation theory
H(t) = H
0
+ H
1
(t), H
0
= time independent unperturbed Hamiltonian
H
1
(t) = small time-dependent perturbation acting from time t
0
E
0
n
, |n
0
unperturbed energies and eigenstates
E
1
n
, |n
1
rst order corrections to energies and eigenstates, etc
Expand solution as |(t) =

n
d
n
(t)e
iE
0
n
t/
|n
0

To rst order: d
f
(t) =
fi

_
t
t
0
f
0
|H
1
(t

)|i
0
e
i
fi
t

dt

,
fi
= (E
0
f
E
0
i
)/
Fermis golden rule: Transition rate due to a periodic perturbation H
1
(t) =
H
1
e
it
, averaged over long times: R
if
=
2

|f
0
|H
1
|i
0
|
2
(E
0
f
E
0
i
)
5
18 Pictures
Schrodinger picture
|
S
(t) = U
S
(t, t
0
)|
S
(t
0
)
i
d
dt
U
S
(t) = H
S
U
S
(t)
i
d
dt

S
(t)|O
S
|
S
(t) =
S
(t)|[O
S
, H
S
]|
S
(t)
Heisenberg picture
|
H
= U

S
(t, t
0
)|
S
(t) = |
S
(t
0
)

H
(t) = U

S
U
S
, i
d
H
dt
= [
H
, H
H
]
Interaction picture
|
I
(t) = U
0
S
(t, t
0
)|
S
(t)

I
= U
0
S

S
U
0
S
i
d
I
dt
= [
I
, H
0
I
]
i
d
dt
U
I
= H
1
I
U
I
U
I
(t, t
0
) = 1
i

_
t
t
0
H
1
I
(t

)U
I
(t

, t
0
)dt

= 1
i

_
t
t
0
H
1
I
(t

)dt

+ O(H
2
I
)
19 Quantization of EM eld
Classical Lagrangian: L =
1
2
mv
2
+
q
c
A v q
Canonical momentum: mv = p qA/c
Classical Hamiltonian: H =
1
2m
(p qA/c)
2
+ q
Coulomb gauge: A = 0
Vector potential operator and its canonical conjugate
A =

_
_
c
2
4
2

_
1/2
[a(k)(k)e
ikr
+ a

(k)(k)e
ikr
]d
3
k
=

_
1
i
_

64
4
c
2
_
1/2
[a(k)(k)e
ikr
a

(k)(k)e
ikr
]d
3
k
Commutator: [a(k), a

(k

)] =
,

3
(k k

)
(k) polarization vector, = photon energy
p = k = photon momentum
Hamiltonian: H =

_
[a

(k)a(k) + 1/2] d
3
k
20 Scattering theory
Lippmann-Schwinger eq: | = | +
1
EH
0
+i
V |
(x) incoming state, (x) scattered state
(x) =
1
(2)
3/2
_
e
ikx
+
e
ikr
r
f(k

, k)
_
f scattering amplitude,
d
d
= |f|
2
dierential cross section
6
First order Born approximation:
f
(1)
(k

, k) =
1
4
2m

2
_
d
3
x

e
i(kk

)x

V (x

)
Partial wave expansion: for r
e
ikz

l=0
i
l
(2l + 1)
sin(kr l/2)
kr
P
l
(cos )
(r)

l=0
C
l
sin(kr l/2 +
l
(k))
kr
P
l
(cos )
f() =

l=0
(2l + 1)
e
i
l
k
sin(
l
)P
l
(cos )
Optical theorem: Im f( = 0) =
k
4

21 Constants
elementary charge e = 1.602 10
19
C
electron volt 1 eV = 1.602 10
19
J
mass m
e
= 9.11 10
31
, m
p
= 1.67 10
27
, m
n
= 1.67 10
27
kg
magnetic moment
e
= 9.28 10
24
,
p
= 1.41 10
26
,
n
= 4.49 10
26
Am
2
Planck const = h/2 = 1.05 10
34
Js
light speed c = 3.00 10
8
m/s
ne structure const = e
2
/4
0
c = 1/137
Coulomb law const 1/4
0
10
7
c
2
Bohr radius a
0
= 5.29 10
11
m
Boltzmann const k
B
= 1.38 10
23
J/K
22 Special functions
Harmonic oscillator wave functions
(a
2
= /m)
0
(x) =
1
(a

)
1/2
e
x
2
/2a
2
,
1
(x) =
1
(2a

)
1/2
2
_
x
a
_
e
x
2
/2a
2

2
(x) =
1
(8a

)
1/2
_
2 4
_
x
a
_
2
_
e
x
2
/2a
2
,
3
(x) =
1
(48a

)
1/2
_
12
_
x
a
_
8
_
x
a
_
3
_
e
x
2
/2a
2
7
Spherical harmonics
Y
0
0
=
_
1
4
, Y
0
1
=
_
3
4
cos , Y
1
1
=
_
3
8
sin e
i
Y
0
2
=
_
5
16
(3 cos
2
1), Y
1
2
=
_
15
8
sin cos e
i
, Y
2
2
=
_
15
32
sin
2
e
2i
Legendre polynomials
P
l
(cos ) =
_
4
2l + 1
_
1/2
Y
0
l
Hydrogen-like radial wave functions
R
10
= 2
_
Z
a
0
_
3/2
e
Zr/a
0
, R
20
= 2
_
Z
2a
0
_
3/2
_
1
Zr
2a
0
_
e
Zr/2a
0
R
21
=
1

3
_
Z
2a
0
_
3/2
Zr
a
0
e
Zr/2a
0
, R
30
= 2
_
Z
3a
0
_
3/2
_
1
2Zr
3a
0
+
2(Zr)
2
27a
2
0
_
e
Zr/3a
0
R
31
=
4

2
9
_
Z
3a
0
_
3/2
Zr
a
0
_
1
Zr
6a
0
_
e
Zr/3a
0
, R
32
=
2

2
27

5
_
Z
3a
0
_
3/2
_
Zr
a
0
_
2
e
Zr/3a
0
Spherical Bessel and Neumann functions
j
0
() =
sin

, j
1
() =
sin

2

cos

, j
2
() =
_
3

3

1

_
sin
3 cos

0
() =
cos

,
1
() =
cos

2

sin

,
2
() =
_
3

3

1

_
cos
3 sin

2
8