MP463_Ch0

MP463 QUANTUM MECHANICS

IntroductionQuantum theory of angular momentumQuantum theory of a particle in a central potential- Hydrogen atom- Three-dimensional isotropic harmonic oscillatorNon-relativistic quantum theory of electron spinAddition of angular momentaStationary perturbation theoryTime-dependent perturbation theorySystems of identical particles

REFERENCES

Claude Cohen-Tannoudji, Bernard Liu, and Franck LaloeQuantum Mechanics I and IIJohn Wiley & Sons

Lecture notes - online access:http://www.thphys.nuim.ie/Notes/MP463/

REQUIREMENTS

The total mark consists of:

Examination (constitutes 80% of the total mark):duration: 90 minutes,structure: 3 questions, each with several sub-questions,requirements: answer 2 (and only 2) of the 3 questions in writing,maximum mark: 100 points.

Continuous Assessment - 2 quizzes (20% of the total mark):duration of each quiz: 30 minutes,maximum mark of each quiz: 10 points.

CHAPTER 0: THE POSTULATES OF QUANTUM MECHANICS

(From Cohen-Tannoudji, Chapters II & III)

FIRST POSTULATE

At a fixed time t, the state of a physical system is defined by specifying a ket |ψ(t)�belonging to the state space E.

The state space is a space of all possible states of a given physical system, and it isa Hilbert space, i.e. it is:

1. a vector space over the field of complex numbers C

|ψ� , |φ1� , |φ2� ∈ E, c1, c2 ∈ C (0.1)

|ψ� = c1 |φ1� + c2 |φ2� (0.2)

Definition: A vector space over the field of complex numbers C is a set of el-ements, called vectors, with an operation of addition, which for each pair ofvectors |ψ� and |φ� specifies a vector |ψ� + |φ�, and an operation of scalar multi-plication, which for each vector |ψ� and a number c ∈ C specifies a vector c |ψ�such that (s.t.)1) |ψ� + |φ� = |φ� + |ψ�2) |ψ� + (|φ� + |χ�) = (|ψ� + |φ�) + |χ�3) there is a unique zero vector s.t. |ψ� + 0 = |ψ�4) c(|ψ� + |φ�) = c |ψ� + c |φ�5) (c + d) |ψ� = c |ψ� + d |ψ�6) c(d |ψ�) = (cd) |ψ�7) 1. |ψ� = |ψ�8) 0. |ψ� = 0Example: a set of N-tuples of complex numbers.

2. with an inner (scalar) product.Dirac bra-ket notation:

|ψ� , |φ� ∈ E (0.3)�φ|ψ� ∈ C (0.4)

A bra �φ| is the adjoint of a ket |φ�, e.g.

if |ψ� = c1 |φ1� + c2 |φ2� , (0.5)then �ψ| = c∗1 �φ1| + c∗2 �φ2| (0.6)

We call |φ1� and |φ2� a basis (or basis elements) of E if and only if

span{|φ1� , |φ2�} = E (0.7)and

�φi|φ j�= δi j (0.8)

where δi j is the Kronecker delta-symbol.

3. And with a norm and metric induced by the inner product.

(a) Norm:

e.g.�φi|φ j�= δi j i.e. (0.9)

�φ1|φ1�1/2 = �φ1� = 1 (0.10)

≡ the norm of |φ1�

If the norm is 1, the state is normalized, i.e. its length equals 1.

Two vectors are orthogonal if their inner product is zero. Mutually orthogonalvectors of unit length (norm) are called orthonormal.

(b) Metric: a metric is a map which assigns to each pair of vectors |ψ�, |φ� ascalar ρ ≥ 0 such that

i. ρ (|ψ� , |φ�) = 0 iff |ψ� = |φ�;

ii. ρ (|ψ� , |φ�) = ρ (|φ� , |ψ�)

iii. ρ (|ψ� , |χ�) ≤ ρ (|ψ� , |φ�) + ρ (|φ� , |χ�) (triangle identity)

We say that the metric is induced by the norm if

ρ (|ψ� , |φ�) = �|ψ� − |φ�� (0.11)

So the Hilbert space is normed and a metric space. What else?

4. It is also a complete space so every Cauchy sequence of vectors, i.e.

�|ψn� − |ψm�� → 0 as m, n→ ∞ (0.12)

converges to a limit vector in the space.(We need this condition to be able to handle systems with infinite-dimensionalHilbert spaces, i.e. with infinite degrees of freedom.)

Can we be more concrete about quantum states? What really is a ket |ψ�?

Now, we need the concept of representation.Let us say we have the Hilbert space E and the basis

B = {|φ1� , |φ2�} (0.13)

and we have a ket

|ψ� ∈ E (0.14)

which we wish to express in the representation given by the basis B.We use the completeness relation

�

i|φi� �φi| = 1 (0.15)

as follows

|ψ� =�

i|φi� �φi|ψ�����

a number∈C

(0.16)

=�

ici |φi� (0.17)

Our state becomes a specific superposition of the basis set elements, i.e. we haveexpanded |ψ� in terms of {|φi�}.

What about a representation in a continuous case (e.g. a free particle)?

The completeness relation:The coordinate operator X has the spectral decomposition

X =

�∞

−∞

x |x� �x| dx (0.18)

where x are eigenvalues and |x� are eigenstates, i.e.

X |x� = x |x� (0.19)

Then the completeness relation is�∞

−∞

|x� �x| dx = 1 (0.20)

Coordinate representation

|ψ� ∈ E (0.21)

|ψ� =

�∞

−∞

|x� �x|ψ� dx

=

�∞

−∞

ψ(x) |x� dx (0.22)

{ψ(x)} are coefficients of the expansion of |ψ� using the basis given by the eigenvec-tors of the operator X, called wavefunction

Inner product in coordinate representation

�φ1|φ2� =

�∞

−∞

φ∗1(x)φ2(x) dx (0.23)

SECOND POSTULATE

Every measurable physical quantityA is described by an operator A acting on E; thisoperator is an observable.————–An operator A : E→ F such that

���ψ��= A |ψ� for

|ψ� ∈ E����domain D(A)

(0.24)

and���ψ��∈ F����

range R(A)

(0.25)

Properties:

1. Linearity A�

i ci |φi� =�

i ciA |φi�

2. Equality A = B iff A |ψ� = B |ψ� and D(A) = D(B)

3. Sum C = A + B iff C |ψ� = A |ψ� + B |ψ�

4. Product C = AB iff

C |ψ� = AB |ψ�= A

�B |ψ��= A���Bψ�

(0.26)

5. Functions A2 = AA, then An = AAn−1 and if a function f (ξ) =�

n anξn, then bythe function of an operator f (A) we mean

f�A�=�

nanAn (0.27)

e.g.

eA =

∞�

n=0

1n!

An (0.28)

Commutator and anticommutatorIn contrast to numbers, a product of operators is generally not commutative, i.e.

AB � BA (0.29)

———–For example: three vectors |x�, |y� and |z� and two operators Rx and Ry such that:

Rx |x� = |x� , Ry |x� = − |z� ,Rx |y� = |z� , Ry |y� = |y� ,Rx |z� = − |y� , Ry |z� = |x�

(0.30)

then

RxRy |z� = Rx |x� = |x� � (0.31)RyRx |z� = −Ry |y� = − |y� (0.32)

—————

An operator�A, B�= AB − BA is called commutator.

We say that A and B commute iff�A, B�= 0 in which case also

�f (A), f (B)

�= 0.

An operator�A, B�= AB + BA is called anticommutator.

Basic properties:�A, B�= −

�B, A�

(0.33)�A, B�=�B, A�

(0.34)�A, B + C

�=�A, B�+�A, C�

(0.35)�A, BC

�=�A, B�C + B

�A, C�

(0.36)

the Jacobi identity:�A,�B, C��+�B,�C, A��+�C,�A, B��= 0 (0.37)

Types of operators (examples)

1. A is bounded iff ∃β > 0 such that���A |ψ�

��� ≤ β �|ψ�� for all |ψ� ∈ D(A). Infimum ofβ is called the norm of A

2. A is symmetric if�ψ1|Aψ2

�=�Aψ1|ψ2

�for all |ψ1� , |ψ2� ∈ D(A).

3. A is hermitian if it is bounded and symmetric.

4. Let A be a bounded operator (with D(A) dense in E); then there is an adjoint operator A†

such that�ψ1|A†ψ2

�=�Aψ1|ψ2

�(0.38)

i.e.�ψ1|A†ψ2

�=�ψ2|Aψ1

�∗(0.39)

for all |ψ1� , |ψ2� ∈ D(A).

Properties:����A†���� =

���A��� (0.40)

�A†�†= A (0.41)

�A + B

�†= A† + B† (0.42)

�AB�†= B†A† (the order changes) (0.43)

�λA�†= λ∗A† (0.44)

How can we construct an adjoint?E.g. Let us have an operator in a matrix representation (so it is also a matrix)then

A† =�AT�∗ = transpose & complex conjugation (0.45)

5. A is selfadjoint if A† = A.This is the property of observables!Their eigenvalues are real numbers, e.g. X |x� = x |x�

6. A is positive if �ψ| A |ψ� ≥ 0 for all |ψ� ∈ E

7. A is normal if AA† = A†A i.e.�A, A†

�= 0

������������������commutator

8. Let A be an operator. If there exists an operator A−1 such that AA−1 = A−1A = 1(identity operator) then A−1 is called an inverse operator to AProperties:

�AB�−1

= B−1A−1 (0.46)�A†�−1

=�A−1�† (0.47)

9. an operator U is called unitary if U† = U−1, i.e. UU† = U†U = 1.

Formal solution of the Schrodinger equation leads to a unitary operator: if H isthe Hamiltonian (total energy operator),

i�ddt|ψ(t)� = H |ψ(t)� (0.48)

⇒

� t

0

d���ψ(t�)

�

|ψ(t�)�= −

i�

� t

0Hdt� (0.49)

If the Hamiltonian is time independent then

|ψ(t)� = e−i�Ht|ψ(0)� = U |ψ(0)� (0.50)

10. An operator P satisfying P = P† = P2 is a projection operator or projectore.g. if

���ψk�

is a normalized vector then

Pk =���ψk� �ψk��� (0.51)

is the projector onto one-dimensional space spanned by all vectors linearly de-pendent on

���ψk�.

Composition of operators (by example)

1. Direct sum A = B ⊕ CB acts on EB (2 dimensional) and C acts on EC (3 dimensional)Let

B =�

b11 b12b21 b22

�and C =

c11 c12 c13c21 c22 c23c31 c32 c33

(0.52)

A =

b11 b12 0 0 0b21 b22 0 0 00 0 c11 c12 c130 0 c21 c22 c230 0 c31 c32 c33

(0.53)

Acts on EB ⊕ EC

Properties:

Tr�B ⊕ C

�= Tr

�B�+ Tr�C�

(0.54)

det�B ⊕ C

�= det

�B�

det�C�

(0.55)

2. Direct product A = B ⊗ C:

|ψ� ∈ EB, |φ� ∈ EC, |χ� ∈ EB ⊗ EC (0.56)

A |χ� =�B ⊗ C

�(|ψ� ⊗ |φ�)����������������

|ψ�|φ� to simplify the notation(0.57)

= B |ψ� C |φ� (0.58)

A = (0.59)

b11c11 b11c12 b11c13 b12c11 b12c12 b12c13b11c21 b11c22 b11c23 b12c21 b12c22 b12c23b11c31 b11c32 b11c33 b12c31 b12c32 b12c33b21c11 b21c12 b21c13 b22c11 b22c12 b22c13b21c21 b21c22 b21c23 b22c21 b22c22 b22c23b21c31 b21c32 b21c33 b22c31 b22c32 b22c33

(0.60)

Eigenvalues and eigenvectors

Solving a quantum mechanical system means to find the eigenvalues and eigenvec-tors of the complete set of commuting observables (C.S.C.O.)

1. The eigenvalue equation

A |ψα� = α����eigenvalue

|ψα�����eigenvector

(0.61)

If n > 1 vectors satisfy the eigenvalue equation for the same eigenvalue α, wesay the eigenvalue is n-fold degenerate.

2. The eigenvalues of a self-adjoint operator A, which are observables and repre-sent physical quantities, are real numbers

α �ψα|ψα� =�ψα|Aψα

�(0.62)

=�Aψα|ψα

�∗= α∗ �ψα|ψα� (0.63)

⇒ α = α∗ ⇒ α ∈ R (0.64)

3. Eigenvectors of self-adjoint operators corresponding to distinct eigenvalues areorthogonal.Proof: if β � α is also an eigenvalue of A then

�ψα|Aψβ

�= β

�ψα|ψβ

�(0.65)

and also�ψα|Aψβ

�=�ψβ|Aψα

�∗(0.66)

= α∗�ψβ|ψα

�∗= α�ψα|ψβ

�(0.67)

which implies�ψα|ψβ

�= 0 (0.68)

4. Matrix representationOperator is uniquely defined by its action on the basis vectors of the Hilbertspace.Let B =

����ψ j��

be a basis of E (= D(A))

A���ψ j�=�

k

���ψk� �ψk��� A���ψ j�

(0.69)

=�

kAk j���ψk�

(0.70)

where Ak j =�ψk��� A���ψ j�

are the matrix elements of the operator A in the matrixrepresentation given by the basis B.For practical calculations

A =�

k j

���ψk� �ψk��� A���ψ j� �ψ j��� =�

k jAk j���ψk� �ψ j��� (0.71)

5. Spectral decomposition (of eigenrepresentation)Assume that the eigenvalues of A define a basis B =

����ψ j��

,

then Ak j =�ψk��� A���ψ j�= α jδk j.

Operator in this basis is a diagonal matrix with eigenvalues on the diagonal

A =�

k jAk j���ψk� �ψ j��� (0.72)

=�

jα j���ψ j� �ψ j��� (0.73)

=�

jα jE j (0.74)

E j is a projector onto 1-dim. space spanned by���ψ j�⇒ Spectral decomposition!

Generalization to the continuous spectrum

A |α� = α |α� (0.75)�α�|α�= δ(α − α�) (0.76)

δ-function [Cohen-Tannoudji II Appendix II]Spectral decomposition

A =

� αmax

αminα |α� �α| dα (0.77)

Completeness relation� αmax

αmin|α� �α| dα = 1 (0.78)

Wavefunction

ψ (α) = �α|ψ� (0.79)

the inner product

�ψ1|ψ2� =

� αmax

αminψ∗1 (α)ψ2 (α) dα (0.80)

Coordinate and momentum operatorsIn coordinate representation (x-representation)

X =�∞

−∞

x |x� �x| dx spectral decomposition (0.81)

and�∞

−∞

|x� �x| dx = 1 completeness relation (0.82)

|φ� =

�∞

−∞

|x� �x|φ� dx =�∞

−∞

φ(x) |x� dx (0.83)

What about P (momentum op.)?It has to satisfy the canonical commutation relation

�X, P�|φ� = XP |φ� − PX |φ� (0.84)= i� |φ� (0.85)

which in coordinate representation is

xP(x)φ(x) − P(x)xφ(x) = i�φ(x) (0.86)

This is satisfied by

P(x) = −i�∂

∂x(0.87)

In momentum representation

B = {|p�} : P =�∞

−∞

p |p� �p| dp (0.88)

and X = i�∂

∂p(0.89)

More on coordinate and momentum representationCoordinate representation

X =�∞

−∞

x |x� �x| dx X |x� = x |x� (0.90)

P(x) = −i�∂

∂x⇐�X, P�= i� (0.91)

For all p ∈ R, there is a solution to the eigenvalue equation

−i�ddxψp(x) = pψp(x) (0.92)

where ψp(x) is the eigenstate of the momentum operator (in coordinate representa-tion) corresponding to eigenvalue p

P |p� = p |p� |p� =�∞

−∞

|x� �x|p� dx =�∞

−∞

ψp(x) |x� dx (0.93)

and every solution depends linearly on function

ψp(x) =1√

2π�e

i�px = �x|p� (0.94)

which satisfies the normalization condition�∞

−∞

ψ∗p�(x)ψp(x) dx = δ�p − p�

�(0.95)

Similarly�∞

−∞

ψ∗p�x��ψp(x) dp = δ

�x − x�

�(0.96)

Momentum representation

P =

�∞

−∞

p |p� �p| dp (0.97)

The completeness relation�∞

−∞

|p� �p| dp = 1 (0.98)

|φ� =

�∞

−∞

|p� �p|φ� dp =�∞

−∞

momentum representation��������φ(p)(p) |p� dp (0.99)

How is the wavefunction φ(p)(p), which describes the ket |φ� in the momentum rep-resentation, related to φ(x) which describes the same vector in the coordinate repre-sentation?

φ(p)(p) =�∞

−∞

�p|x� �x|φ� dx =1√

2π�

�∞

−∞

e−i�pxφ(x) dx (0.100)

φ(p)(p) is the Fourier transform of φ(x)

φ(x) is the inverse F.T. of φ(p)(p)

φ(x) =1√

2π�

�∞

−∞

e+i�pxφ(p)(p) dp (0.101)

(Cohen-Tannoudji Q.M. II Appendix I)The Parseval-Plancharel formula

�∞

−∞

φ∗(x)ψ(x) dx =�∞

−∞

φ(p)∗(p)ψ(p)(p) dp (0.102)

F.T. in 3 dimensions:

φ(p) ��p�=

1(2π�)3/2

�e−

i��p·�rφ

��r�

d3r (0.103)

δ-”function”

1. Principal propertiesConsider δ�(x):

δ�(x) =� 1� for − �2 ≤ x ≤ �20 for |x| > �2

(0.104)

and evaluate�∞

−∞δ�(x) f (x) dx ( f (x) is an arbitrary function defined at x = 0)

if � is very small (� → 0)�∞

−∞

δ�(x) f (x) dx ≈ f (0)�∞

−∞

δ�(x) dx (0.105)

= f (0) (0.106)

the smaller the �, the better the approximation.

For the limit � = 0, δ(x) = lim�→0 δ�(x).

�∞

−∞

δ(x) f (x) dx = f (0) (0.107)

More generally�∞

−∞

δ�x − x0

�f (x) dx = f

�x0�

(0.108)

2. Properties(i) δ(−x) = δ(x)(ii) δ(cx) = 1

|c|δ(x)and more generally

δ�g(x)�=�

j

1����g��x j�����δ�x − x j

�(0.109)

{x j} simple zeroes of g(x) i.e. g(x j) = 0 and g�(x j) � 0(iii) xδ(x − x0) = x0δ(x − x0)and in particular xδ(x) = 0and more generally g(x)δ(x − x0) = g(x0)δ(x − x0)

(iv)�∞

−∞

δ(x − y)δ(x − z) dx = δ(y − z) (0.110)

3. The δ-”function” and the Fourier transform

ψ(p)(p) =1√

2π�

�∞

−∞

e−i�pxψ(x) dx (0.111)

ψ(x) =1√

2π�

�∞

−∞

ei�pxψ(p)(p) dp (0.112)

The Fourier transform δ(p)(p) of δ(x − x0):

δ(p)(p) =1√

2π�

�∞

−∞

e−i�pxδ�x − x0

�dx (0.113)

=1√

2π�e−

i�px0 (0.114)

The inverse F.T.

δ�x − x0

�=

1√

2π�

�∞

−∞

ei�pxδ(p)(p) dp (0.115)

=1√

2π�

�∞

−∞

ei�px 1√

2π�e−

i�px0 dp (0.116)

=1

2π�

�∞

−∞

ei�p(x−x0) dp (0.117)

=1

2π

�∞

−∞

eik(x−x0) dk (0.118)

Derivative of δ(x)�∞

−∞

δ��x − x0

�f (x) dx = (0.119)

−

�∞

−∞

δ�x − x0

�f �(x) dx = − f �

�x0�

(0.120)

THIRD POSTULATE(Measurement I)

The only possible result of the measurement of a physical quantity A is one of theeigenvalues of the corresponding observable A.

FOURTH POSTULATE(Measurement II)

1. a discrete non-degenerate spectrum:When the physical quantity A is measured on a system in the normalized state|ψ�, the probability P(an) of obtaining the non-degenerate eigenvalue an of thecorresponding physical observable A is

P (an) = |�un|ψ�|2 (0.121)

where |un� is the normalised eigenvector of A associated with the eigenvalue an.

2. a discrete spectrum:

P (an) =gn�

i=1

�����ui

n|ψ�����

2(0.122)

where gn is the degree of degeneracy of an and {���ui

n�} (i = 1, . . . , gn) is an or-

thonormal set of vectors which forms a basis in the eigenspace En associatedwith the eigenvalue an of the observable A.

3. a continuous spectrum:the probability dP(α) of obtaining result included between α and α + dα is

dP(α) = |�vα|ψ�|2 dα (0.123)

where |vα� is the eigenvector corresponding to the eigenvalue α of the observ-able A.

FIFTH POSTULATE(Measurement III)

If the measurement of the physical quantity A on the system in the state |ψ� givesthe result an, the state of the system immediately after the measurement is the mor-malized projection

Pn |ψ���ψ| Pn |ψ�

=Pn |ψ����Pn |ψ�

���(0.124)

of |ψ� onto the eigensubspace associated with an.

SIXTH POSTULATE(Time Evolution)

The time evolution of the state vector |ψ(t)� is governed by the Schrodinger equation

i�ddt|ψ(t)� = H(t) |ψ(t)� (0.125)

where H(t) is the observable associated with the total energy of the system.—————-Classically

H(�r, �p) =�p2

2m+ V��r�

(0.126)

Quantum mechanics

�r → �R�p→ �P

H =

�P2

2m+ V��R�

(0.127)

Canonical quantization (in the coordinate rep.)

�R → �r (0.128)

Pi → −i�∂

∂xi=�−i��∇

�i (0.129)

⇒ H = −�2

2m∇

2

��������kinetic energy

+ V��r�

����potential energy

(0.130)