The Quantum Mechanics Solver 2 docx

Definition of the State of a System; Pure Case The state of a physical system is completely defined at any time t by a vector of the Hilbert space, normalized to 1, noted|ψt.. Owing to the

XIV Contents

27 Bloch Oscillations 277

27.1 Unitary Transformation on a Quantum System 277

27.2 Band Structure in a Periodic Potential 277

27.3 The Phenomenon of Bloch Oscillations 278

27.4 Solutions 281

27.5 Comments 285

Author Index 287

Subject Index 289

In the following pages we remind the basic definitions, notations and results

of quantum mechanics

1 Principles

Hilbert Space

The first step in treating a quantum physical problem consists in identifying the appropriate Hilbert space to describe the system A Hilbert space is a complex vector space, with a Hermitian scalar product The vectors of the space are called kets and are noted |ψ
The scalar product of the ket |ψ1

and the ket|ψ2
is noted ψ2|ψ1
It is linear in |ψ1
and antilinear in |ψ2

and one has:

ψ1|ψ2
= (ψ2|ψ1
) ∗

Definition of the State of a System; Pure Case

The state of a physical system is completely defined at any time t by a vector

of the Hilbert space, normalized to 1, noted|ψ(t)
Owing to the superposition

principle, if|ψ1
and |ψ2
are two possible states of a given physical system,

any linear combination

|ψ
∝ c1|ψ1
+ c2|ψ2
,

where c1and c2are complex numbers, is a possible state of the system These coefficients must be chosen such thatψ|ψ
= 1.

2 Summary of Quantum Mechanics

Measurement

To a given physical quantity A one associates a self-adjoint (or Hermitian)

operator ˆA acting in the Hilbert space In a measurement of the quantity A,

the only possible results are the eigenvalues a αof ˆA.

Consider a system in a state|ψ
The probability P(a α) to find the result

a α in a measurement of A is

P(a α) =


ˆP

α |ψ



2

,

where ˆP αis the projector on the eigensubspaceE αassociated to the eigenvalue

a α

After a measurement of ˆA which has given the result a α, the state of the system is proportional to ˆP α |ψ
(wave packet projection or reduction).

A single measurement gives information on the state of the system after the measurement has been performed The information acquired on the state before the measurement is very “poor”, i.e if the measurement gave the result

a α, one can only infer that the state|ψ
was not in the subspace orthogonal

to E α

In order to acquire accurate information on the state before measurement,

one must use N independent systems, all of which are prepared in the same

state |ψ
(with N  1) If we perform N1 measurements of ˆA1 (eigenval-ues {a 1,α }), N2 measurements of ˆA2 (eigenvalues {a 2,α }), and so on (with

p

i=1 N i = N ), we can determine the probability distribution of the a i,α, and therefore the  ˆ P i,α |ψ
2 If the p operators ˆ A i are well chosen, this determines unambiguously the initial state |ψ
.

Evolution

When the system is not being measured, the evolution of its state vector is given by the Schr¨odinger equation

i¯h d

dt |ψ
= ˆ H(t) |ψ(t)
,

where the hermitian operator ˆH(t) is the Hamiltonian, or energy observable,

of the system at time t.

If we consider an isolated system, whose Hamiltonian is time-independent, the energy eigenstates of the Hamiltonian |φ n
are the solution of the time

independent Schr¨odinger equation:

ˆ

H |φ n
= E n |φ n

They form an orthogonal basis of the Hilbert space This basis is particu-larly useful If we decompose the initial state |ψ(0)
on this basis, we can

immediately write its expression at any time as:

|ψ(0)
=

n

α n |φ n
→ |ψ(t)
=

n

α ne−iE n t/¯ h |φ n

The coefficients are α n =φ n |ψ(0)
, i.e.

|ψ(t)
=

n

e−iE n t/¯ h |φ n
φ n |ψ(0)

Complete Set of Commuting Observables (CSCO)

A set of operators{ ˆ A, ˆ B, , ˆ X } is a CSCO if all of these operators commute

and if their common eigenbasis{|α, β, , ξ
} is unique (up to a phase factor).

In that case, after the measurement of the physical quantities{A, B, , X},

the state of the system is known unambiguously If the measurements have

given the values α for A, β for B, , ξ for ˆ X, the state of the system is

|α, β, , ξ
.

Entangled States

Consider a quantum system S formed by two subsystems S1 and S2 The Hilbert space in which we describe S is the tensor product of the Hilbert

spacesE1 andE2respectively associated withS1 andS2 If we note{|α m
} a

basis ofS1 and{|β n
} a basis of S2, a possible basis of the global system is

{|α m
⊗ |β n
}.

Any state vector of the global system can be written as:

|Ψ
=

m,n

C m,n |α m
⊗ |β n

If this vector can be written as|Ψ
= |α
⊗ |β
, where |α
and |β
are vectors

ofE1 andE2 respectively, one calls it a factorized state

In general an arbitrary state |Ψ
is not factorized: there are quantum

correlations between the two subsystems, and|Ψ
is called an Entangled state.

Statistical Mixture and the Density Operator

If we have an incomplete information on the state of the system, for instance because the measurements are incomplete, one does not know exactly its state vector The state can be described by a density operator ˆρ whose properties

are the following:

• The density operator is hermitian and its trace is equal to 1.

• All the eigenvalues Π n of the density operator are non-negative The den-sity operator can therefore be written as

ˆ

ρ =

n

Π n |φ n
φ n | ,

4 Summary of Quantum Mechanics

where the|φ n
are the eigenstates of ˆρ and the Π ncan be interpreted as a probability distribution In the case of a pure state, all eigenvalues vanish except one which is equal to 1

• The probability to find the result a α in a measurement of the physical

quantity A is given by

P(a α) = Tr

 ˆ

P α ρˆ



n

Π n φ n | ˆ A |φ n

The state of the system after the measurement is ˆρ  ∝ ˆ P α ρ ˆˆP α

• As long as the system is not measured, the evolution of the density operator

is given by

i¯h d

dt ρ(t) = [ ˆˆ H(t) , ˆ ρ(t)]

2 General Results

Uncertainty Relations

Consider 2N physical systems which are identical and independent, and are

all prepared in the same state |ψ
(we assume N  1) For N of them, we

measure a physical quantity A, and for the N others , we measure a physical quantity B The rms deviations ∆a and ∆b of the two series of measurements

satisfy the inequality

∆a ∆b ≥ 1

2



ψ|[ ˆ A, ˆ B] |ψ


Ehrenfest Theorem

Consider a system which evolves under the action of a Hamiltonian ˆH(t), and

an observable ˆA(t) The expectation value of this observable evolves according

to the equation:

d

dt a
= i¯1h ψ|[ ˆ A, ˆ H] |ψ
+ ψ| ∂ ˆ ∂t A |ψ

In particular, if ˆA is time-independent and if it commutes with ˆ H, the

expec-tation valuea
is a constant of the motion.

3 The Particular Case of a Point-Like Particle; Wave Mechanics

The Wave Function

For a point-like particle for which we can neglect possible internal degrees of freedom, the Hilbert space is the space of square integrable functions (written

in mathematics as L2(R3))

The state vector|ψ
is represented by a wave function ψ(r) The quantity

|ψ(r)|2

is the probability density to find the particle at point r in dimensional space Its Fourier transform ϕ(p):

ϕ(p) = (2π¯ h)13/2



e−ip·r/¯h ψ(r) d3r

is the probability amplitude to find that the particle has a momentum p.

Operators

Among the operators associated to usual physical quantities, one finds:

• The position operator ˆr ≡ (ˆx, ˆy, ˆz), which consists in multiplying the wave

function ψ(r) by r.

• The momentum operator ˆp whose action on the wave function ψ(r) is the

operation−i¯h∇.

• The Hamiltonian, or energy operator, for a particle placed in a potential

V (r):

ˆ

H = pˆ

2

2M + V (ˆ r) → Hψ(r) = −ˆ ¯h

2

2M ∇2

ψ(r) + V (r)ψ(r) ,

where M is the mass of the particle.

Continuity of the Wave Function

If the potential V is continuous, the eigenfunctions of the Hamiltonian ψ α(r)

are continuous and so are their derivatives This remains true if V (r) is a step

function: ψ and ψ  are continuous where V (r) has discontinuities.

In the case of infinitely high potential steps, (for instance V (x) = + ∞

for x < 0 and V (x) = 0 for x ≥ 0), ψ(x) is continuous and vanishes at the

discontinuity of V (ψ(0) = 0), while its first derivative ψ  (x) is discontinuous.

In one dimension, it is interesting to consider potentials which are Dirac

distributions, V (x) = g δ(x) The wave function is continuous and the

discon-tinuity of its derivative is obtained by integrating the Schr¨odinger equation

around the center of the delta function [ψ (0

+)− ψ (0

− ) = (2M g/¯ h2) ψ(0) in

our example]

Position-Momentum Uncertainty Relations

Using the above general result, one finds:

[ˆx, ˆ p x] = i¯h → ∆x ∆p x ≥ ¯h/2 ,

and similar relations for the y and z components.

6 Summary of Quantum Mechanics

4 Angular Momentum and Spin

Angular Momentum Observable

An angular momentum observable ˆJ is a set of three operators { ˆ J x , ˆ J y , ˆ J z }

which satisfy the commutation relations

[ ˆJ x , ˆ J y ] = i¯ h ˆ J z , [ ˆJ y , ˆ J z ] = i¯ h ˆ J x , [ ˆJ z , ˆ J x ] = i¯ h ˆ J y

The orbital angular momentum with respect to the origin ˆL = ˆ r × ˆ p is an

angular momentum observable

The observable ˆJ2 = ˆJ2

x+ ˆJ2

y + ˆJ2

z commutes with all the components ˆ

J i One can therefore find a common eigenbasis of ˆJ2 and one of the three components ˆJ i Traditionally, one chooses i = z.

Eigenvalues of the Angular Momentum

The eigenvalues of ˆJ2are of the form ¯h2j(j + 1) with j integer or half integer.

In an eigensubspace of ˆJ2corresponding to a given value of j, the eigenvalues

of ˆJ z are of the form

¯

hm , with m ∈ {−j, −j + 1, , j − 1, j} (2j + 1 values)

The corresponding eigenstates are noted|α, j, m
, where α represents the other

quantum numbers which are necessary in order to define the states completely The states|α, j, m
are related to |α, j, m±1
by the operators ˆ J ±= ˆJ x ±i ˆ J y:

ˆ

J ± |α, j, m
=j(j + 1) − m(m ± 1) |α, j, m ± 1

Orbital Angular Momentum of a Particle

In the case of an orbital angular momentum, only integer values of j and m are allowed Traditionally, one notes j =  in this case The common eigenstates

ψ(r) of ˆ L2 and ˆL z can be written in spherical coordinates as R(r) Y ,m (θ, ϕ), where the radial wave function R(r) is arbitrary and where the functions Y ,m

are the spherical harmonics, i.e the harmonic functions on the sphere of radius one The first are:

Y 0,0 (θ, ϕ) = √1

4π , Y 1,0 (θ, ϕ) =

3

4π cos θ ,

Y 1,1 (θ, ϕ) = −

3

8π sin θ e

iϕ , Y 1, −1 (θ, ϕ) =

3

8π sin θ e

−iϕ .

In addition to its angular momentum, a particle can have an intrinsic angular

momentum called its Spin The spin, which is noted traditionally j = s, can

take half-integer as well as integer values

The electron, the proton, the neutron are spin s = 1/2 particles, for which

the projection of the intrinsic angular momentum can take either of the two

values m¯ h: m = ±1/2 In the basis |s = 1/2 , m = ±1/2
, the operators ˆ S x, ˆ

S y, ˆS z have the matrix representations:

ˆ

S x= ¯h

2

0 1

1 0

, Sˆy= ¯h

2

0−i

i 0

, Sˆz= ¯h

2

1 0

0−1

.

Addition of Angular Momenta

Consider a systemS made of two subsystems S1andS2, of angular momenta ˆ

J1and ˆJ2 The observable ˆJ = ˆ J1+ ˆJ2is an angular momentum observable

In the subspace corresponding to given values j1 and j2 (of dimension (2j1+ 1)× (2j2+ 1)), the possible values for the quantum number j corresponding

to the total angular momentum of the system ˆJ are:

j = |j1− j2| , |j1− j2| + 1 , · · · , j1+ j2 ,

with, for each value of j, the 2j + 1 values of m: m = −j, −j + 1, · · · , j.

For instance, adding two spins 1/2, one can obtain an angular momentum 0 (singlet state j = m = 0) and three states of angular momentum 1 (triplet states j = 1, m = 0, ±1).

The relation between the factorized basis |j1, m1
⊗ |j2, m2
and the

to-tal angular momentum basis |j1, j2 ; j, m
is given by the Clebsch-Gordan

coefficients:

|j1, j2; j, m
= 

m1m2

C j j,m

1,m1;j2,m2|j1, m1
⊗ |j2, m2

5 Exactly Soluble Problems

The Harmonic Oscillator

For simplicity, we consider the one-dimensional problem The harmonic

po-tential is written V (x) = mω2x2/2 The natural length and momentum scales

are

x0=

¯

h

mω , p0=

√

¯

hmω

By introducing the reduced operators ˆX = ˆ x/x0 and ˆP = ˆ p/p0, the Hamil-tonian is:

8 Summary of Quantum Mechanics

ˆ

H = ¯hω

2

 ˆ

P2+ ˆX2



, with [ ˆX, ˆ P ] = i

We define the creation and annihilation operators ˆa † and ˆa by:

ˆ

a = √1

2

 ˆ

X + i ˆ P



, ˆa †=√1

2

 ˆ

X − i ˆ P



, [ˆa, ˆ a † ] = 1

One has

ˆ

H = ¯ hω

ˆ

a †ˆa + 1/2 .

The eigenvalues of ˆH are (n + 1/2)¯ hω, with n non-negative integer These

eigenvalues are non-degenerate The corresponding eigenvectors are noted|n
.

We have:

ˆ

a † |n
= √ n + 1 |n + 1

and

ˆ

a |n
= √ n |n − 1
if n > 0 ,

= 0 if n = 0

The corresponding wave functions are the Hermite functions The ground state|n = 0
is given by:

ψ0(x) = 1

π 1/4 √ x

0

exp(−x2/ 2x2

0)

Higher dimension harmonic oscillator problems are deduced directly from these results

The Coulomb Potential (bound states)

We consider the motion of an electron in the electrostatic field of the proton

We note µ the reduced mass (µ = m e m p /(m e + m p)  m e) and we set

e2= q2/(4π0) Since the Coulomb potential is rotation invariant, we can find

a basis of states common to the Hamiltonian ˆH, to ˆ L2and to ˆL z The bound

states are characterized by the 3 quantum numbers n, , m with:

ψ n,,m(r) = Rn, (r) Y ,m (θ, ϕ) , where the Y ,m are the spherical harmonics The energy levels are of the form

E n=− E I

n2 with E I = µe

4

2¯h2  13.6 eV

The principal quantum number n is a positive integer and  can take all integer values from 0 to n − 1 The total degeneracy (in m and ) of a given energy

level is n2(we do not take spin into account) The radial wave functions R n,

are of the form:

R n, (r) = r  P n, (r) exp( −r/(na1)) , with a1= ¯h

2

µe2  0.53 ˚ A

P n, (r) is a polynomial of degree n −  − 1 called a Laguerre polynomial The

length a1 is the Bohr radius The ground state wave function is ψ 1,0,0 (r) =

e −r/a1/

πa3

6 Approximation Methods

Time-Independent Perturbations

We consider a time-independent Hamiltonian ˆH which can be written as ˆ H =

ˆ

H0+ λ ˆ H1 We suppose that the eigenstates of ˆH0 are known:

ˆ

H0|n, r
= E n |n, r
, r = 1, 2, , p n where p n is the degeneracy of E n We also suppose that the term λ ˆ H1 is sufficiently small so that it only results in small perturbations of the spectrum

of ˆH0

Non-degenerate Case In this case, p n= 1 and the eigenvalue of ˆH which

coincides with E n as λ → 0 is given by:

˜

E n = E n + λ n| ˆ H1|n
+ λ2

k=n

|k| ˆ H1|n
|2

E n − E k + O(λ3)

The corresponding eigenstate is:

|ψ n
= |n
+ λ

k=n

k| ˆ H1|n

E n − E k |k
+ O(λ2)

Degenerate Case In order to obtain the eigenvalues of ˆH at first order in

λ, and the corresponding eigenstates, one must diagonalize the restriction of

λ ˆ H1to the subspace of ˆH0associated with the eigenvalue E n , i.e find the p n

solutions of the “secular” equation:









n, 1|λ ˆ H1|n, 1
− ∆E n, 1|λ ˆ H1|n, p n

n, r|λ ˆ H1|n, r
− ∆E

n, p n |λ ˆ H1|n, 1
n, p n |λ ˆ H1|n, p n
− ∆E









= 0

The energies to first order in λ are ˜ E n,r = E n +∆E r , r = 1, , p n In general, the perturbation is lifted (at least partially) by the perturbation

The eigenvalues of ˆH are (n + 1 /2) ¯ hω, with n non-negative integer These

eigenvalues are non-degenerate The corresponding eigenvectors...

exp(−x2< /small>/ 2x2< /small>

0)

Higher dimension harmonic oscillator problems are deduced directly from these results

The Coulomb... ϕ) , where the Y ,m are the spherical harmonics The energy levels are of the form

E n=− E I

n2< /small>