Theoretical Physics III Quantum MechanicsAxel Groß23 May 2005

29 3 Quantum Dynamics 31 3.1 Time Evolution and the Schr¨odinger Equation.. 2.5 Measurements, Observables And The Uncertainty Relation2.5 Measurements, Observables And The Uncertainty Re

Theoretical Physics III Quantum Mechanics

Axel Groß

23 May 2005

Theoretical Physics 3 Master Quantum Mechanics

Prof Dr Axel Groß

1 Introduction – Wave Mechanics

2 Fundamental Concepts of Quantum Mechanics

• J.J Sakurai, Modern Quantum Mechanics, Benjamin/Cummings 1985

• G Baym, Lectures on Quantum Mechanics, Benjamin/Cummings 1973

• F Schwabl, Quantum Mechanics, Springer 1990

Criteria for getting the Schein:

• 50% of the points in the homework sets (at most two students can turn in thehomework sets)

• Passing the final exam

These lecture notes are based on the class “Theoretical Physics – Quantum chanics” in the sommer semester 2002 at the Technical University Munich I am verygrateful to Maximilian Lein who provided a LATEX version of the original notes whichhave been the basis for this text; furthermore, he created many of the figures Withouthis efforts this version of the lecture notes would not have been possible

1.1 Postulates of Wave Mechanics 1

1.2 One-dimensional problems 2

1.2.1 Bound states 2

1.2.2 Transmission-Reflection Problems 4

1.2.3 Tunneling Through a Potential Barrier 5

2 Fundamental Concepts of Quantum Mechanics 9 2.1 Introduction 9

2.2 Kets, Bras, and Operators 10

2.2.1 Kets 10

2.2.2 Bra space and inner product 11

2.3 Operators 12

2.3.1 Multiplication of Operators 12

2.3.2 Outer Product 12

2.3.3 Base Kets and Matrix Representations 13

2.3.4 Eigenkets as Base Kets 13

2.3.5 Resolution of the Identity, Completeness Relation, or Closure 13

2.4 Spin1/2System 14

2.5 Measurements, Observables And The Uncertainty Relation 15

2.5.1 Compatible Observables 15

2.5.2 Uncertainty Relation 18

2.5.3 Change Of Basis 20

2.5.4 Diagonalization 22

2.6 Position, Momentum, And Translation 22

2.6.1 Digression On The Dirac Delta Function 23

2.6.2 Position and momentum eigenkets 23

2.6.3 Canonical Commutation Relations 25

2.7 Momentum-Space Wave Function 27

2.7.1 Gaussian Wave Packets 28

2.7.2 Generalization To Three Dimensions 29

3 Quantum Dynamics 31 3.1 Time Evolution and the Schr¨odinger Equation 31

3.1.1 Time Evolution Operator 31

3.1.2 Derivation of the Schr¨odinger Equation 32

3.1.3 Formal Solution for U (t, t0) 32

3.1.4 Schr¨odinger versus Heisenberg Picture 36

3.1.5 Base Kets and Transition Amplitudes 38

3.1.6 Summary 39

3.2 Harmonic Oscillator 39

3.2.1 Heisenberg Picture 43

3.3 Schr¨odinger’s Wave Equation 45

3.4 Harmonic Oscillator using Wave Mechanics 47

3.4.1 Symmetry of the Wave Function 48

4 Angular Momentum 49 4.1 Rotations and Angular Momentum 49

4.2 Spin1 2 Systems and Finite Rotations 51

4.3 Eigenvalues and Eigenstates of Angular Momentum 54

4.3.1 Matrix Elements of Angular Momentum Operators 56

4.3.2 Representations of the Rotation Operator 57

4.4 Orbital Angular Momentum 57

4.5 The Central Potential 59

4.5.1 Schr¨odinger Equation for Central Potential Problems 61

4.5.2 Examples for Spherically Symmetric Potentials 62

4.6 Addition of Angular Momentum 63

4.6.1 Orbital Angular Momentum and Spin 1 2 63

4.6.2 Two Spin 1 2 Particles 64

4.6.3 General Case 65

5 Approximation Methods 67 5.1 Time-Independent Perturbation Theory: Non-Degenerate Case 67

5.1.1 Harmonic Oscillator 70

5.2 Degenerate Perturbation Theory 71

5.2.1 Linear Stark Effect 72

5.2.2 Spin-Orbit Interaction and Fine Structure 73

5.2.3 van-der-Waals Interaction 75

5.3 Variational Methods 76

5.4 Time-Dependent Perturbation Theory 77

6 Symmetry in Quantum Mechanics 83 6.1 Identical Particles 83

6.2 Two-Electron System 85

6.3 The Helium Atom 87

6.3.1 Ground State 87

6.3.2 Excited States 88

7 Scattering Theory 89 7.1 Wave Packets 89

7.2 Cross Sections 91

7.3 Partial Waves 91

7.4 Born Approximation 92

8 Relativistic Quantum Mechanics 93 8.1 Relativistic Spin Zero Particles 93

8.2 Klein’s Paradox 96

8.3 Dirac Equation 98

1 Introduction - Wave mechanics

We will start by recalling some fundamental concepts of quantum wave mechanicsbased on the correspondence principle

1.1 Postulates of Wave Mechanics

1 The state of a system is described by its wave function Ψ(x, t) The probability

Figure 1.1: Square-well potential

1.2 One-dimensional problems

For the sake of simplicity, we consider piecewise continuous potentials Assume thatthe potential has a step at a The time-independent Schr¨odinger equation can bereformulated as

e+κxis not normalizable for x > a, analogously e−κxfor x < a

Furthermore, since V (x) is an even potential, the solutions can be characterizedaccording to their symmetry

If we have even symmetry, the solution will be

Ψ(x) =

(

A cos qx |x| ≤ a

1.2 One-dimensional problemsFor odd symmetry, we get

Ψ0 has to be continuous, too From that, we get

Aq sin qa = κe−κa

⇒ tan aq = κ

This is a transcendental equation that cannot be solved analytically

Now assume odd symmetry

B sin qa = e−κa Bq cos qa = −κe−κa

The lowest energy state is always even Whenever tan qa = κ/q or − cot qa = κ/q, we

have a solution There is at least one crossing point The number of states is given by

NS = 2a√2mV0

π~



(1.19)with [α] nearest integer greater than α Even and odd states alternate

Figure 1.2: Graphical solution of the square well problem

Now let us assume that the potential walls are infinitely high, i.e V0→ ∞

Ψn = (−1)k

(cos(k +1/2)πxa n = 2k + 1sinkπ

2

non-In order to see whether our results make sense, we consider a limiting case Let

E → ∞(E >> V0), then k0 → k and R → 0 and T → 1

Let us now consider an energy less than the potential step, i.e., E < V0

Ψ00− κ2Ψ = Ψ00+ (iκ)2Ψ = 0 x > 0, κ = 1

~

p2m(V0− E)The solution here can be obtained from the solution in case 1 where E > V0

is called tunneling Its amplitude decreases exponentially.

1.2.3 Tunneling Through a Potential Barrier

If the potential barrier has a finite width, then particles can be transmitted even withenergies below the barrier height This is a typical quantum phenomenon and notpossible in classical mechanics The potential we consider is given by

The general solution is given by

~p2m(V0− E) are called wave numbers.

First of all, we write down the matching conditions at x = −a

Ae−ika+ Beika= Ceκa+ De−κa (1.41)ikAe−ika− Beika = −κCeκa− De−κa



=

eκa e−κaiκ



M (a)CD



=AB



(1.43)Here M (a) is given by



(1.46)where M−1(−a)is given by

cosh 2κa +iε

2 sinh 2κa
ei2ka iη



(1.48)with ε = κ

k−k

κ and η = κ

k +κk.The incoming wave amplitude is given by A, the reflected wave amplitude is given

by B and the transmitted flux is given by F

The transmission amplitude is given by t(E) ≡ F

A.t(E) = F

A =

FF

cosh 2κa +iε

Continuous Potential Barrier

If we have a continuous potential, then approximate V (x) by individual square barriers

of width dx, i.e replace step width 2a in Eq (1.52) by dx

2 Fundamental Concepts of Quantum

Mechanics

2.1 Introduction

Let us start with first discussing the Stern-Gerlach experiment performed in 1922.

Figure 2.1: Diagram of the Stern-Gerlach-Experiment

The magnetic moment of the silver atoms is proportional to the magnetic moment

of the 5s1electron, the inner electron shells do not have a net magnetic moment TheForce in z-direction in an inhomogeneous magnetic field is given by

or S = −~/2ez– the electron spin is quantitized.

Historically, more sophisticated experiments followed Instead of using just onemagnet, several magnets are used in series, so that sequential Stern-Gerlach experi-ments can be performed:

They show that the spin is quantized in every direction by the amount above, ±~/2

It also suggests that selecting the Sx+component after a Stern-Gerlach experiment

in x-direction completely destroys any previous information about Sz There is infact an analogon in classical mechanics – the transmission of polarized light throughpolarization filters

The following correspondence can be made

Sz± atoms ↔ x−, y − polarized light

Sx± atoms ↔ x0−, y0− polarized light (2.2)where x0 and y0-axes are x and y axes rotated by 45◦

Notation We write the Sz+ state as |Sz; +i or |Sz; ↑i; similarly, the Sz− state sponds to |Sz; ↓i We assume for Sxstates superposition of Szstates

2.2 Kets, Bras, and Operators

Consider a complex vector space of dimension d which is related to the nature of thephysical system

The space of a single electron spin is two-dimensional whereas for the description

of a free particle a vector space of denumerably infinite dimension is needed

2.2.1 Kets

The vector space is called Hilbert Space The physical state is represented by a state

vector Following Dirac, a state vector is called ket and denoted by |αi.

They suffice the usual requirements for vector spaces (commutative, associative dition, existence of null ket and inverse ket, and scalar multiplication)

ad-One important postulate is that |αi and c · |αi with c 6= 0 correspond to the samephysical state Mathematically this means that we deal with rays rather than vectors

A physical observable can be represented by an operator Operators act on kets from

the left

2.2 Kets, Bras, and Operators

In general, applying an operator to a ket cannot be expressed as a scalar multiplication,i.e.,

with c any complex number

Analogously to eigenvectors, there are eigenkets

A(|αi) = a |αi , A(|α0i) = a0|α0i , (2.10)with eigenvalues a, a0,

Example

Spin1/2System Sz|Sz, ↑i = +~

2|Sz, ↑i, Sz|Sz, ↓i = −~

2|Sz, ↑i

The bras belong to the dual vector space

2.2.2 Bra space and inner product

The Bra space is the vector space dual to the ket space It is spanned by the eigenbras{ha0|} which correspond to the eigenkets {|a0i} There is an isomorphism that assignseach ket onto its bra

hα| ↔ |αi cαhα| + cβhβ| ↔ c∗α|αi + c∗β|βi (2.11)Note the complex-conjugated coefficients

Now we introduce the inner or scalar product1

2.3 Operators

X and Y are said to be equal, X = Y , if X |αi = Y |αi Operator addition is tative and associative,

commu-X + Y = Y + commu-X ; commu-X + (Y + Z) = (commu-X + Y ) + Z (2.16)Operators are usually linear, that is,

X(a1|α1i + a2|α2i) = a1X |α1i + a2X |α2i (2.17)

An exception is for example the time-reversal operator which is antilinear.

An operator acts on a bra from the right side.

Note X |αiand hα| X are in general not dual to each other The correspondence is

2.3 Operators2.3.3 Base Kets and Matrix Representations

0 = hα00|A|α0i − hα00|A|α0i

= a0hα00|α0i − a00∗hα00|α0i = (a0− a00∗) hα00|α0iThis is true if either (for the same state) a0= a00∗or, if the two eigenvalues are not the

Usually, we will assume that eigenkets are normalized, i e hαi|αji = δij Thus, theeigenkets form an orthogonal set

2.3.4 Eigenkets as Base Kets

Normalized eigenkets of A form a complete orthonormal set, i e an arbitrary ket |βican be expressed as a linear combination of eigenkets

2.3.5 Resolution of the Identity, Completeness Relation, or Closure

(2.28) can be extremely useful

k|ck|2= 1 Each summand |αki hαk| selects the portion

of |βi parallel to |αki Thus, it is a projection operator; it is denoted by Λk = |αki hαk|

Therefore, every operator can be represented in a matrix via X = 1X1; the bra index

is the row index, the ket index is the column index

2.5 Measurements, Observables And The Uncertainty Relation

2.5 Measurements, Observables And The Uncertainty Relation

Consider a state |αi =P

kck|αki =P

k|αki hαk|αi According to the quantum theory

of measurement, after a measurement of the observable A the system is ’thrown’ into

These probabilities P (αk)can be determined with a large number of experiments

performed on an ensemble of identically prepared physical systems, a so-called pure

ensemble.

If the system already is in an eigenstate αk, then the probability to measure αk is 1

The expectation value of an operator A with respect to state α is

This corresponds to the average measured value which can be derived from

Definition Compatible Observables

Observables A and B are defined to be compatible, if the correspondingoperators commute, i e

and incompatible, if [A, B] 6= 0

If the observables A and B are compatible, then A measurements and B measurements

do not interfere, as we will see below

An important example for incompatible observables are Sxand Sy, but Szand S2≡P

kS2are compatible

Theorem Representation of Compatible Observables

Suppose that A and B are compatible observables and the eigenvalues of

A are nondegenerate, i e ai 6= aj ∀i 6= j, then the matrix elements

hαi|B|αji are all diagonal

Thus, both operators have a common set of eigenkets, their corresponding matrixrepresentations can be diagonalized simultaneously

2.5 Measurements, Observables And The Uncertainty Relation

Immediately we see that A and B are diagonalized simultaneously Suppose B acts on

form a maximal set of compatible observables which means that we cannot add any

more observables to our list without violating (2.46) Then the (collective) index

Ki= (ai, bi, ci, )uniquely specifies the eigenket

|Kii = |ai, bi, ci, i (2.47)The completenes relation implies that

What does it mean when two operators are compatible or not?

Consider a successive measurement of compatible observables

|αi−A→ |ai, bii−B→ |ai, bii−→ |aA i, bii (2.50)Thus, A and B measurements do not interfere, if A and B are compatible observables

Now, imagine an experiment with a sequential selective measurement of

incompati-ble observaincompati-bles.

|αi−A→ |aii−B→ |bji−C→ |cki (2.51)

The probability to find |cki (provided that |cki is normalized) is given by

P (ck)and P0(ck)are different (double sum vs single sum)! The important result is

that it matters whether or not B is switched on

2.5 Measurements, Observables And The Uncertainty Relation

4|h[A, B]i|2 (2.60)

If the observables do not commute, then there is some inherent “fuzziness” in themeasurements

For the proof, we need two lemmas

Theorem Schwarz Inequality

⇒ hα|αi −hα|βi hβ|αi

expecta-The proofs are trivial.

Now we are in a position to prove the Uncertainty Relation

Proof

We use the Schwarz Inequality

In conjunction with the Hermiticity of A and B, we get

Consider two incompatible observables A and B (if both are compatible, they have

a common spectrum of eigenkets!) The ket space may either be spanned by theeigenkets {|aii} and {|bii}

How are these basis related? We want to find the operator that connects the sentation of A with the representation of B

repre-Theorem

Given two sets of base kets, both satisfying orthonormality (and

complete-ness), there exists an unitary operator U such that

2 similarly, we can decompose any real function in an odd and an even part

2.5 Measurements, Observables And The Uncertainty Relation

Unitary means that the Hermitian of the operator is the inverse

The other direction follows directly from U U†= (U†U )† = 

The matrix representation of U in the old {|aii} basis is

hai|U |aji = hai|bji , (2.75)i.e it corresponds to the inner product of old base bras and new base kets

Let us consider an arbitrary ket |αi

Quantities that are invariant under transformations are of special importance in

physics The trace of the operator is such an invariant quantity.

A matrix in its eigenbasis is diagonalized Finding the eigenvalues and the eigenkets

of an operator B is equivalent to finding the unitary matrix U that diagonalizes B (Uconsists of the eigenkets of B!)

2.6 Position, Momentum, And Translation

The observables so far have been assumed to exhibit discrete eigenvalue spectra ever, in quantum mechanics observable can have a continuous spectrum (i.e a contin-uous set of eigenvalues) An example is pz, the z component of the momentum.Consider the eigenvalue equation with a continuous spectrum

2.6 Position, Momentum, And Translation

where W is an operator and w is simply a number What is the analogy to discretespectra? A vector space spanned by eigenkets with a continuous spectrum has aninfinite dimension Many of the results derived for a finite-dimensional vector spacewith discrete eigenvalues can be generalized Replace the Kronecker symbol δij bythe Dirac function δ(w) Replace the sum over the eigenvalues by an integral over acontinuous variable

hai|aji = δij ⇐⇒ hw0|w00i = δ(w0− w00) (2.87)X

i

|aii hai| = 1 ⇐⇒

Z

|w0i hw0| dw0= 1 (2.88)There are further analogies:

The Dirac δ function is in fact a distribution with the properties

Z

R

δ(x − x0)f (x0) dx0= f (x) f arbitrary function (2.94)Z

R

The dimension of δ is given by the inverse dimension of its argument, so if for example

xis a length, then δ(x) has the dimension 1/length Furthermore, by doing partialintegration, we get an expression for the derivative

The eigenkets |x0i of the position operator ˆxsatisfy

R

Now, the probability to find the particle in an intervall dx is

In this formalism, the wave function is identified with the inner product

2.6 Position, Momentum, And Translation2.6.3 Canonical Commutation Relations

Recall the Poisson brackets in classical mechanics:

=

Z Z

ψβ∗(x) hx|A|x0i ψα(x0) dx dx0 (2.115)Consider for example A = ˆx2:

hx00|[ˆx, ˆp]|x0i = hx00|i~|x0i = i~δ(x0− x00)

= hx00|ˆxˆp − ˆpˆx|x0i = x00hx00|ˆp|x0i − hx00|ˆp|x0i x0 (2.118)

This shows that hx00|[ˆx, ˆp]|x0i vanishes for x00 6= x0 Note that ˆpstill acts on both, |x0iand x0 A possible solution is

−x00 ∂

∂x00δ(x00− x0) + ∂

∂x00δ(x00− x0)x00= δ(x0− x00) (2.121)Let f (x00)be an arbitrary function of x00

 ∂

∂x00f (x00)

δ(x00− x0)x00

is also a possible solution of 2.118 But one can show that this only causes a differentoverall phase for the wave function which still leads to the same state

In fact, from (2.119) immediately follows that

hx|ˆp|αi =

Zhx|ˆp|x0i hx0|αi dx0

= ~i

Z ∂

∂xδ(x − x

0)ψα(x0)dx0

= ~i

∂

∂x

Zδ(x − x0)ψα(x0)dx0

= ~i

∂

2.7 Momentum-Space Wave FunctionFurther matrix elements:

n

∂n

∂x0nψα(x0) dx0 (2.128)This means that in this derivation we do not postulate the correspondence p → ~

i∇,

we simply require that the fundamental canonical commutation relations hold Westill obtain the same representation of the momentum operator

2.7 Momentum-Space Wave Function

Instead of the position representation we can work equally well in the momentumrepresentation Haw are the wave functions in both representations related? To make

a long story short, the answer is that both are the Fourier transform of each other.Let’s start by considering the basis of position space The δ functions are the eigen-functions of the position operator in position space There must also be an eigenfunc-tion for the position operator

hx0|p0i induces a transformation function from the x representation to the p sentation

hx0|p0i = Ce~ip0x0 , (2.130)where C is a normalization constant

δ(x0− x00) = hx0|x00i =

Z

hx0|p0i hp0|x00i dp0

= |C|2Z

e~ip0(x0−x 00 )dp0= |C|2· 2π~δ(x0− x00) (2.131)

⇒ |C|2= 1

Ccan be a complex number; but we can choose any phase that we want, so we set C

to be a real positive number

C = √12π~

⇒ hx0|p0i = √1

2π~e

i

This is just a plane wave.

ψα(x0) = hx0|αi =

Z

hx0|p0i hp0|αi dp0

= √12π~

Z

e~ip0x0Φα(p0) dp0 (2.134)Now we will decompose the state α in p space

Φα(p0) = hp0|αi =

Z

hp0|x0i hx0|αi dx0

=√12π~

Z

e−~ip 0 x 0

Ψα(x0) dx0 (2.135)Thus, Ψαand Φαare related by a Fourier transformation!

A very important kind of wave functions are the Gaussian wave packets.

2.7.1 Gaussian Wave Packets

A Gaussian wave packet is a wave function enveloped by a Gaussian

R

|hx0|αi|2x02dx0 =√1

πdZ

The Gaussian wave packet is a minimum uncertainty wave packet.

Let’s take a look at momentum space

hp0|αi = √1

2π~

1

π1 / 4√dZ

R

e−~ip0x0+ikx0− x02

2d2 dx0

=√12π~

1

π1√dZ

2.7 Momentum-Space Wave Function

The Fourier transformation of the Gaussian function in real space is a Gaussian inmomentum space! It is centered at ~k – it is the mean momentum of the wave packet.The width of Gaussian wave packets in real space and momentum space is inverselyproportional

2.7.2 Generalization To Three Dimensions

The position and momentum operator can be generalized to three dimensions as lows

3 Quantum Dynamics

3.1 Time Evolution and the Schr¨ odinger Equation

3.1.1 Time Evolution Operator

We first specify the notation Assume that a physical system is in state α at a point intime t0 Then the time evolution of the state is denoted by

|α, t0i = |αi−−−−−−−−→ |α, ttime evolution 0; ti (3.1)The change of the system can be described by the time evolution operator U (t, t0)

|α, t0; t0+ dti = U (t0+ dt, t0) |α, t0i (3.6)This operator must reduce to the unity operator as dt goes to zero; furthermore, itshould be of first order in dt These requirements are fulfilled by

U (t0+ dt, t0) : = 1 − iΩdt , (3.7)where Ω is a Hermitian operator

Let’s take a look at the composition property.

3.1.2 Derivation of the Schr¨ odinger Equation

Use the composition property (3.5)

by |α, t0i, we obtain

i~∂t∂ U (t, t0) |α, t0i = HU (t, t0) |α, t0i (3.14)Since |α, t0i does not depend on t, this can be rewritten (using (3.2)) as

i~∂t∂ |α, t0; ti = H |α, t0; ti (3.15)

This is the Schr¨ odinger Equation for a state ket Any time evolution of a state ket is

given by the time evolution operator U (t, t0) In order to determine U (t, t0), we have

to first consider the formal solutions of the Schr¨odinger equation for the time evolutionoperator (3.13)

3.1.3 Formal Solution for U (t, t0)

There are three cases

3.1 Time Evolution and the Schr¨odinger EquationCase 1 H is independent of time Then U is

Alternative proof: We regard U as a compound of infinitesimal time-evolution ators

Case 3 We have a time-dependent evolution and the Hamiltonians do not commute

at different points in time Then the general solution is given by

Expand the time-evolution operator.

Then the time evolution is simply given by

|α, t0= 0; ti = e−iEj t/~|aji (3.29)Thus, the system remains in the state |aji at all times, just the phase is modulated If

an observable is compatible with the Hamiltonian H, then it is a constant of motion.Time Dependence of Expectation Values

It is easy to show that the expectation value of any observable with respect to an

eigenket does not change under time evolution:

hBi = hα, t0= 0; t|B|α, t0= 0; ti

=Daj|eiEjt/~Be−iEj t/~|ajE

Therefore eigenkets of H are also called stationary kets.

Consider now the decomposition of a ket into stationary kets

c∗icjhai|B|aji e−i(E j −E i )t/~ (3.31)

This is a sum over oscillatory terms with frequency

ωij = Ej− Ei