Exercises for Quantum Mechanics

(TFFY54)

Johan Henriksson and Patrick Norman

Department of Physics, Chemistry and Biology,

Linkoping University, SE-581 83 Linkoping, Sweden

Spring Term 2007

1

For a Hermitian operator Ω, we know that∫

ψ∗Ωψ dr =

∫

(

Ωψ)∗ψ dr, ∀ψ.

Show that∫

ψ∗1Ωψ2 dr =

∫

(

Ωψ1

)∗ψ2 dr, ∀ψ1, ψ2.

Hint: Consider a linear combination Ψ = c1ψ1 + c2ψ2, where c1, c2 ∈ C.

2

Consider a one-dimensional harmonic oscillator with mass m and characteristicfrequency ω. At time t = 0, the state is given by

Ψ(x) =1√2(ψ0(x) + ψ1(x)),

where ψn(x) are eigenstates to the Hamiltonian with energies En = ~ω(n+1/2).Determine the time-dependent state vector Ψ(x, t) for t > 0.

3

Let ψ(x) be a solution to the time-independent Schrodinger equation with apotential V (x) that is symmetric with respect to the origin, i.e., V (−x) = V (x).

a) Show that ψ(−x) also is a solution with the same energy eigenvalue.

b) If the energy levels are nondegenerate (i.e., there is at most one eigen-function associated with a given energy), show that ψ(−x) = ψ(x) orψ(−x) = −ψ(x), i.e., the eigenfunctions are either symmetric or antisym-metric with respect to the origin.

4

A particle (mass m) is incident from the left towards the potential step

V (x) =

0 x ≤ 0

V0 x > 0

The energy of the particle is E = 2V0, V0 > 0.

a) Solve the time-independent Schrodinger equation.

Note: Since the particle is unbounded it is not possible to normalize thewave function.

b) Calculate the probability current density j.

c) Define and calculate the transmission T using the result in b).

d) Define and calculate the reflection R using the result in b).

e) Calculate R and T and check that R+ T = 1.

1

5

Show that the expectation value of the momentum operator 〈p〉 is real for thewave packet

ψ(x) =1√2π~

∞∫

−∞

eixp/~Φ(p) dp.

6

Determine the wave function in x-space corresponding to

a)

ϕ(k) =

(2ξ)−1/2 |k| ≤ ξ0 otherwise

b)

ϕ(k) =1√2πσ

exp

(

− k2

2σ2

)

7

Consider the time-dependent Schrodinger equation

i~∂

∂tψ(r, t) = Hψ(r, t).

If the potential is time-independent, i.e., V (r) 6= V (r, t), show that it is possibleto find solutions separable in space and time, i.e., ψ(r, t) = ψ(r)f(t). Find theexplicit form of f(t) and show that ψ(r) is a solution of an eigenvalue problem.

8

A particle of mass m in a one-dimensional box

V (x) =

0 0 ≤ x ≤ a∞ otherwise

is in a mixed state composed of the ground state and the first excited state. Thenormalized wave function can be written as

Ψ(x) = c1ψ1(x) + c2ψ2(x),

where c1 and c2 are constants and ψ1(x) and ψ2(x) are eigenfunctions cor-responding to the ground state and the first excited state, respectively. The

average value of the energy is π2~2

ma2 . What can be said about c1 and c2?

2

9

If 〈ψ|Ω|ψ〉 is real for all ψ, show that

〈ψ1|Ω|ψ2〉 = 〈ψ2|Ω|ψ1〉∗

for all ψ1 and ψ2. N.b., solve the problem without assuming that Ω is Hermitian.Hint: Consider the linear combinations Ψ = ψ1 + ψ2 and Ψ = ψ1 + iψ2, respec-tively.

10

Let ψn be a complete set of orthonormal functions which are solutions to thetime-independent Schrodinger equation Hψn = Enψn. At t = 0 the system isdescribed by the wave function

Ψ(x) =1√2eiαψ1(x) +

1√3eiβψ2(x) +

1√6eiγψ3(x).

a) Write down Ψ(x, t).

b) At time t a measurement of the energy of the system is performed. Whatis the probability to obtain the result E2?

c) Calculate 〈H〉

d) Is the mean value of the energy equal to any of the possible outcomes ofa measurement?

11

A particle of mass m is moving in the one-dimensional potential

V (x) =

0 0 ≤ x ≤ a∞ otherwise

.

At a certain time the particle is in a state given by the wave function

Ψ(x) = Nx(a− x)

where N is a normalization constant.

a) Calculate the probability that a measurement of the energy yields theground state energy.

b) Calculate the probability that a measurement of the energy yields a result

between 0 and 3~2π2

ma2 .

3

12

Consider a particle (mass m) in a one-dimensional box (0 ≤ x ≤ a). At timet = 0, the particle is described by the wave function

Ψ(x) = N

[

√

2

asin(π

ax)

+

√

2

asin

(

4π

ax

)

]

.

a) Determine N and Ψ(x, t).

b) Calculate 〈x〉t = 〈Ψ(x, t)|x|Ψ(x, t)〉.

Hint: sin(ϕ) · sin(4ϕ) = [cos(3ϕ)− cos(5ϕ)]/2

13

Verify the following relations for matrix exponentials.

a) exp(A)† = exp(A†)

b) B exp(A)B−1 = exp(BAB−1)

c) exp(A + B) = exp(A) exp(B) if [A,B] = 0

d) exp(−A) exp(A) = 1

e) ddλ exp(λA) = A exp(λA) = exp(λA)A, A 6= A(λ)

f) exp(−A)B exp(A) = B + [B,A] + 12 [[B,A],A] + 1

3! [[[B,A],A],A] + . . .

Hint: Consider the Taylor expansion of exp(−λA)B exp(λA) around λ = 0.

14

Define the trace of an operator as

Tr(Ω) =∑

i

〈i|Ω|i〉 =∑

i

Ωii

and the density operator, commonly used in many applications, as ρ = |ψ〉〈ψ|.

a) Show that Tr(ΩΛ) = Tr(ΛΩ).

b) If the basis |i〉 is transformed by a unitary transformation, i.e., |i′〉 = U |i〉,show that the trace of the operator is unchanged in the new basis.

c) Show that Tr(ρ) = 1.

d) Show that it is possible to use ρ to express the expectation value of anoperator as 〈Ω〉 = Tr(ρΩ).

Comment: This means that expectation values of observables are not affectedby the choice of representation (basis) we make for our wave functions since thetrace is invariant under unitary transformations.

4

15

In a three-dimensional vector space, assume that we have found the commutingoperators Ω and Λ corresponding to some physical observables. We choose abasis |n〉, n = 1, 2, 3, for which none of the operators are diagonal but givenby the matrix representations

Ω =

2 0 i0 1 0−i 0 2

and Λ =1

2

3 −i√

2 i

i√

2 2√

2

−i√

2 3

.

a) Solve the eigenvalue problem Ω|ω〉 = ω|ω〉 to find which values of theobservable Ω we can measure.

b) Since one eigenvalue is degenerated, the eigenstates are not uniquely de-fined through the eigenvalues ω. To resolve this problem, we can use thecommuting operator Λ. Show that Λ is block diagonal in the basis |ω〉.

c) Diagonalize the 2 × 2 block in Λ to find a basis in which both Ω and Λ

are diagonal.

d) The pairs of eigenvalues |ω, λ〉 uniquely defines the eigenstates. Which arethe three pairs of eigenstates?

Comment: This exercise is closely related to real problems such as the hydrogenatom where one of the observables usually is the Hamiltonian and you encounterdegenerate energy levels.

16

Consider a Hermitian operator Ω.

a) Show that exp(iΩ) is unitary.

b) Given the result in a), show that a wave function normalized at t = t0 willremain normalized at any t > t0.

c) Show that nondegenerate eigenstates of Ω are orthogonal.

d) Show that eigenvalues and expectation values of Ω are real.

17

In a three-dimensional vector space the operator Ω can be represented as

Ω =

2 0 i0 1 0−i 0 2

.

Find the matrix representation of the operator√

Ω, i.e., the operator whichwhen squared yields the operator Ω.

5

18

Let U(a) be a unitary operator defined as

U(a) = e−iap/~,

where a is a real number of dimension length. Furthermore, define the transfor-mation of an arbitrary operator Ω as

Ω = U †(a)ΩU(a).

a) What does this transformation correspond to in your laboratory?

Note: The wave function will be left unchanged in this case.

b) Determine the transformed coordinate and momentum operators x and p.

c) If you got the correct answers in b), it is trivial to determine the expecta-tion values 〈x〉 and 〈p〉. These averages should reflect your answer in a).Determine these expectation values.

d) If we, instead of transforming the operators, transform our state vectorsaccording to

|ψ〉 = U(a)|ψ〉,what does |ψ〉 correspond to in your laboratory? Note that the observ-ables, of course, will be unaltered, i.e., 〈ψ|Ω|ψ〉 = 〈ψ|Ω|ψ〉.

19

A harmonic oscillator of mass m is in a state described by the wave function

Ψ(x, t) =1√2eiβψ0(x)e

− i~

E0t +1√2e−iβψ1(x)e

− i~

E1t,

where β is a real constant, ψ0 and ψ1 are the ground and the first excitedstates, respectively, and E0 and E1 are the corresponding energies. Determinethe expectation values of H and x.

20

Consider a harmonic oscillator of massm with eigenstates |ψn〉 and energy levelsEn = ~ω(n+ 1

2 ). The Hamiltonian for this system is H = p2/2m+mω2x2/2.For this system, show that

∂En

∂ω=

⟨

ψn

∣

∣

∣

∣

∣

∂H

∂ω

∣

∣

∣

∣

∣

ψn

⟩

.

Comment: This is a direct result of the much more general Hellman–Feynmantheorem in quantum mechanics.

6

21

A particle of mass m is located in the potential V (x) = mω2x2/2 . The particleis not in a stationary state and is at time t = 0 described by the wave function

|Ψ(0)〉 =∞∑

n=0

cn|n〉.

We can assume that the wave function is real at t = 0, i.e., all cn are realnumbers. Show that the time-dependent average value of the position 〈x〉t is

〈x〉t = 〈x〉t=0 cosωt.

Comment: The average of the position is oscillating with a frequency ω (muchlike a classical particle). The frequency ω is related to the energy in the usualway, i.e., En = ~ω(n+ 1/2).

22

Consider a one-dimensional harmonic oscillator (mass m) in state ψn.

a) Show that the uncertainty product in this state is given by

(∆x)n(∆p)n =

(

n+1

2

)

~, n = 0, 1, 2, . . .

b) What is so special with the case n = 0?

c) Show that

〈T 〉n = 〈V 〉n =1

2En =

1

2〈H〉n.

23

Consider a harmonic oscillator in the energy basis |n〉, with H |n〉 = En|n〉and En = ~ω(n + 1/2). In order to make a transition from energy basis to arepresentation in ordinary space, ψn(x), we can exploit the properties of thecreation and annihilation operators, a†|0〉 = |1〉 and a|1〉 = |0〉, where

a† =1√2

(

ξ − d

dξ

)

and a =1√2

(

ξ +d

dξ

)

.

Determine ψ0(ξ) and ψ1(ξ).

24

A particle in a system with V (x) = V (−x) is described by the wave function

Ψ(x) = σ1/2π−1/4e−σ2x2

2 .

7

If Ψ(x) is expanded in eigenfunctions to the Hamiltonian,

Ψ(x) =∞∑

n=0

cnψn(x),

half of the coefficients can be determined using a simple symmetry argument.How?

25

A particle is moving in a central potential, V (r), corresponding to a potentialfunction approaching zero as the distance r to the center approaches infinity.The particle is in a stationary state where the time-independent part of thewave function is given by

Ψ(x, y, z) = Nxye−αr,

where N is a normalization constant and α is a given positive constant.

a) Calculate the possible results when measuring L2 and Lz and state thecorresponding probabilities.

b) Determine the potential V (r).

26

A hydrogen atom is in the 2p state with ml = 0 and ms = 1/2. The system isthus represented by the wave function

ψ2,1,0, 12

(r, θ, φ, t) = R2,1(r)Y1,0(θ, φ)

(

1

0

)

e−iE2t/~.

At time t = 0, a measurement of the orbital angular momentum along the x-axisis performed.

a) Motivate why the magnitude of the orbital angular momentum remainsunchanged by the measurement, i.e., the state at time t > 0 will still bean eigenstate of L2 with eigenvalue l = 1.

b) Determine the possible values and corresponding probabilities in the mea-surement performed at time t = 0.

27

a) For a centrosymmetric system with V = V (r), show that

[Lz, V ] = 0 and [Lz, p2] = 0.

b) Show furthermore that if H = p2

2m + V , it follows that

[Lz, H ] = 0 and [L2, H] = 0.

c) What consequences do these results have?

8

28

For a system with l = 1, determine the matrix representation of L2x.

29

Study a state |ψ〉 given by 〈r, θ, φ|ψ〉 = ψ(r, θ, φ) in coordinate basis. In this

basis, determine the transformed state e−iφ0Lz/~|ψ〉.Hint: The effect of Lz in the coordinate basis is −i~∂/∂φ.

30

Consider ψ(r, θ, φ) = f(r)Yl,m(θ, φ), where Yl,m(θ, φ) is a spherical harmonic.We want to make a simultaneous measurement of Lx and Ly, but an uncertaintyin the measurement is unavoidable.

a) Why?

b) For a given (fixed) value of the quantum number l, find the value of mwhich leads to the largest possible accuracy if Lx and Ly are measuredsimultaneously.

Hint: Minimize D(m) = (∆Lx)2

+ (∆Ly)2

c) What is the smallest possible accuracy if Lx, and Ly is measured simul-taneously?

d) What happens in the special case of l = 0?

31

Consider a particle of mass m moving in the potential

V (x) =

0 0 ≤ x ≤ a∞ otherwise

.

In nonrelativistic quantum mechanics (i.e., v ≪ c) the energy levels, as we allknow, are determined by

En =n2π2~2

2ma2, n = 1, 2, 3, . . .

and the corresponding normalized wave functions are given by

ψn(x) =

√

2a sin

(

nπxa

)

0 ≤ x ≤ a0 otherwise.

We want to make an approximation of the relativistic correction to the energyEn using perturbation theory. For the kinetic energy we have (according to thetheory of relativity)

Ek =√

(pc)2 + (mc2)2 −mc2 = mc2

[√

1 +( p

mc

)2

− 1

]

9

If this is expanded for small linear momenta this yields

Ek =p2

2m− p4

8m3c2+ . . . ,

which gives us a correction − p4

8m3c2 to the nonrelativistic expression p2

2m for thekinetic energy.

a) What is, quantum mechanically, the perturbative part of the Hamiltonian?

b) Use this Hamiltonian to make a first -order perturbation theory calculation

of the relativistic correction to the unperturbed energy level E(0)n and

compare quantitatively the correction term with E(0)n .

32

In a hydrogen atom, the electrostatic interaction between the electron and theproton in the nucleus results in the potential energy given by (in spherical co-ordinates)

V (r, θ, φ) = V (r) = − e2

4πǫ0

1

r.

The time-independent Schrodinger equation is separable, Ψ(r, θ, φ) = ψn(r)Yl,m(θ, φ),where Yl,m(θ, φ) are spherical harmonics. In spherical coordinates, the Hamil-tonian is given by

H = − ~2

2µ

1

r

d2

dr2r − e2

(4πǫ0)r+L2

2I,

where µ is the reduced mass of the electron and the proton, I is the moment ofinertia, and L is the angular momentum operator.

a) Use the variational principle to estimate the ground state energy for theelectron in a hydrogen atom.

Hint: Due to the fact that the potential energy does not depend on theangles θ and φ, a suitable trial function for the ground state is φ = φ(r) =N exp(−αr), where α ≥ 0 is the variational parameter.

b) How good/bad is the estimate? Why so?

33

Consider a particle with mass m moving in the two-dimensional potential

V (x, y) =

0 0 ≤ x ≤ a and 0 ≤ y ≤ a∞ otherwise

.

Using first-order perturbation theory, show that the degeneracy of the first ex-cited state is lifted by the perturbation

H ′ = ǫx2,

where ǫ is small. Visualize the energy levels schematically in a graph. Whichare the proper zeroth-order eigenfunctions?

10

34

A hydrogen atom in state |1, 1, 0,+ 12 〉 is subjected to a time-dependent mag-

netic field, B(r, t) = B0ex sin(ωt). Neglecting terms quadratic in the magneticfield, the interaction between the electron and the external magnetic field, B, isdescribed by the Hamiltonian

H ′ =µB

~(L + 2S) ·B,

where µB is the Bohr magneton. Determine the possible final states. Commenton these final states in comparison with those of an perturbing electric field?

35

Using the variational principle, estimate the ground state energy for a one-dimensional anharmonic oscillator with potential V (x) = λx4. Make the as-sumption

φ(x) =

√

β√πe−β2x2/2, 〈φ|φ〉 = 1.

36

A one-dimensional harmonic oscillator (mass m, force −mω2xex) is perturbedby a small force −mǫxex where 0 < ǫ ≪ ω. Using perturbation theory, deter-mine the corrections to the unperturbed energy levels to second order. Further-more calculate the exact energy levels of the perturbed system and compare theresults.

37

A particle with spin 12 is in the state |α〉 =

(

10

)

. If the spin is measured alongthe en-direction forming an angle θ with the z-axis, what are the probabilitiesof getting the results +~

2 and −~

2 , respectively?

38

Consider a particle of spin s = 1/2 in a time-dependent magnetic field B =B cos(ωt)ez. The Hamiltonian becomes H = µBBσz cos(ωt). Assume that thespin state at time t = 0 is |χ(0)〉 = |βx〉.

a) Calculate |χ(t)〉.

b) Calculate 〈χ(t)|Sx|χ(t)〉.

11

39

In spin space it is possible to describe an infinitesimal rotation δα around they-axis by the unitary operator

U(δαey) = I − i

~δαSy.

a) Show that U(αey) = exp(−iασy/2).

b) Let U(αey) operate on an arbitrary state χ and show that this is equivalentto a rotation of the spin vector by an angle α/2 in spin space.

c) What is the expected result of a rotation by the angle 2π? Make a 2πrotation, and compare to the expected result. How many turns must aspin vector be rotated in order to get back to its original state?

40

In magnetic resonance, the magnetic moment, m, of a particle interacts withthe applied magnetic field. This interaction is described by the Hamiltonian

H = −m ·B,

where the components of the magnetic dipole moment operator are mi = −µBσi

and µB is the Bohr magneton. Assume that a constant magnetic field B = Bez

is applied. At time t = 0, the spin is measured to be +~/2 in the x-direction,i.e., |χ(0)〉 = |αx〉.

a) Calculate |χ(t)〉 by using the propagator

|χ(t)〉 = e−iHt/~|χ(0)〉.

b) Calculate |χ(t)〉 by solving the time-dependent Schrodinger equation usingthe assumption

|χ(t)〉 =

(

a(t)b(t)

)

.

c) Let t > 0. Calculate the probability to obtain the result −~/2 in a mea-surement of the spin along the x-direction.

d) Let t > 0. Calculate the probability to obtain the result +~/2 in a mea-surement of the spin along the z-direction.

41

Determine the eigenvalues of the operator en · S, where en = (sin θ, 0, cos θ) and

S is the spin operator for a particle with spin 12 . The eigenvalues correspond to

the possible measured values when measuring spin along the direction en.

12

42

Consider the spin operators

Sx =~

2

(

0 11 0

)

, Sy =~

2

(

0 −ii 0

)

, Sz =~

2

(

1 00 −1

)

.

a) Show that S2x = S2

y = S2z = ~

2

4 I.

b) Determine the eigenvalues and eigenvectors to Sx and Sy.

c) Show the anticommutation relation [Sx, Sy]+ = SxSy + SySx = 0.

43

The spin and angular dependent part of the wave function for an electron isgiven by:

Φ(θ, φ) =

√

2l

2l+ 1Yl,l−1(θ, φ)|α〉 +

√

1

2l+ 1Yl,l(θ, φ)|β〉

where Yl,m(θ, φ) are normalized spherical harmonics (eigenfunctions to L2 and

Lz) and

|α〉 =

(

1

0

)

and |β〉 =

(

0

1

)

are eigenstates to S2 and Sz.Show that Φ(θ, φ) is an eigenfunction to the z-component of the angular

momentum operator, Jz = Lz +Sz, and calculate the corresponding expectationvalue.

44

At time t = 0, the spin state of an electron is given by

|χ(0)〉 =1

2

( √2

1 + i

)

.

a) Determine the direction

en =

sin θ cosφsin θ sinφ

cos θ

along which to apply a magnetic field, i.e., B = B0en, such that, at timet = t0, the spin state is given by

|χ(t0)〉 =

(

10

)

.

b) How long time will you need to apply the magnetic field?

Hint: The Hamiltonian of the system is H = µBB · σ, where µB is theBohr magneton.

13

45

From the point of view of the electron, the orbital motion of the proton inhydrogen creates an magnetic field of considerable strength—the numerical valueof the magnetic field experienced by the electron is in the order of 0.4 Tesla.Due to this strong internal magnetic field, the energy of the atom will dependon the orientation of the electron spin.

The interaction operator that needs to be added to the nonrelativistic Hamil-tonian in order to describe spin-orbit interaction is

HSO =~α

2m2c

1

r3L · S,

where m is the electron mass, c is the speed of light, and α is the dimensionlessfine-structure constant (α ≈ 1/137).

Consider hydrogen in its 2p-state. Without consideration made to the spin-orbit interaction this state is sixfold energy degenerate.

a) Give a complete set of commuting observables for hydrogen when spin-orbit interaction is accounted for.

b) Describe qualitatively how the energy degeneration of the six states willchange in this case.

c) Determine numerical values of any energy splittings you have proclaimedabove. Also, give the energy separation between the 1s and 2p levels inorder to make sure that the corrections are small in absolute terms.

46

At time t = t0 a hydrogen atom, described by the wave function

ψ(x, y, z, t0) = Nx exp

(

−√

x2 + y2 + z2

2a0

)

(

1−i

)

,

where N is a normalization constant and a0 is the Bohr radius, is placed ina constant external magnetic field B = B0ez. The Hamiltonian describingthe interaction with the external field is H = µBB0σz. Calculate the time-dependent expectation values of the spin projection along the x-, y-, and z-axes.

47

At time t0, the wave function of a hydrogen atom is given by

ψ(x, y, z, t0) = Nx exp

(

−√

x2 + y2 + z2

2a0

)

(

1−i

)

,

where N is a normalization constant and a0 is the Bohr radius.

a) If we measure the orbital angular momentum projection along the z-axis,which values will we measure and what are their respective probabilities?

b) If we measure the projection of the spin along the x-, y, and z-axes, whichvalues will we measure and what are their respective probabilities?

14

48

a) Using the variational method, show that the virial theorem

2〈T 〉+ 〈V 〉 = 0,

where 〈T 〉 is the kinetic energy and 〈V 〉 is the potential energy, is satisfiedfor a hydrogen atom.

Hint: Assume that the true normalized wave function is Ψ(x, y, z) and useas a normalized variational wave function

ψη(x, y, z) = η3/2ψ(ηx, ηy, ηz)

where η is a scaling parameter. Here η = 1 corresponds to the exactsolution.

b) For a hydrogen-like system in its ground state, calculate

1

〈r〉 ,⟨

1

r

⟩

, 〈p2x〉, 〈p2

y〉, 〈p2z〉, and 〈p2〉.

49

Consider a spinless particle with mass m and charge −e in the potential

V (x, y) =

12mω

2x2, y ∈ [0, a]

∞, otherwise

where

ω =3π2~

2ma2.

The system is in the first excited state when a weak electric field is appliedaccording to E = Een, en = cos θ ex + sin θ ey. To first order, determine theeffect of the perturbing electric field on the energy of the system. Comment onthe θ-dependence of the perturbation.Hint:

∫ a

0

x sin2(nπx

a

)

dx =a2

4

50

A hydrogen atom is in the 2p state with ml = 0 and ms = 1/2, and the systemis thus represented by the wave function

ψ2,1,0,1/2(r, θ, φ) = R2,1(r)Y1,0(θ, φ)

(

10

)

.

Determine the expectation value of the total angular momentum J2.

15

51

A complete set of commuting observables for the hydrogen atom is given bythe set of operators H , L2, Lz, S

2, and Sz and the set common eigenkets canbe denoted as |n, l,ml, 1/2,ms〉. The 2p-level of the hydrogen atom is sixfolddegenerate due to the different values ml and ms.

Another complete set of commuting observables is given by the set of op-erators H , J2, Jz, L

2, and S2 and the set common eigenkets can in this casebe denoted as |n, j,mj , l, 1/2〉. In terms of the above “old” eigenkets, deriveexplicit expressions for the six wave functions that correspond to the 2p-level inthis “new” basis .

52

The spin-orbit interaction operator is given by

HSO = A L · S,

where A is a scalar that is independent of (θ, φ). By re-writing the spin-orbitoperator, motivate why this problems involving this operator should be solvedin the basis |j,mj〉 that are eigenvectors of J2 and Jz.

53

Assume that the electron in hydrogen is in a d-orbital, i.e., l = 2. If one considersthe interaction between the spin and orbital motion, this level will no longer be10-fold energy degenerate. The energy operator that describes the interactionis

HSO = A L · S,where A is a scalar that is independent of (θ, φ).

a) The possible values of j are l ± 1/2, what are the interaction energies inthese two cases?

b) Express |j,mj〉 = |5/2, 5/2〉 in terms of the“old”basis vectors |l,ml, s,ms〉.Show with an explicit calculation that your linear combination of old basisvectors is an eigenstate of J2 and Jz .

54

Let us consider a hydrogen atom described by the wave function

ψ(r) =1√3R2,1(r)

[ √2Y1,0(θ, φ)Y1,1(θ, φ)

]

.

a) Is this state an eigenstate to total angular momemtum operators J2 andJz, and, if so, what are the values of the corresponding quantum numbersj and mj?

16

b) With use of first-order perturbation theory, determine the shift in energyrelative to the 2p-level in hydrogen as due to spin-orbit coupling.

Hint: The spin-orbit coupling operator is HSO = ~α2m2c

1r3 L · S, where m is the

electron mass, c is the speed of light, and α is the dimensionless fine-structureconstant (α ≈ 1/137). The following integral may be needed 〈r−3〉R2,1 =

a−30 /24, where a0 is the Bohr radius.

55

At t = 0 the wave function for the electron in hydrogen is given by

ψ(x, y, z, 0) = Nyze−r/3a0

(

12i

)

,

where N is a normalization constant. Determine the possible values and prob-abilities in measurements of Jz , Lz and Sz.

17

Answers

2.

Ψ(x, t) = 1√2(ψ0(x)e

−iωt/2 + ψ1(x)e−i3ωt/2)

4.

a) Ψ(x, t) =

Aiei(2

√mV0x/~−ωt) +Are

−i(2√

mV0x/~+ωt), x ≤ 0(

Ar =√

2−1√2+1

Ai

)

Atei(√

2mV0x/~−ωt), x > 0(

At = 2√

2√2+1

Ai

)

b) jx =

2√

V0

m (|Ai|2 − |Ar|2) = 8√

23+2

√2

√

V0

m |Ai|2, x ≤ 0√

2V0

m |At|2 = 8√

23+2

√2

√

V0

m |Ai|2, x > 0

c) T = 4√

23+2

√2

d) R = 3−2√

23+2

√2

6.

a) ψ(x, 0) = (πξ)−1/2 sin(ξx)/x

b) ψ(x, 0) = (2π)−1/2 exp(−σ2x2/2)

7.

a) N = 2−1/2

Ψ(x, t) = a−1/2(

sin(πx/a)e−iπ2~t/2ma2

+ sin(4πx/a)e−i8π2~t/ma2

)

8.

|c1|2 = 23

|c2|2 = 13

10.

a) Ψ(x, t) = 1√2ei(α−E1t/~)ψ1 + 1√

3ei(β−E2t/~)ψ2 + 1√

6ei(γ−E3t/~)ψ3

b) P2 = 1/3

c) 〈H〉 = E1

2 + E2

3 + E3

6

d) No

11.

P(E1) = |c1|2 = 960π6

P(E2) = |c2|2 = 0

P(0 ≤ E ≤ 3~2π2

ma2 ) = |c1|2 + |c2|2 = 960π6

18

12.

a) N = 2−1/2

Ψ(x, t) = a−1/2(

sin(

πxa

)

exp(

−i π2~t

2ma2

)

+ sin(

4πxa

)

exp(

−i 8π2~t

ma2

))

b) 〈x〉t = a2 − 2a

π2

[

19 − 1

25

]

cos(

E4−E1

~t)

15.

a) ω1 = 1←→ |ω1〉 = 2−1/2(1, 0, i)T

ω2 = 1←→ |ω2〉 = (0, 1, 0)T

ω3 = 3←→ |ω3〉 = 2−1/2(1, 0,−i)T

b) U†ΩU =

1 −i 0i 1 00 0 2

c) |ω1, λ1〉 = 2−1/2(|ω1〉 − i|ω2〉)|ω2, λ2〉 = 2−1/2(|ω1〉+ i|ω2〉)|ω3, λ3〉 = |ω3〉

d) |ω, λ〉 = |1, 0〉, |1, 2〉, |3, 2〉

17.

√Ω = 1

2

1 +√

3 0 (√

3− 1)i0 2 0

(1−√

3)i 0 1 +√

3

18.

a) The system is fixed, but you are performing measurement w.r.t. to a trans-lated coordinate system.

b) x = x+ a and p = p

c) 〈x〉 = 〈x〉+ a and 〈p〉 = 〈p〉

d) The system is moved within the laboratory.

19.

〈E〉 = 12 (E0 + E1) = ~ω where ω = E0+E1

2~.

〈x〉 =[

~

2mω

]1

2 cos(ωt+ 2β)

23.

ψ0(ξ) = 1π1/4

e−ξ2/2

ψ1(ξ) = 21/2

π1/4ξe−ξ2/2

24.

cn = 0 if n is an odd number.

19

25.

a) The only possible measurement result for L2 is 6~2 (with probability 1).Possible results when measuring Lz are ±2~, each with probability 1

2 .

b) The potential is V (r) = − 3~2α

mr .

26.

a) L2 and Lz are commuting observables.

b) The measured values are ±~, both with probability 1/2.

28.

L2x = ~

2

2

1 0 10 2 01 0 1

29.

e−iφ0Lz/~|ψ〉 = ψ(r, θ, φ− φ0)

30.

a) [Lx, Ly] 6= 0

b) D(m) = l(l+ 1)~2 −m2~

2

c) ~2l(l+ 1)

d) D(m) = 0, i.e., it is possible to determine both Lx and Ly simultaneously.

31.

b) Epertn = E0

n

(

1− E0

n

2mc2

)

32.

a) E0 ≤ − µe4

32π2~2ε2

0

33.

The shifts in energy are[

13 − 1

2π2

]

ǫa2 and[

13 − 1

8π2

]

ǫa2, respectively.

34.

∆l = 0,

∆ml = ±1, ∆ml = 0

∆ml = 0, ∆ml = −1

35.

Emin = 34/3

4

(

~2

2m

)2/3

λ1/3

36.

Epertn = Eexact

n = ~ω(

n+ 12

)

− ǫ2

2mω2

20

37.

Pα = cos2 θ2

Pβ = sin2 θ2

38.

a) |χ(t)〉 =[

exp(

−iµBB~ω sin(ωt)

)

|αz〉 − exp(

iµBB~ω sin(ωt)

)

|βz〉]

/√

2

b) 〈χ(t)|Sx|χ(t)〉 = −~

2 cos(

2µBB~ω sin(ωt)

)

40.

a) |χ(t)〉 = [exp(−iµBBt/~)|αz〉+ exp(+iµBBt/~)|βz〉] /√

2

c) Pαx = cos2(

µBBt~

)

d) Pαz = 12

41.

λ = ±~2

2

42.

b) Sx: +~

2 ↔ 1√2

(

11

)

and −~

2 ↔ 1√2

(

1−1

)

.

Sy: +~

2 ↔ 1√2

(

1i

)

and −~

2 ↔ 1√2

(

1−i

)

.

43.

〈Jz〉 = (l − 12 )~

44.

a) en = 2−1/2(1,−1, 0)T

b) t0 = π~

4µBB

45.

a) H, J2, Jz, L2, and S2

(c) ∆E ≈ 4.5 · 10−5 eV

46.

〈Sx〉t = ~

2 sin(

µBB~

(t− t0))

〈Sy〉t = −~

2 cos(

µBB~

(t− t0))

〈Sx〉t = 0

21

47.

a) Measuring Lz yields ±~, both with probability 1/2.

b) Measuring Sx yields ±~/2, both with probability 1/2.Measuring Sy yields −~/2.Measuring Sz yields ±~/2, both with probability 1/2.

48.

b) 1〈r〉 = 2Z

3a0

⟨

1r

⟩

= Za0

〈p2x〉 = 〈p2

y〉 = 〈p2z〉 = 1

3

(

~Za0

)2

49. First excited state is double degenerate with E1 = E2 = 11π2~2/(4ma2).The first-order energy correction is the same for both states and equals ∆E1 =∆E2 = 1

2aeE sin θ.

50. 〈J2〉 = 114 ~

2

51. The |j,mj , l, 1/2〉 kets take the form

Y l,s=1/2j=l±1/2,mj

(θ, φ) =

±√

l±mj+1/22l+1 Yl,mj−1/2(θ, φ)

√

l∓mj+1/22l+1 Yl,mj+1/2(θ, φ)

.

53. a)A~2

2×

l; j = l + 1/2−l − 1; j = l − 1/2

54.

a) j = 3/2,mj = 1/2

b) 〈HSO〉 = ~3α

96m2ca3

0

55. The measurements of angular momenta give:

Operator Value Probability

Sz ~/2 1/5−~/2 4/5

Lz ~ 1/2−~ 1/2

Jz 3~/2 1/10~/2 4/10−~/2 1/10−3~/2 4/10

22

Summary

Commutator relations

[A, BC] = [A, B]C + B[A, C]

eABe−A = B + [A, B] +1

2!

[

A, [A, B]]

+1

3!

[

A,[

A, [A, B]]]

+ . . .

Time evolution

|ψ(t)〉 = e−iHt/~|ψ(0)〉 d

dt〈Ω〉 = 1

i~

⟨

[Ω, Λ]⟩

+

⟨

∂Ω

∂t

⟩

Uncertainty relations12 |〈[Ω, Λ]〉| ≤ ∆Ω ∆Λ

Momentum basis

〈x|p〉 =1√2π~

exp(

ipx

~

)

Translations and rotations

UT (aen) = exp

(

− ia~

en · p)

UR(αen) = exp

(

− iα~

en · J)

Harmonic oscillator

x =

√

~

2mω

(

a† + a)

p = i

√

m~ω

2

(

a† − a)

a|n〉 =√n|n− 1〉 a†|n〉 =

√n+ 1|n+ 1〉

Angular momentum

J× J = i~J

J± = Jx ± iJy Jx = 12 (J+ + J−) Jy = 1

2i(J+ − J−)

J±|j,m〉 = ~√

j(j + 1)−m(m± 1)|j,m±1〉 = ~√

(j ∓m)(j ±m+ 1)|j,m±1〉Spin angular momentum

|αn〉 =

(

cos θ2e

−i ϕ2

sin θ2e

+i ϕ2

)

|βn〉 =

(− sin θ2e

−i ϕ2

cos θ2e

+i ϕ2

)

Variational principle

E0 ≤〈Ψ|H |Ψ〉〈Ψ|Ψ〉

Perturbation theory

Time-independent, nondegenerate:

E(1)n =

⟨

ψn

∣

∣

∣H ′∣

∣

∣ψn

⟩ ∣

∣

∣φ(1)n

⟩

= −∑

m 6=n

〈ψm| H ′ |ψn〉E

(0)m − E(0)

n

|ψm〉 E(2)n =

⟨

ψn

∣

∣

∣H ′∣

∣

∣φ(1)n

⟩

Time-independent, degenerate:

det

H′n − E

(1)j I

= 0

Time-dependent:

d(1)n (t) =

1

i~

t∫

0

dt′eiωnit′

H ′ni(t

′) Pfi(t) =1

~2

∣

∣

∣

∣

∣

∣

t∫

0

dt′eiωfit′

H ′fi(t

′)

∣

∣

∣

∣

∣

∣

2

H ′(t) = A sin(ωt) =⇒ Pfi(t, ω) =|Afi|24~2

∣

∣

∣

∣

1− ei(ωfi+ω)t

ωfi + ω− 1− ei(ωfi−ω)t

ωfi − ω

∣

∣

∣

∣

2

24