# Exercises for Quantum Mechanics (TFFY54) 2010-04-06آ Exercises for Quantum Mechanics (TFFY54)...

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Exercises for Quantum Mechanics

(TFFY54)

Johan Henriksson and Patrick Norman

Department of Physics, Chemistry and Biology,

Linköping University, SE-581 83 Linköping, Sweden

Spring Term 2007

1

For a Hermitian operator Ω̂, we know that ∫

ψ∗Ω̂ψ dr =

∫

(

Ω̂ψ )∗ ψ dr, ∀ψ.

Show that ∫

ψ∗1Ω̂ψ2 dr =

∫

(

Ω̂ψ1

)∗ 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 characteristic frequency 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 Schrödinger equation with a potential 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) or ψ(−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 Schrödinger equation.

Note: Since the particle is unbounded it is not possible to normalize the wave 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 〈p̂〉 is real for the wave 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| ≤ ξ 0 otherwise

b)

ϕ(k) = 1√ 2πσ

exp

(

− k 2

2σ2

)

7

Consider the time-dependent Schrödinger equation

i~ ∂

∂t ψ(r, t) = Ĥψ(r, t).

If the potential is time-independent, i.e., V (r) 6= V (r, t), show that it is possible to find solutions separable in space and time, i.e., ψ(r, t) = ψ(r)f(t). Find the explicit 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. The normalized 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 π 2 ~ 2

ma2 . What can be said about c1 and c2?

2

9

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

〈ψ1|Ω̂|ψ2〉 = 〈ψ2|Ω̂|ψ1〉∗

for all ψ1 and ψ2. N.b., solve the problem without assuming that Ω̂ 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 the time-independent Schrödinger equation Ĥψn = Enψn. At t = 0 the system is described by the wave function

Ψ(x) = 1√ 2 eiαψ1(x) +

1√ 3 eiβψ2(x) +

1√ 6 eiγψ3(x).

a) Write down Ψ(x, t).

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

c) Calculate 〈Ĥ〉

d) Is the mean value of the energy equal to any of the possible outcomes of a 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 the ground 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 time t = 0, the particle is described by the wave function

Ψ(x) = N

[

√

2

a sin (π

a x )

+

√

2

a sin

(

4π

a x

)

]

.

a) Determine N and Ψ(x, t).

b) Calculate 〈x〉t = 〈Ψ(x, t)|x̂|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] + 13! [[[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(Ω̂) = ∑

i

〈i|Ω̂|i〉 = ∑

i

Ωii

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

a) Show that Tr(Ω̂Λ̂) = Tr(Λ̂Ω̂).

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

c) Show that Tr(ρ̂) = 1.

d) Show that it is possible to use ρ̂ to express the expectation value of an operator as 〈Ω̂〉 = Tr(ρ̂Ω̂).

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

4

15

In a three-dimensional vector space, assume that we have found the commuting operators Ω̂ and Λ̂ corresponding to some physical observables. We choose a basis |n〉, n = {1, 2, 3}, for which none of the operators are diagonal but given by the matrix representations

Ω =

2 0 i 0 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 Ω̂|ω〉 = ω|ω〉 to find which values of the observable 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 the commuting operator Λ̂. 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 are the three pairs of eigenstates?

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

16

Consider a Hermitian operator Ω̂.

a) Show that exp(iΩ̂) is unitary.

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

c) Show that nondegenerate eigenstates of Ω̂ are orthogonal.

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

17

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

Ω =

2 0 i 0 1 0 −i 0 2

.

Find the matrix representation of the operator √

Ω̂, i.e., the operator which when squared yields the operator Ω̂.

5

18

Let Û(a) be a unitary operator defined as

Û(a) = e−iap̂/~,

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

Ω̃ = Û †(a)Ω̂Û(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 x̃ and p̃.

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

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

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

19

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

Ψ(x, t

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