PHYS*7010 — Quantum mechanics IProblems for Chapter 2

1. (Credit: Cohen-Tannoudji.) In a one-dimensional problem, a particle is in a state represented by the wavefunction

ψ(x) = Neip0x/~

√x2 + a2

,

where N , p0, and a are real constants.

(a) Determine N to ensure that the wave function is normalized.

(b) What is the probability that when the particle’s position is measured, it lies in the interval between −a/√

3

and +a/√

3?

(c) What is the expectation value of the momentum operator in this state?

2. (Credit: Cohen-Tannoudji.) In a one-dimensional problem, the wave function of a free particle at t = 0 is givenby

ψ(0, x) = N

∫ +∞

−∞

e−|k|/k0eikx dk,

where N and k0 are real constants.

(a) Calculate P(0, k1), the probability that at t = 0, the particle’s momentum lies in the interval between −~k1

and +~k1. Provide a plot of P as a function of k1/k0.

(b) What is the state at a time t 6= 0, and what is the probability P(t, k1)? Give an interpretation to yourresult.

(c) Evaluate ψ(0, x) explicitly and provide a plot of |ψ(0, x)|2. Describe qualitatively the time-evolution of thewave packet.

(d) Calculate 〈x〉, 〈p〉, ∆x, and ∆p at t = 0. What is ∆x∆p for this initial state?

3. (Credit: Cohen-Tannoudji.) In a one-dimensional problem, a particle moves in a potential V (X) = −fX , wheref is a positive constant and X is the position operator.

(a) State Ehrenfest’s theorem for this case, and integrate the differential equations for 〈x〉 and 〈p〉. How do thesolutions compare to the classical motion?

(b) Show that ∆p does not vary in time.

(c) Write down the Schrodinger equation in the |p〉 representation.

(d) From your result in part (c), deduce a relation between

∂

∂t|〈p|ψ(t)〉|2 and

∂

∂p|〈p|ψ(t)〉|2.

Integrate this equation and give a physical interpretation to the result.

4. (Credit: Cohen-Tannoudji.) Let j(t,x) be the probability current associated with a wave function ψ(t,x).

(a) Show that 〈P 〉 = m∫

j d3x, where 〈P 〉 is the expectation value of the momentum operator.

(b) The orbital angular momentum operator L is defined by L = X ×P . (Is this operator self-adjoint?) Showthat 〈L〉 = m

∫

(x × j) d3x.

5. (Credit: Cohen-Tannoudji.) We consider a particle of mass m placed in an infinite potential well of length a.The energy eigenstates for this particle are |n〉, where n = 1, 2, 3, · · ·, and the energy eigenvalues are

En =n2π2

~2

2ma2= n2E1.

The state of the particle at t = 0 is

|ψ(0)〉 =1√4

(

|1〉 + |2〉 + |3〉 + |4〉)

.

2

(a) The energy of the particle is measured. What is the probability that its value will be found to be smallerthan 6E1?

(b) Calculate 〈H〉 and ∆H in the initial state. Express your results in terms of E1.

(c) Calculate |ψ(t)〉 for t 6= 0. How do your answers in parts (a) and (b) change when t 6= 0?

(d) The energy of the particle is measured, and the value 16E1 is found. What is the state of the system afterthe measurement? What value will be found if the energy is measured again?

6. (Credit: Bransden and Joachain.) We consider a particle of mass m placed in an infinite potential well of lengtha. The energy eigenfunctions are

φn(x) =

√

2

asin

nπx

a,

where 0 ≤ x ≤ a and n = 1, 2, 3, · · ·. The particle’s state at t = 0 is represented by

ψ(x) = Nx(a− x),

where N is a normalization constant.

(a) Calculate the probability P (En) that a measurement of the energy will return the eigenvalue En.

(b) Calculate 〈H〉, the expectation value of the Hamiltonian operator.

7. (Credit: Cohen-Tannoudji.) In a two-dimensional problem, a particle of mass m is placed in an infinite potentialwell of length a in both directions (x and y). The particle’s Hamiltonian is H = Hx +Hy, where

Hx =P 2

x

2m+ V (X), Hx =

P 2y

2m+ V (Y ).

The potential function V (x) is zero when 0 ≤ x ≤ a and is infinite elsewhere; the same is true of V (y).

(a) What are the energy eigenvalues and eigenstates for this system? (Use the |x〉|y〉 representation, and expressthe eigenvalues in terms of E1 = π2

~2/2ma2.)

A particle is in a state represented by the wave function

ψ(x, y) = N cosπx

acos

πy

asin

2πx

asin

2πy

a,

where N is a constant, and x and y are both limited to the interval between 0 and a.

(b) Calculate 〈H〉 for this state. Express your result in terms of E1.

(c) If the total energy H is measured, what values may be returned and with what probabilities?

(d) If Hx is measured, what values may be returned and with what probabilities?

8. (Credit: Cohen-Tannoudji.) We consider a physical system with a three-dimensional state space. We select anorthonormal basis {|1〉, |2〉, |3〉} in this space, and we find that the Hamiltonian H and the two observables Aand B are represented by the matrices

H = H0

1 0 00 2 00 0 2

, A = A0

1 0 00 0 10 1 0

, B = B0

0 1 01 0 00 0 1

,

where H0, A0, and B0 are real constants. The state of the system at t = 0 is

|ψ(0)〉 =1√2|1〉 +

1

2|2〉 +

1

2|3〉.

(a) The energy of the system is measured at t = 0. What values may be returned and with what probabilities?

(b) Calculate 〈H〉 and ∆H at t = 0.

(c) Calculate the state |ψ(t)〉 for t 6= 0.

3

(d) Calculate the expectation values 〈A〉(t) and 〈B〉(t).(e) The observable A is measured at time t. What values may be returned and with what probabilities?

(f) The observable B is measured at time t. What values may be returned and with what probabilities?

9. (Credit: Cohen-Tannoudji.) A density operator ρ(t) evolves in time under the influence of a Hamiltonian H(t).Show that Tr(ρ2) cannot vary as a function of time. Then answer the following question: Is it possible for aclosed quantum system to evolve from a pure state to a statistical mixture of states? Explain your answer.

10. (Credit: Sakurai.) Let X(t) be the Heisenberg-picture position operator for a free particle moving in onedimension. Calculate the commutator [X(t), X(0)].

11. (Credit: Sakurai.) Let |1〉 and |2〉 be eigenstates of a self-adjoint operator A, with eigenvalues a1 and a2,respectively. We assume that a1 6= a2, and the states are orthonormal. In this basis the Hamiltonian operatoris given by

H = H0(|1〉〈2| + |2〉〈1|),

where H0 is a real constant.

(a) What are the eigenvalues of the Hamiltonian, and the corresponding eigenstates?

(b) Suppose the state of the system at t = 0 is |ψ(0)〉 = |1〉. What is the state of the system at t 6= 0?

(c) If the system is in the state of part (b), what is the probability of finding it in a state |2〉 at time t 6= 0?