Quantum Mechanics Assignment 5 Solution

Example Sheet 5
Problem : Fourier Transforms and Expectation Values (15 points)
Function f(x) and its Fourier transform f˜(k) are related by,

By definition, the expectation is defined as :
Z Z
⟨x⟩ = dxP(x)x = dx|ψ(x)|2x
Using the definition of expectation values of (powers of) the momentum operator,
Z
⟨pˆn⟩ = dxψ∗(x)pˆnψ(x)
the form of the momentum operator,

and the definition of the Fourier transform,
1. (5 points) : Show that

2. (5 points) : and that
,
3. (5 points) : and, in general, that
.
The relation P(k) = |ψ˜(k)|2, as discussed in lecture, thus follows from the Born relation,
P(x) = |ψ(x)|2.
Problem : Superposition in the Infinite Well (20 points)
Verify the results for the eigenvalues of the energy operator for the infinite potential well of width L,
(
0 0 ≤ x ≤ L
V (x) = ∞ else
for which the energy eigenvalues are,

Now suppose that at t = 0 we place a particle in an infinite well in the state

Note : Each step below requires relatively little computation. You will not need the functional form of the energy eigenfunctions φn(x) to complete them, only the energy eigenvalues.
1. (2 points) : How does ψA evolve with time? Write down the expression for ψA(x,t).
2. (3 points) : Calculate the expectation value of the energy, < E >ˆ , for the particle described by ψA(x,t). Write your answer in terms of E0. Does this quantity change with time?
3. (2 points) : What is the probability of measuring the energy to equal < E >ˆ as a result of a single measurement at t = 0? At a later time, t = t1 ?
5. (2 points) : The energy of the particle is found to be E2 as a result of a single measurement at t = t1. Write down the wave function ψA(x,t), which describes the state of the particle for t > t1. What energy values will be observed and with what probabilities at a time t2 > t1 ?
6. (2 points) : Construct another normalized wave function ψB(x,0) which is linearly independent of ψA(x,0) but yields the same value of < E >ˆ as well as the same set of measured energies with the same probabilities.
Problem 3 : A Hard Wall [5 points]
A particle of mass m is moving in one dimension, subject to the potential V (x) :
(
0, for x > 0,
V (x) =
∞, for x ≤ 0.
Find the stationary states and their energies. These states cannot be normalized.
Problem 4 : A Step Up on the Infinite Line [10 points]
A particle of mass m is moving in one dimension, subject to the potential V (x) :
(
V0, for x > 0,
V (x) =
0, for x ≤ 0.
Find the stationary states for energies (0 < E < V0).
Problem 5 : A Wall and Half of a Finite Well [10 points]
A particle of mass m is moving in one dimension, subject to the potential V (x) :

∞,

V (x) = −V0,
0, for x < 0, for 0 < x < a, for x > a. (V0 > 0)
In this case, find the stationary states corresponding to bound states (E < 0). Is there always a bound state? Find the minimum value of z0 given by
,
For which there are three bound states. Explain the precise relation of this problem to the problem of the finite square well of width 2a.
Problem 6 : Evaluate the following integrals :
1.
2.
3.
Problem 8 : Delta Potential
Consider the double delta-function potential
V (x) = −α[δ(x + a) + δ(x − a)],
where α and a are positive constants.
1. Sketch this potential.
2. How many bound states does it posses? Find the allowed energies for α = ℏ2/ma and α = ℏ2/4ma and sketch the wavefunctions.