Name:
Score:
/out of 100 possible points
OPTI 511R, Spring 2015
Mid-Term Exam 1
Prof. Ewan Wright
In-class exam.
2:00 PM - 3:15 PM
Notes for the exam:
1. Exam is 75 minutes. This is a closed-book, closed-notes exam. Calculators are
allowed. Some solutions may require a numerical value. If you do not have time to calculate the final
numerical value for a given problem, simplify as much as possible to receive maximum partial credit.
2. Show your work and answers on the exam paper in the space following each question.
An extra blank sheet is provided on the last page. If you need more blank sheets, please ask. Staple
all sheets together when you are finished, including any scratch paper.
3. Some of the following information may be useful but is not necessarily needed on this exam.
Constants, conversions, useful formulas
e = 1.6 × 10−19 C (charge of an electron is −e)
a0 = 0.5 × 10−10 m (Bohr radius)
kB = 1.4 × 10−23 J/K (Boltzman constant)
ε0 = 8.9 × 10−12 C2 /(J·m)
¯h = 1.1 × 10−34 J·s
1 eV = 1.6 × 10−19 J
√
π=
+∞
R
2
e−u du
−∞
Euler’s formula: eix = cos(x) + isin(x)
∂
pˆx = −i¯h ∂x
Heisenberg’s Uncertainty Principal
ˆ
For any two physical quantities (observables) A and B and their associated linear operators Aˆ and B,
2 2
σA
σB ≥ (
1 ˆ ˆ 2
h[A, B]i) .
2i
1
Section I: Short answers (25 points total).
1. (4 points) Suppose that the ground state of an arbitrary time-independent potential well is written
in the form Ψ0 (x, t) = A · f (x) · exp (−iE0 t/¯h), where f (x) is a dimensionless function and E0 is the
ground-state energy eigenvalue.
(a) What is the probability that the particle will be found in the first excited state at a later time
t = t1 ?
(b)Write an integral expression for the probability of finding the particle’s location between two
points, x1 and x2 at time t = 0?
ˆ
2. (4 points) Suppose that there is a quantum mechanical system for which the Hamiltonian H
ˆ
and another operator Q share a common set of orthonormal eigenstates (orthogonal and normalized),
ˆ Q]=0.
ˆ
but the two operators do not have the same eigenvalues. We could thus conclude that [H,
Let
ˆ
Q be an operator that corresponds to an unspecified physical observable.
ˆ be real, imaginary, or is either option possible?
(a) Will the eigenvalues of Q
(b) Please comment on any inherent limitations there may be in determining the system’s energy
ˆ
and the physical observable corresponding to Q?
2
3. (2 points) Write down the de Broglie wavelength for a particle with momentum p.
2
2
4. (5 points) A particle at time t=0 is described by the wavefunction: Ψ(x, 0)= A · e−x /a · e−ikx ,
where A is a normalization constant. Calculate the expectation value of the particle’s momentum,
hpx i, at t=0.
5. (2 points) A particle of mass m is trapped in a 1-dimensional simple harmonic oscillator. The
strength of the potential well is characterized by the classical oscillation frequency ω. The wavefunction for the particle is not known. If a measurement of the particle’s energy is made, what are the
possible results one might measure? Give your answer in terms of the quantities described above.
3
ˆ A measurement of this physical
6. (4 points) A physical observable is represented by the operator Q.
observable is performed. What do the postulates of quantum mechanics tell us about the possible
values that can be measured and the resulting quantum state of the system after this measurement?
2
2
7. (4 points) Suppose the wavefunction for a free particle at time t = 0 is given by Ψ(x, 0)= A·e−x /a ,
where A is a normalization constant. For times t > 0, determine whether or not the following test
2
2
wavefunction correctly describes its time evolution: Ψ(x, t)= A·e−x /a ·e−iEt/¯h , where E is the energy
of the particle. Justify your answer using the time-dependent Schrodinger equation.
4
Section II: Longer questions (75 points total)
8. (15 pts) A particle of mass m is trapped in a 1-dimensional simple harmonic oscillator potential of the form V (x) = (1/2) · m · ω 2 · x2 .
(a) Suppose that a single measurement of the particle’s energy yields the result Em1 = (3/2)¯
hω.
Make a sketch of the corresponding wavefunction for the particle immediately after the measurement.
Be sure to also plot the potential well to indicate the position of the wavefunction relative to the
potential.
(b) A measurement is now made of the particle’s position. Is the probability to measure the particle within the classically allowed region equal to or less than 1? To answer this question, solve for
the classically allowed region of space given that the total energy before the measurement is known
to be (3/2)¯
hω (hint: solve for the case total energy = potential energy). Then indicate with a simple
sketch and/or integral expression why the probability may be 1 or less than 1.
(c) Some time after the measurement of the particle’s position, we make yet another measurement
of the particle’s energy. Are we certain to measure the same value Em1 = (3/2)¯
hω from the first
measurement again? Please justify your answer in a sentence or two.
5
9. Infinite Square Well Potential (25 points).
Consider a particle of mass m in an infinite square well of width a defined by the potential
(
V (x) =
0 0<x<a
∞ otherwise
)
,
(a) Write down the normalized energy eigenstates for this system inpterms of kn , where kn =
(Hint: the normalization coefficient for each energy eigenstate is 2/a.)
√
2mEn /¯
h.
(b) Write down, or derive, the allowed energy eigenvalues, En , in terms of the well width a and
particle mass m.
Suppose the particle has the initial wavefunction: Ψ(x, 0) = A sin3 (πx/a) for (0< x < a), where
A is a constant.
(c) Using the trigonometric identity: sin(3θ) = 3sin(θ) − 4sin3 (θ), express this initial state as a
superposition of energy eigenstates for the potential well.
6
(d) Is this state an energy eigenstate of the potential well?
(e) Determine the normalization coefficient A in terms of a.
(f) Write down the time dependent wavefunction Ψ(x, t) for this superposition state.
(g) What is the average energy of the particle, hEi, at time t = 0?
(h) Will hEi change at times t > 0?
7
10. Relative phases in quantum mechanics (15 points). Assume we can write an arbitrary
quantum state |ψi as a superposition of two orthonormal energy eigenstates |φ1 i and |φ2 i having
energy eigenvalues E1 and E2 , respectively. For this problem, define two superposition states as
|ψA i =
|ψB i =
√1 |φ1 i +
2
√1 |φ1 i +
2
√1 |φ2 i, and
2
√1 eiθ |φ2 i.
2
Here, θ is a real unitless number we can label as a phase.
(a) Evaluate hψA |ψB i as defined above, writing your final expression in terms of θ.
(b) Give a value of θ for which |ψA i and |ψB i are orthogonal.
(c) Give a value of θ for which |ψA i and |ψB i are “aligned,” so that they are described by identical superpositions of the basis states |φ1 i and |φ2 i.
(d) Evaluate the energy expectation value for each state, hEiA and hEiB . Can measurements of
energy be used to distinguish between states |ψA i and |ψB i ? Please explain.
8
11. (20 points). Consider a particle trapped in an unspecified potential well. The orthonormal
ˆ are denoted by |φn i with n any non-negative integer. The eneigenstates of the Hamiltonian H
ergy eigenvalues are En . The full time-dependent expressions for the stationary states are given by
|φn ie−iEn t/¯h . Now suppose the particle is in an arbitrary quantum state represented by the normalized
P
−iEn t/¯
h , where c are complex numbers. For the following questions,
superposition |Ψi = ∞
n
n=0 cn |φn ie
simplify your answers as much as possible. (You do not have to show math or reasoning for
the cases where you can immediately state the correct answer.)
(a) Is |Ψi a stationary state?
ˆ ni
(b) Evaluate H|φ
ˆ
(c) Evaluate H|Ψi
(d) Evaluate hΨ|Ψi
9
(e) Evaluate
P∞
2
n=0 |cn |
(f) Evaluate hφn |φm i, where n and m are independent indices for energy eigenstates
(g) Evaluate hφn |Ψi
(h) Evaluate the energy expectation value of the particle in state |Ψi
10
USE THIS BLANK PAGE FOR SCRATCH WORK IF NEEDED. If you remove this
sheet, be sure to re-staple it to your exam answers.
11
USE THIS BLANK PAGE FOR SCRATCH WORK IF NEEDED. If you remove this
sheet, be sure to re-staple it to your exam answers.
12