Name:
Exam 1 - PHY 4604 - Fall 2013
October 3, 2013
8:20-10:20PM, NPB 1002
Directions: Please clear your desk of everything except for pencils and pens. The exam is
closed book, and you are not allowed calculators or formula sheets. Leave substantial space
between you and your neighbor. Show your work on the space provided on the exam. I can
provide additional scratch paper if needed.
Unless otherwise noted, all parts (a), (b), . . . are worth 5 points, and the entire exam is
100 points.
Harmonic oscillator:
1
(−ip + mωx)
2h̄mω
1
a− = √
(+ip + mωx)
2h̄mω
s
h̄
(a+ + a− )
x =
2mω
a+ = √
s
h̄mω
(a+ − a− )
2
mω 1/4
mω 2
ψ0 (x) =
x
exp −
πh̄
2h̄
1
ψn = √ (a+ )n ψ0
n!
p = i
Delta function potential V (x) = α δ(x):
dϕ(0+ )
dϕ(0− )
2mα
−
= 2 ϕ(0).
dx
dx
h̄
1
1. Short answer section
(a) Write down the time dependent Schrodinger equation in one dimension.
(b) What are the energy eigenvalues for the harmonic oscillator?
(c) What does it mean for a set of wave functions φm (x), for m = 1, 2, 3, . . . to be
orthonormal? (Give an equation.)
(d) What is the general form of the solution to the time independent Schrodinger
equation in a region where the potential is constant and greater than the energy,
Vo > E?
(e) Write down the continuity equation for the probability.
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2. General properties
At t = 0 the wave function of a particle in an infinite square well between x = 0 and
x = a is
ψ(x, 0) = C for 0 < x < a/2
ψ(x, 0) = −C for a/2 < x < a.
(1)
(2)
(a) What is the constant C so that the wave function is normalized?
(b) Compute the expectation values hxi, hx2 i, as well as σx for ψ(x, 0). You may use
the symmetry in the problem.
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(c) Express ψ(x, 0) as a linear combination of the eigenstates of the infinite square
well, ψn (x),
ψ(x, 0) =
∞
X
cn ψn (x),
n=1
by calculating the coefficients cn . Are any of the cn zero? If so, which ones?
(d) Use the result of part (c) to write down an expression for ψ(x, t).
(e) Is there a time, τ , for which ψ(x, τ ) = −ψ(x, 0)? If so what is that time?
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(3)
3. Harmonic oscillator
At t = 0 the wave function of a particle in a harmonic oscillator potential is given by
1
eiπ/4
√
ψ(x, 0) =
ψ1 (x) + √ ψ2 (x).
2
2
(a) What is ψ(x, t)?
(b) What is the expectation value of x for ψ(x, t)?
(c) What is the expectation value of p for ψ(x, t)?
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(d) What is the expectation value of the energy for ψ(x, t)?
(e) Determine σx σp for ψ(x, t) and show that the uncertainty principle is satisfied.
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4. Piecewise constant and delta function potentials
For this problem use the one dimensional time independent Schrodinger equation with
potential
V (x) = 0 for x < −a
V (x) = Vo < 0 for − a < x < 0
V (x) = ∞ for x > 0.
(4)
(5)
(6)
The energy of the particle is in the range Vo < E < 0.
(a) For Vo < E < 0, what is the general form of the solution? Make sure to define all
the variables that you introduce.
(b) What is the boundary condition for an infinite square well? This is the same
boundary condition as we have in this problem at x = 0. Apply this boundary
condition to eliminate one of the coefficients in part (a) in the region −a < x < 0.
(c) Which of the solutions in the region x < −a is physical and why?
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(d) What are the boundary conditions at x = −a applied to the wave function of
parts (a)-(c)?
(e) Derive a condition for there to be a bound states solution and solve it graphically.
What is the condition that there be one bound state solution?
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