Physics 137A: Quantum Mechanics I, Spring 2006
Midterm I 2/23/06
Directions: The allotted time is 80 minutes. The 4 problems count equally. Two sides of
your own notes are allowed. No books or calculators are allowed, and please ask for help only if a
question’s meaning is unclear.
1. Consider the harmonic oscillator ground state wavefunction ψ(x) = Ae−mωx2/2¯hwith A=
mω
π¯h1/4, which makes this a normalized wavefunction.
For this problem, you may wish to use the Gaussian integrals
Z∞
−∞
dx e−ax2=rπ
a
Z∞
−∞
dx x2e−ax2=rπ
4a3.(1)
(a) (5 points) Write an integral for the probability to find the particle between x=band x=c.
You do not need to do the integral for this part of the problem.
(b) (5 points) Is this wavefunction an eigenstate of kinetic energy? Justify your answer.
(c) (15 points) What is the expected value of kinetic energy of a free particle with this wave-
function? Try at least to write an integral; do the integral if you can.
2. At t= 0, suppose that a state in the infinite square well with potential
V(x) = ∞if x < 0 or x > a
0 if 0 ≤x≤a(2)
has the wavefunction
ψ(x, t = 0) = 1
√2ψ1(x) + 1
√2ψ2(x),(3)
where the normalized eigenstates are ψn=q2
asin(nπx/a).
(a) (10 points) Is h(x−a/2)ipositive or negative in this state (that is, is the particle on the
left side or the right side of the well)? You do not need to calculate the integral if you can give a
convincing answer without it.
(b) (15 points) Find a later time Tat which h(x−a/2)i(t=T) = −h(x−a/2)i(t= 0), so that
the particle has moved to the other side of the well.
3. (a) (8 points) Prove that the expected value of the momentum operator is constant in time,
if the system starts in an energy eigenstate.
