PHY4604–Introduction to Quantum Mechanics
Fall 2004
Practice Test 2
October 31, 2004
These problems are similar but not identical to the actual test. One or two parts will
actually show up.
1. Short answer.
• You are told that a potential can be written V (x) =
10
X
Vn x2n . All Vn ≥ 0.
n=1
What can you say about the solutions ψ to Hψ = Eψ?
• What is tunnelling? Draw a picture describing the process, and describe
explicitly a situation which distinguishes classical physics from quantum
mechanics.
• Write the Dirac expression hα|H|αi as an explicit integral in three dimensions, assuming that hr|αi represents a wave function ψα (r). If |αi is an
eigenvector of H with eigenvalue Eα , evaluate the integral.
• Identify: Ehrenfest’s theorem.
• If A and B are self-adjoint, is the combination A + iB self-adjoint, antiself-adjoint, unitary, antiunitary, none of the above, or more than one of
the above? Defend your answer.
• Prove that if (ψ, Oψ) = (Oψ, ψ) for all ψ, then O is self-adjoint.
• Prove that eigenvectors corresponding to distinct eigenvalues are orthogonal.
• Find the expansion coefficient of the first excited SHO state in the function
(x2 + x20 )−2
• Draw a picture of a finite attractive square well in 1D, V = −V0 inside and
V = 0 outside, and sketch the form of the ground and 1st excited states of
negative energy, assuming such states exist. Be sure to indicate how these
functions differ from the analogous eigenfunctions of the infinite square
well problem. What happens to the number of bound states (E < 0) when
you make the well in this problem deeper?
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2. Electrons incident on metal surface. Consider a metal occupying a halfspace x > 0, and model it as a region of constant potential −V0 . On the left for
x < 0 is a true vacuum with V = 0. Electrons are incident on the metal surface
from the left with initial energy E.
(a) Write down the general solution to Schrödinger’s equation on the left and
the right. Take the amplitudes of the incident wave, reflected wave, and
transmitted wave to be A, B, and C.
(b) State the boundary conditions at the interface and use them to extract
relations between A, B, and C.
(c) Calculate the probability current densities in the incident, transmitted,
and reflected waves, and verify probability conservation.
(d) Calculate the reflection probability of electrons if E = 0.1eV and V0 = 8eV .
3. 2-level system. A Hamiltonian for a 2-level system is written as
H = E0 ((|1ih1| − |2ih2| + |2ih1| + |1ih2|)
where the vectors |1i and |2i represent two orthonormal states of the system
which span the Hilbert space of the problem.
(a) Write down the matrix H of the problem in the |1i, |2i basis.
(b) Find the eigenvectors |ai, |bi and eigenvalues Ea and Eb of the Hamiltonian.
(c) The system is initially in state |1i. Find the probability that it is still in
state |1i after a time t.
(d) Suppose you are given another Hermitian operator Q = q1 |2ih2|. Determine whether it is possible to find a set of states which are simultaneous
eigenvectors of Q and H.
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