Physics 481
Spring 2004
Midterm Exam
Spring 2004
1. Consider a certain quantum system with only three states. In the basis
⎛
⎞
1
⎜
ψ1 = ⎝ 0 ⎟
⎠
0
⎛
⎞
0
⎜
ψ2 = ⎝ 1 ⎟
⎠
0
⎛
⎞
0
⎜
ψ3 = ⎝ 0 ⎟
⎠
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the state ψ1 has an energy Ea and the states ψ2 and ψ3 have a higher energy Eb . A
perturbation with the following matrix elements is introduced:
ψ1 |H |ψ1 ψ2 |H |ψ2 ψ3 |H |ψ3 ψi |H |ψj =
=
=
=
2
0
0
for i = j
(a) Write the Hamiltonian matrix for the unperturbed system (H 0 ) as well as the
Hamiltonian matrix for the perturbed system (H) in the basis given. (3 points)
(b) Use first order perturbation theory to find the corrections to the energy of all
three states. (6 points)
(c) What are the wave functions for the excited energy states in zeroth order
perturbation theory ? (3 points)
2. Consider three non-interacting particles in a one-dimensional infinite square potential well.
0 for − a2 ≤ x ≤ + a2
V =
∞ otherwise
A measurement of the total energy of the three particles yields 11E1 , where E1 is
the single-particle ground state energy.
(a) Write the single particle wave functions and single particle energies. (2 points)
(b) Give an upper limit on the single particle ground state energy using a trial
2
wave function ψ(x) = N ( a4 − x2 ). Compare the result to the true ground state
energy. (6 points)
(c) Give the degeneracy of the three-particle state and the wave function(s) where
possible in case the particles are i) distinguishable, ii) identical fermions, or iii)
identical bosons (ignore spin for this problem). (5 points)
Good luck !
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