PHY 4605
April 25, 2012
Final Quiz
In some sort of crude sense which no vulgarity, no humour, no overstatement can quite
extinguish, the physicists have known sin; and this is a knowledge which they cannot lose.
J. Robert Oppenheimer
h 2 2i
(1) The wave function ψ(x) = A exp − b 2x , where A and b are real constants, is a
normalized eigenfunction of the Schrödinger equation for a particle of mass M and
energy E in a one dimensional potential V (x) such that V (x) = 0 at x = 0. Which of
the following is correct? (20 points)
2 4
b
(A) V = h̄2M
2 4 2
b x
(B) V = h̄ 2M
2 6 4
b x
(C) V = h̄ 2M
(D) E = h̄2 b2 (1 − b2 x2 )
2 4
b
(E) E = h̄2M
(2) In perturbation theory, what is the first order correction to the energy of a hydrogen
atom (Bohr radius a0 ) in its ground state due to the presence of a static electric field
E? (20 points)
(A) Zero
(B) eEa0
(C) 3eEa0
(D)
8e2 Ea30
3
(E)
8e2 E 2 a30
3
~1
(3) Two ions, 1 and 2, at fixed separation, with spin and angular momentum operators S
~2 , have the interaction Hamiltonian H = −J S
~1 · S
~2 , where J > 0. The values of
and S
~1 k2 and kS
~2 k2 are fixed at S1 (S1 + 1) and S2 (S2 + 1), respectively. Which of the
kS
following is the energy of the ground state of the system? (25 points)
(A) 0
(B) −JS1 S2
(C) −J[S1 (S1 + 1) − S2 (S2 + 1)]
(D) −(J/2)[(S1 + S2 )(S1 + S2 + 1) − S1 (S1 + 1) − S2 (S2 + 1)]
h
i
S1 (S1 +1)+S2 (S2 +1)
(E) − J2 (S
+S )(S +S +1)
1
2
1
2