Download 1 Problem 1 (10 points): (a) (3 points) An electron bound to a proton

1
Problem 1 (10 points):
(a) (3 points) An electron bound to a proton has possible eigenstates ψn,`,m where n is the
principal quantum number, ` is the orbital angular momentum quantum number, and m is the
magnetic quantum number (we are ignoring spin in this problem). Circle those of the following
wave functions that you think are possible:
ψ1,2,3
ψ−3,2,1
ψ1,−2,3
ψ3,2,1
ψ1,2,−2 ,
ψ2,1,2
ψ1,2,2 ψ2,−1,2
ψ2,2,1
ψ4,1,2
ψ10,3,2
ψ2,2,−1
ψ4,4,2
ψ10,3,−2
ψ10,−6,8
ψ4,0,0
ψ10,−6,−8
ψ10,4,−3
ψ4,1,2
ψ10,4,3
(b) (2 points) Is it possible for any of the allowed states you found in part (a) to decay to another
one of the allowed states listed above with the emission of a photon? Circle one:
YES NO.
If you circled “YES”, then draw a line joining these two states, with an arrow showing the
direction of the decay, for each possible pair of states.
(c) (1 point)
h̄2 ?
What is the square of the angular momentum, L2 , for the state ψ5,3,0 in units of
(d) (2 points) A photon of circularly polarized light carries a spin angular momentum in the
direction of travel of the photon of (circle one of the following):
0
1
h̄
2
± 12 h̄,
0 or ± 12 h̄
h̄
± h̄
0 or ±h̄
(e) (2 points) What is the degeneracy of the energy level for which n = 3? [Again, ignore spin.]
2
Problem 2 (10 points)
(a) (3 points) A quantum-mechanical particle of mass M with kinetic energy E is incident
on a potential step of height U0 . Sketch on the adjacent figure roughly how the probability T of
transmission of the particle past the step varies with the energy E in the range 0 < E < 3U0 .
1
T
E
U0
0
E
U0
(b) (3 points) A quantum-mechanical particle of mass M with kinetic energy E is incident on
a potential barrier of height U0 and width L. In this particular case, U0 = π 2 h̄2 /2M L2 . Sketch on
the adjacent figure roughly how the probability T of transmission of the particle past the barrier
varies with the energy E in the range 0 < E < 3U0 .
1
T
E
U0
0
E
U0
c) (4 points) A quantum-mechanical particle of mass M is in a well of depth U0 and width
a. There are only three bound states, of energies E1 , E2 , and E3 , where E1 < E2 < E3 . The
energy E3 is infinitesimally less than U0 . Sketch the form of the wavefunction corresponding to
the energy E3 , and evaluate a in terms of U0 , M , and Planck’s constant h.
E3
a
ψ(x)
U0