Physics 3101, Introduction to Modern Physics, Fall 2004
EXAM #2
Monday, November 8, 2004
Instructions
Read each problem carefully before starting. Choose 3 of the following 4 problems.
Remember to work each problem as fully as possible. Partial credit will be given so
SHOW ALL OF YOUR WORK!!
Constants:
c = 3 x 108 m/s
h = 6.63 x 10-34 J s = 4.14 x 10-15 eV s
1
k=
= 8.99 x 109 N m2/C2
hc = 1240 eV nm
4πε 0
k = Boltzmann’s constant = 1.38 x 10-23 J/K = 8.61 x 10-5 eV/K
h = h/2π
me = 9.11 x 10-31 kg
1 MeV/c2 = 1.783 x 10-30 kg
Useful relationships:
sin 2 x =
1
(1 − cos 2 x)
2
∫
∞
0
x n e − ax dx =
n!
n ⋅ ( n − 1)... ⋅ 2 ⋅ 1
=
n +1
a
a n +1
1. Consider a one dimensional infinite square well in which V = 0 in the range 0<x<L.
(a) Assuming that a particle of mass m obeys the Schrodinger equation, write down a
general expression for the normalized wave functions of the well states. Also
write down an expression for the energies of the well quantum states. (12 points)
(b) Compute the expectation value <p2> for the nth wave function. Using this result,
compute the expectation value <Ek> for the nth state. Compare your results for
the energy with part (a). (11 points)
(c) Assuming that the particles are Fermions with ½ integer spins, i.e., they obey the
Pauli exclusion principle, how many particles will it take to fill up the well to the
nth level? (10 points)
2. An electron of energy E is traveling from the left toward a one dimensional step
potential as shown in the figure below.
V(x)
E
V(x) = V0
V(x) = 0
0
I
II
x
(a) Write down the Schrodinger equation for electron in region I (x < 0) and region II (x
> 0). You do not have to solve the equations. (11 points)
(b) Sketch the character of the wave functions in regions I and II, and explicitly calculate
the de Broglie wavelength in terms of E and V0. (11 points)
(c) If the incident energy E is 5 eV and V0 is 2 eV, compute the number of electrons
reflected back at x = 0 if 106 electrons are traveling toward the step. Classically, how
many electrons would be transmitted? (11 points)
3. The ground state wave function for the hydrogen atom can be written as
Ψ100 (r ,θ , φ ) =
1
1
π a0
3/ 2
e − r / a0
(a) Write down the expression for the probability of finding an electron between r
and r + dr (10 points)
(b) Prove that an electron in the ground state is most likely to be found at r = a0. (12
points)
(c) Compute the expectation value <r2> of the ground state. Hint: the integral you
need is in the ‘useful integral’ section. (11 points)
4. The diagram below displays the energy level diagram for the hydrogen atom.
E
C
D
A
B
(a) List in spectroscopic notation the n=1, 2, and 3 states and find the total number of
electron states for n=1,2,3, listing the possible values of angular momentum, zcomponent of angular momentum, and spin. (15 points)
(b) For the n=3, l = 2 state, compute the total angular momentum of the electron. Do
the same for the n = 5, l = 3 state. (10 points)
(c) Which of the energy transitions shown in the figure are allowed? (8 points)