Physics 3101, Introduction to Modern Physics, Spring 2005
EXAM #2
Wednesday, March 30, 2005
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
λc = electron Compton wavelength = 2.34 x 10-15 m
a0 = Bohr radius = 0.0529 nm
1 MeV/c2 = 1.783 x 10-30 kg
Useful relationships:
∞
∫x
−∞
2
e − ax dx =
2
1
π a −3 / 2
2
1. (a) An electron has a de Broglie wavelength of 0.5 nm. Compute its momentum and
energy. (10 points)
(b) An electron has a de Broglie wavelength of 0.5 fm (1 fm = 10-15 m). Compute its
momentum and energy. (10 points)
(c) Prove that the phase velocity of a wave packet is v/2, where v is the classical velocity
of the particle. Also prove that the group velocity of the particle is v. (13 points)
2. The wavefunction for the first excited state of a harmonic oscillator in a parabolic
potential is given by:
ψ 1 (x ) = C1 x e −m ω x
2
2h
where m is mass of the particle and ω is the oscillation frequency in the well.
(a) Compute the value of the normalization constant C1. (15 points)
(b) Show that the expectation value <xn> for ψ1(x) is zero for n = any odd integer.
Hint: you do not need to explicitly compute the integrals. It suffices to recognize
something about the symmetry of the integrand and the limits of integration. (10
points)
(c) Write down the energy of the first excited state. From this, compute the classical
turning points of the motion and sketch the wave function. (8 points)
3. The first excited state wave function (n=2) for the hydrogen atom with zero angular
momentum can be written as
Ψ200 (r ,θ , φ ) =

r  − r / 2 a0
1 −
 e
3
a
2
8πa0 
0 
1
(a) Write down the expression for the probability of finding an electron between r
and r + dr (10 points)
(b) What is the probability that the electron is likely to be found in the range r = a0 to
r = 1.01 a0? What is the probability that the electron is likely to be found in the
range r = 5a0 to r = 5.01 a0? (12 points)
(c) To what other levels (n,l,m) can this state make transitions to? (11 points)
4. (a) For the n=4, l = 3 state, compute the total angular momentum of the electron. Draw
a vector model diagram accurately illustrating the possible orientation of the orbital
angular momentum vector and its z-component. (15 points)
(b) Consider an infinite 3 dimensional square well with widths L, 2L and 5L along x,y,
and z. (18 points)
(i)
(ii)
Compute the five lowest energies and their respective quantum numbers
How many electrons would it take to fill up the five lowest levels?