Physics 3101, Introduction to Modern Physics, Fall 2004
EXAM #1
Friday, October 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 integrals:
∞
π4
x3
dx
=
∫0 e x − 1
15
∞
∫e
−u 2
du = π
−∞
1. A neutral ‘K’ particle (rest mass m) has a momentum 2mc decays into two photons γ1
and γ2, with one photon emitted in the original direction of the ‘K’particle and the other
in the opposite direction.
Before:
After:
‘K’
γ2
γ1
(a) What is the total energy (in terms of m) of the ‘K’ particle before it decays? (10
points)
(b)
How fast (in units of c) is the ‘K’ particle traveling? (10 points)
(c)
Using conservation of energy and momentum, determine the momentum of each
photon (in units of mc). (13 points)
2. The expression for the energy density spectral distribution function found by Planck is
given by:
8πhcλ −5
u (λ ) = hc / λkT
e
−1
where k is the Boltzmann constant.
(a) If the relationship between E and u(λ) is given by u(λ) = 8π E λ−4 , what is the
expression for the average energy per oscillator E ? Recall that ε = hc/λ. (10
points)
(b) What is the average energy E for an oscillator that has energy ε = kT ln2 ? (ln is
the natural logarithm.) Recall that ε = hc/λ. How does this compare to the
classical value of the average energy? (10 points)
(c) The total energy density is found by integrating u(λ) over all wavelengths:
∞
U = ∫ u (λ )dλ
0
Find U in terms of k, h, c, and T. Hint: Let x = hc/λkt and take it from there…(13
points)
3. Uranium has an atomic number of Z = 92. It is possible to create U91+, i.e., a uranium
ion that has one bound electron. Using the Bohr model, answer the following questions:
(a) What is the radius of the ground state (n=1) orbit of U91+ ? (12 points)
(b) What is the ground state energy of U91+ ? (11 points)
(c) What is the wavelength of a photon emitted from the n=3 to n=1 transition in
U91+? What part of the electromagnetic spectrum does this correspond to? (10
points)
4. (a) Determine the de Broglie wavelength for a lambda particle (rest mass 1116
MeV/c2) whose kinetic energy is 2220 MeV. Is this larger or smaller than the Compton
wavelength of the lambda particle? (12 points)
(b) A particle has a wave function given by Ψ(x) = A e −4x defined over the x-axis.
2
i)
What is the value of A? (8 points)
ii)
What value of x is the particle most likely to be found? (7 points)
iii)
Explain the interpretation of the wave function. (6 points)