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Instructor(s): D. Reitze
Physics 3101, Introduction to Modern Physics
PHYSICS DEPARTMENT
Springl 2006, EXAM #2
Name (print, last first):
Friday, March 10, 2006
Signature:
On my honor, I have neither given nor received unauthorized aid on this examination.
YOUR TEST NUMBER IS THE 5-DIGIT NUMBER AT THE TOP OF EACH PAGE.
(1) Code your test number on your answer sheet (use lines 76–80 on the answer sheet for the 5-digit
number). Code your name on your answer sheet. DARKEN CIRCLES COMPLETELY. Code your UFID
number on your answer sheet.
(2) Print your name on this sheet and sign it also.
(3) Do all scratch work anywhere on this exam that you like. Circle your answers on the test form. At the end of
the test, this exam printout is to be turned in. No credit will be given without both answer sheet and printout with
scratch work most questions demand.
(4) Blacken the circle of your intended answer completely, using a #2 pencil or blue or black ink. Do not
make any stray marks or some answers may be counted as incorrect.
(5) The answers are rounded off. Choose the closest to exact. There is no penalty for guessing.
(6) Hand in the answer sheet separately.
Instructions
Read each problem carefully before starting. There are 6 multiple choice problems each worth 10 points and 1 partial credit
problem worth 40 points. Show all of your work on the partial credit problem!!
Constants:
c = 3 × 108 m/s
h̄ = h/2π
h = 6.63 × 10−34 J s = 4.14 × 10−15 eV s
1
hc = 1240 eV nm
k = 4π²
= 8.99 × 109 N m2 /C2 R = Rydberg constant = 1.096776 × 107 m−1
0
k = Boltzmann’s constant = 1.38 × 10−23 J/K = 8.61 × 10−5 eV/K
1 eV = 1.6022 × 10−19 J
−31
2
me = 9.11 × 10
kg
me c = 0.511 MeV
1 MeV/c2 = 1.783 × 10−30 kg
σ = Stefan’s constant = 5.607 × 10−8 W/(m2 K4 )
mproton c2 = 938.27 MeV
Wien’s constant = 2.898 × 10−3 m K
mproton = 1.6726 × 10−27 kg
Useful integrals:
R∞
0
2
x2 e−ax dx =
1√
−3/2
4 πa
Multiple Choice (each worth 10 points)
1. A proton has a De Broglie wavelength that is equal to its Compton wavelength. What is the total energy (in eV) of
the proton?
(1) 1.33 × 109
(2) 9.41 × 108
(3) 3.89 × 108
(4) 6.92 × 104
(5) 7.76 × 109
2. An electron with energy 25π 2 h̄2 /(2me L2 ) is located in an infinite square well of width L with walls at x = 0 and x = L.
At which of the following positions is the probability of finding the electron identically 0?
(1) 3L/5
(2) 3L/4
(3) L/2
(4) L/3
(5) L/6
3. The ground state wave function for the quantum harmonic oscillator is
Ψ0 (x) =
³ mω ´1/4
πh̄
exp(−mωx2 /2h̄)
What is expectation value < x2 > for this wave function?
√
(1) h̄/(2mω)
(2) 2h̄/(mω)
(3) h̄/(mω)
(4) 0
(5) 2h̄/(mω)
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4. An incident beam of α-particles with kinetic energy of 2 MeV is incident on a 2 × 10−6 m gold foil. What fraction of
the particles scatters with an angle greater than 60◦ with respect to their incident direction? The atomic number of
gold is Z=79, and the particle density of gold is 5.9 × 1028 atoms/m3 .
(1) 3.59 × 10−3
(2) 3.92 × 10−8
(3) 5.03 × 10−4
(4) 4.92 × 10−5
(5) 1.96 × 10−8
5. Which of the following statements are true?
I. The quantization of angular momentum in the Bohr model of the atom can be though of as standing matter waves
around the circumference of the electron orbits
II. The transition energy between levels of electron in a potential of the form 12 Kx2 is ∆E = h̄ω.
III. A electron confined in a potential well cannot have zero energy.
(1) I, II, and III
(2) II and III
(3) I and III
(4) III only
(5) I only
6. Light of wavelength 208.4 nm wavelength is emitted from a doubly ionized atomic lithium (Z = 3) gas source. Which
energy levels (initial, final) are responsible for this transition?
(1) (4,3)
(2) (3,1)
(3) (5,4)
(4) (8,1)
(5) (10,2)
Partial Credit Problem (worth 40 points)
Electrons with an energy of 1 eV are traveling toward from the left towards a 2.5 eV potential step barrier as shown
below.
(i) Compute expressions for the wave functions for the electrons in region I and II. Be sure to calculate the wave
numbers. Qualitatively sketch the wave functions in regions I and II. (20 points)
(ii) Write down the expression for the electron density in the barrier region. Compute the effective penetration depth
(in m) that the particles penetrate into the barrier. (20 points)