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Instructor(s): D. Reitze
PHYSICS DEPARTMENT
Physics 3101, Introduction to Modern Physics
Fall 2005, EXAM #2
Name (print, last first):
Wednesday, November 9, 2005
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!
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
0
−23
k = Boltzmann’s constant = 1.38 × 10
J/K = 8.61 × 10−5 eV/K
me = 9.11 × 10−31 kg me c2 = 0.511 MeV
1 MeV/c2 = 1.783 × 10−30 kg
mproton c2 = 938.27 MeV
mproton = 1.6726 × 10−27 kg
Useful Integrals:
Z ∞
n −ax
x e
0
dx =
n!
an+1
Z
Z
π
sin x cos x dx = 0
0
π
sin3 x cos2 x dx =
0
4
15
Multiple Choice (each worth 10 points)
1. Compute the De Broglie wavelength of a proton (in nm) whose total energy is 51 times its rest energy.
(1) 2.59 × 10−8
(2) 1.32 × 10−7
(3) 5.95 × 10−6
(4) 1.97 × 10−9
(5) 5.95 × 10−15
2. An electron with energy 9π 2 h̄2 /(2me L2 ) is located in an infinite square well of width L. What is the probability that
an electron will be found in the region 0.195L − 0.205L?
(1) 1.81 × 10−2
(2) 2.30 × 10−1
(3) 5.11 × 10−3
(4) 1.20 × 10−4
(5) 5.36 × 10−1
3. What is the expectation value for the momentum p (in MeV/c) of an electron in the ground state of an infinite square
well of width L = 1 nm? Hint: pop = (h̄/i)(d/dx)
(1) 0
(2) 2.31 × 10−2
(3) 1.29 × 103
(4) 7.76 × 101
(5) −4.54 × 10−1
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4. An electron of energy E is traveling from the left toward a one dimensional step potential as shown in the figure below.
If the incident energy E is 2.5 eV and V0 is 2 eV, the percentage of electrons that will be reflected back at x = 0 is:
(1) 14.6%
(2) 5.3%
(3) 67.2%
(4) 1.3%
(5) 7.8%
5. An electron is in the n=3 state of the hydrogen atom. Which of the following statements are true?
I. There are 18 possible allowed states for the electron, taking into account l, m, ms .
II. The electron can only make a transition to the n=2 or n=4 state.
III. The electron cannot return to the ground state from this level.
(1) I only
(2) II and III
(3) I and III
(4) III only
(5) I, II, and III
6. The total number of degenerate states for a hydrogen atom with a principal quantum number n is
(1) 2n2
(2) n2
(3) 2n − 1
(4) n < |l|
(5) An infinite number
Partial Credit Problem (worth 40 points)
The wave function of the hydrogen atom is given by:
Ψ321 (r, θ, φ) = C321
r2
sin θ cos θeiφ e−r/3a0
a20
(a) List all of the possible allowed l and m transitions from this state. (15 points)
(b) What are the allowed values of the total angular momentum quantum number j for the n=3, l=2, m=1 state?
What are the allowed values for |J|? List the possible states in spectroscopic notation. (10 points)
(c) Compute C321 . (15 points)