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Unformatted text preview: Physics 3220, Exam 1 Student ID Exam 1 Closed book, one 8.5x11" cheat sheet, and
calculator.
Test time: 90 minutes
Total points: 100 Please do not open the exam until you are asked to. 0 Your exam should have 10 pages, numbered Exam 1.1 thru Exam 1.10. o This exam consists of long—answer problems. 0 Write your answers in the space provided on each page for the written question. Use
the back of the page if needed. SHOW YOUR WORK ON THE WRITTEN
QUESTIONS. Fuli credit will be given only for the correct answer accompanied by I
your reasoning. PLEASE! I 1. Print your name on each page in the space provided.
2. Print your student Identification Number on the page above. I recommend: Don’t waste time erasing on the written problems. Just put a line through defective
reasoning. Ask for more paper if needed. At the end of the exam, check that you have completed all the questi0ns. By handing in this exam, you agree to the following statement: "On my honor, as a
University of Colorado Student, I have neither given nor received unauthorized
assistance on this work" Signature Good luck! Fall 2010 Exam 1.1 NAME: 86% ' ‘ Problem 1. Foundations: Wave functions and operators. (20 pts total) Physics 3220, Exam 1 a) In quantum mechanics, we expect to describe particle behavior Via complex—
valued wave functions. State mathematical] the ‘normalization condition’ we
expect any single particle wave function to satisfy. Then state what it means physically. (5 pts) 63W that) WW :1 b) Given a wave function, Hat), write down the equations you need to evaluate to
find the expectation value of position and squared position. State how these two
values let on calculate the s uared variance of osition. (5 points): <x>= I] ‘i‘kﬂah
<x2>= Santa‘ﬁ‘tainc sz>f<x>1 Fall 2010 Exam 1.2 Physics 3220, Examl ' NAME: §/$M¢ Problem 1. Foundations: Wave functions and operators (continued) c) Given a wave function, SKxJ), write down the equations you need to evaluate to
find the expectation value of momentum and squared momentum. State how these
two values let on calculate the s uared variance of momentum. (5 points). <p2>= 3”¢*t“"3‘%f§z1a d) How do you expect the variance of the position and variance of the momentum to
be related for a typical quantum state? Explain. (5 points). 503. L a
€220} aL Fall 2010 Exam 1.3 Physics 3220, Exam 1 NAMEM Problem 2. Foundations: The Schrodinger Equation (20 points total) 3) Write down the time dependent Schrodinger Equation for a 1—dimensional
quantum problem (depends upon time, t, and position, x). State clearly the
meaning of any constants or functions you use in your equation. (5 points) 2 {lg—LE“ at?“ + V6093 Lime) 1!; ﬂax} I'M/M
Vent) —_ i: —l {"I b) Write down the time independent Schrodinger Equation for a 1d problem
(depends upon position, Jr), State under what conditions you expect solutions of
this equation to be appropriate for a problem, and under what conditions you must
use the full time dependent Schrodinger Equation. If you use new symbols that
you did not have in part (a), deﬁne them. (5 points): : “g %% +VCX)LP0() r. ELiJCK) MTISEZ ‘ “Lev/4”" ’
ii: Wezgsz—DSE E=WWW Fall 2010 Exam 1.4 Physics 3220, Exam 1 NAME: m. Problem 2. Foundations: The Schrodinger Equation (continued) c) Assume that you have solved the time independent Schrodinger Equation for the
eigen functions and eigen energies, £00 and En. Write down the full space and
time dependent wave function for the state, n. (5 points) d) Assume that you have solved the time independent Schrodinger Equation for the eigen functions and eigen energies, ﬂu) and En. Write down the full space and
time dependent wave function for a quantum state that satisﬁes general initial
conditions. If you use any new constants, state clearly either their value or how
you determine them from the initial conditions. (5 points)  46$
2 Ci dime A” h W T— gWS’ C“ 2; g (ﬁrme 09%
W 5‘ 9
M L) CK Fall 2010 Exam 1.5 2”“ Physics 3220, Exam 1 NAME: QM Problem 3. The inﬁnite square well. (30 points total) a) You are asked to solve for the wave functions and eigen energies for the infinite
square potential shown below. The solutions are of the form Asin(kx), so the wave
function is zero at the x = 0 boundary. State the condition at x = d. State the allowed values of k. Write down an equation for the eigen energies. Sketch the
third state at t = 0 on the picture below. (15 points) . V is infinite Well iS hem V is infinite here V=0 here
‘1’ is zero ‘1‘ is zero
here here L193 We) Fall 2010 Exam 1.6 ‘3 )
Physics3220, Exam 1 NAME: 6} Problem 3. The inﬁnite square well. (continued) b) Assume that you prepare a more general state that is a superposition of the first
and the third stationary state so that w( 15,0) 2 A[u1 (x) — i143 . Assuming that the stationary states are normalized already, calculate the normalization constant,
A. (10 points) Fall 2010 Exam 1.7 Physics 3220, Examl ' NAME: 3 obj1&4 Problem 3. The inﬁnite square well. (continued) c) Using your normalized state from (b), and the actual form of the wave functions
for the square well, calculate the expectation value of the squared momentum.
(5 points) <PZ> 2 12%” giuﬁc‘uﬂa‘; M “i Malt ' Wit gay/aﬂ
. W1. Tr?“ <tl>z giétitérii Fa112010 ' I Exam 1.8 Problem 4. The harmonic oscillator (30 points) Physics 3220, Exam 1 3) Think about a simple diatomic molecule like N2. The atoms in the molecule are
bound together by the sharing of electrons and the complicated dance of electronic
potential and kinetic energies. Never the less, it’s a very good approximation to
treat such molecules as though they were masses connected by a spring. Let’s
assume that you can grab the atoms in a nitrogen molecule, and begin pulling
them apart. As you do, the energy of the molecule increases because of the
potential energy you put into the spring. Assume that the energy increases by 1
eV when you stretch the spring by 0.1 nanometers. What is the spring constant
for this spring? (10 points) _zs’PE ~— ieV = in (diam Fall 2010 Exam 1.9 Physics 3220, Exam 1 NAB/[EMMA— Problem 4. The harmonic oscillator (continued) b) You are given a Schrodinger Equation with the following potential: V (x) : émaﬁxz — bx You could go to work trying to solve the Schrodinger Equation for this new
potential, or; you could save yourself lots of time and effort by noticing something
useful. You can rewrite this potential in the following form: 1 2 V(x)=§mwo (x—xor To do so, you must be willing to shift your energy scale and you must also relate
the offset position x0 to the original parameter, b. Show how the reguired energy shift and offset position are related to b. (10 points)
1
Veg: g“ mwozix ZI—XxonrXOZ]
i Z 2_ 1 i i Z '2...
:— :Qjmaa a» max 99376 + grin/ma 9e  + r 5 on
B IZ aﬁe'lf‘rfMﬂﬂvgg gee  : gig/W} 
3%:ng M2 40% . . ~ W M? W 68 MW i ‘ y
% far 1 w [’5 9} c) For the tra tiona harmonic o illator with potential energy of Emaﬁx2 we mulbxz
expect a ground state wave function of the form, 1,110 (x) = Ae 2" . Write down the ground state wavefunction you expect for the potential in part (b). Explain 5
your reasoning. (10 pts) _ ' Fall 2010 Exam 1.10 ...
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