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Unformatted text preview: Name: Exam 1  PHY 4604  Fall 2011
October 5, 2011
8:20—10:20PM, MAEA 303 Directions: Please clear your desk of everything except for pencils and pens. The exam is
closed book, and you are not allowed calculators or formula sheets. Leave substantial space
between you and your neighbor. Show your work on the space provided on the exam. 'I can
provide additional scratch paper if needed. Unless otherwise noted all parts (h), . . . are worth 5 points, and the entire exam is
100 points. Harmonic oscillator: "6
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dm d3: i 112 (10(0) 1. Short answer section (a) Write downthe time dependent Schrodinger equation in one dimension. (b) What is the physical meaning of [w($,t)2? I’lP(x,t)I1d;z is +he PFObabi 1:47 or qtymung
a, parHale be‘i'Wﬁzrb :5 (ﬁx—rd?! (21“ +£me i: (0) Give an expression for the probability current. ‘:i #9 9*
J 2mm 5763—1195213) (d) If the wave function at time t is given in terms of the eigenstates of the time
independent Schrodinger equation as Mr, t) = Z a¢n($)e‘iE“t/E, what are the an in terms of 'IMJZ, 0)? Cm 7” ﬂn*(%)1P(?$ 0) (e) For a free particle (V = 0), give the wave function at time t in terms of its fourier
transform at t = 0, Where ﬁgm—ﬁ _m1/)(:1:,0)e_?krdr (Hz:
4P(X;t)= I Zik" 31.201 t
V22” aao 2. General properties At t = 0 the wave function of a particle in an inﬁnite square well between as : —1 and
:1:=+1 is (Noto:—1 <x< 1 notO <:I:< 1.)
Mam = 0(1 1'2) (1) What is the‘constant C so that the wave function is normalized?
I
.7’ :L 7' 2 2. 4
i = c Java My; =zc [(1275 +7c )dx
 I '0
.. 7 2, I 2 I I [6 w
‘429C (l#—'+—" :ZC ("—— _.....):=’,._.
:5 5 ) + 5 15
.. 15
a C ,6 (b) Compute the expectation values (:17), (2:2), as well as 033 for ¢(:2:, 0). You may use
the symmetry in the problem. 42$): 0 b/ S/mme+ry I
z I a, Q; 2. 4. <76.) = szxtn—x”) c174 =26 f(2<~22< +76 ‘ O #32:. 8
624 = <a¢"?—<%>” =‘/ V? (C) Compute the expectation values (39), (p2), well as (I? for (Mac, 0). P211909 0) ' “Elf; C 0’76") = 26%”
, :6 (d) Use the results of (b) and (0) above to check that the uncertainty principle is
satisﬁed. 0:40;, =gt7éf% ygﬁsg. (e) How is the expectation value {23p} related to the expectation value (pas)? <Xp?—<P)<?=1;'ﬁ 3. Harmonic oscillator
At t = 0 the wave function of a particle in a harmonic oscillator potential is given by (a) What is t)? a ‘
1P(%t)=§W:(>Oa +3L’_£%(X)z (b) What is the expectation value of the momentum for 1,0(35‘, t)? Z
4. “Mat
“" 122 (5:03)“; 6
_‘ ' ._ 'b
=W%(W*+a”w) 5: ,‘gg‘oﬁmw C05 (out) (C) What is the expectation value of the position for 1,11(:c,t) and how is it related to
the momentum of part (b)? aft 5:0 <x>=fdx(§3‘1l’.(><)¢ “gi‘ﬂwezz é) ——— (Q+"'Q—) :32: xii—"JU (é'wvw " hg‘m’z‘m “— )
2 __i ) bad'b :gG (5" fig ' LL
1+ .2. "‘0 "' 2%; §‘)(5>‘/ZZ :: 45 .131 £(e‘éwbaaz’wt) W 25 (d) What is the expectation value of the energy for 1143315)? m = ' (av2w weir—2% “0 ._,'_.——D 50 50 (e) Construct a normalized harmonic oscillator state which has an expectation value
of the energy of (3 / 4)hw. This is not related to 1/; in parts (a)(d). Let 7P: +' ﬂ’lpl )Wl'mfe «1+ ﬁzzﬁ— @ﬁw,3
<E7=or2f—‘9— +ﬂ7’f’ﬁ‘” a1+ ﬂail I
f 2—1 5,——
2ﬂ1=—9_’ ﬁ/G";J 2 25.2 20(5"__.3 H z 4. Piecewise constant and delta function potentials
For this problem consider the one dimensional time independent Schrodinger question with potential V'(:t): Wit?) : V1($)+ V2($) V1(:t:) = 0 for :1: < 0 : l/Ofor0<:cwithl/;,>O : In other words, this is the step potential with a delta function potential at the interface.
Assume that E > V0. For E > V0, what is the general form of the solution for a: > 0 and for :1: < 0.
Make sure to dene all variables that you introduce. I v / r/
Vmon 56”” +52"’“"; MW (b) What is the boundary condition at a: : 0 expressed in terms of your wave functions
 from part (a)? (c) For an incoming wave coming from the left and going to the rightJ which of the
terms in part is zero? / (d) (10 points) Solve for the transmission probability for a wave coming from the left
 and going to the right. A+A’== 5 “2’5 z'tz(A—A’) = 1m“ B
6 r “h”
> damA 3:(2'42’ 2m (2 A ’ k” ' (KL
—A = —— 2
Q,“ $138 I2 1143:;
—> E ____ I
A 2.0+}? +0 12;,
—> 12—): i
(%('*£5))z+(?%)2’
—> T = IE” 2 2': Iz’/f<’« ...
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