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# exam1sol - Name Exam 1 PHY 4604 Fall 2011 October 5 2011...

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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 H 513* NE E A \$3 + | 8'.) L I (L) _ (ﬂy/4 ex <_T_nfﬂ\$2) W i m p 25 1 \$7; 1" m 01+)” 1P0 Delta function potential = 0:6 dcp(0+) (1940‘) _ 2mm 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 qty-mung a, par-Hale 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" 3-1.201 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 [(1-275 +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% ygﬁ-sg. (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‘éwba-az’wt) W 25 (d) What is the expectation value of the energy for 1143315)? m =- ' (av-2w 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+ ﬁzz-ﬁ— @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” > dam-A 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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exam1sol - Name Exam 1 PHY 4604 Fall 2011 October 5 2011...

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