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Unformatted text preview: Question 1: Multiple choice problems: 7X3=21 points Mark your answer on the line to the right of each problem 1. At a ﬁnite potential step, the ﬁrst derivative of the wavefunction, QW/ax, will be: (a) continous (b) show a ﬁnite step (c) an inﬁnite step A see solitme a, my. AM 2. All else constant, increasing mass will cause vibrational zeropoint energy to (a) increase (b) decrease (c) no change B %PE=%(‘%‘)12 =? “IT—727%}, F 3. For a given problem, the energy of a wavefunction that changes sign as a function of
position compared to one that has only one sign will be: (a) higher (b) lower (c) can’t say 5
man, W a kiojlxlr am 4. In quantum mechanics, does the energy of a particle have a deﬁnite value? (a) always (b) sometimes (0) never B amply WHYmﬂ E 4131‘ a~ hwvoﬁm Slack VS. am @‘iwﬁuk
5. Is it possible to simultaneously know the value of position and
momentum of a particle in quantum mechanics? (a) always (b) sometimes (c) never C AKA? "x 1; 6. Classical observables are represented in quantum mechanics as: (a) complex numbers (b) Hermitian operators (0) real functions 8 R04 m\\ﬁaw\ oaks66in: % Peak Vail/€45
7. Which of the following has the longest de Broglie avelength? (a) a baseball pitch (b) a yellow photon (c) a 16V electron B A ~§’ZQ—§%O nm
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A P W OLuﬁLm‘J—[O m’ PW! Question 2. Operators and angular momentum. Classically angular momentum is a
vector:
L = r X p
(a) (8 points) Derive the quantum operators describing the x, y and z components of
angular momentum: 1 3 t 9
:4’; x Y E — 3%“ ‘Sﬁxﬁ‘iﬁ A A y: A3 g 9" ’2’ + vtzécd'mlg (b) (8 points) Evaluate the commutator of the operators for x and py, and z and p:. Note
that a commutator of 2 operators A and B is deﬁned as: [21,3]=211§—é,21
("Mal E!) '1 EV) ﬁ ’ltxéaga‘V‘i’R'BQSCKQ) 7‘ ’~\3V\¥ ya * \kxgz3 ’— 0
=> Cy) : O
[31921F2 [Q’J‘h—g’éB‘F : r\ti%—é+h£§(%$\ = 4’“ izlr‘x’ﬁiqtri/gii/l 53%?
é) [%;Va—l 5 “7i (3 if) (c) (9 points) Explain (with math as needed) why it is possible to know the value of 2
observables at the same time only if their corresponding operators commute. The. UWJCQ‘WCL‘: (Match W3
653 >2 ﬁktmmﬂ
l? MM Wm mw—lﬂ,’ﬂwm (WE: howl M \s 23m) 0% $353 =0] mVlVlg M 1*” “WW
V‘QXmJNMbkiY) GM m can more. balk obwqu‘vA $Wu . 14> M do my} commit, {ﬁrsth «wk we MU
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Woo/MC We.le WWMo’lh/ML amt» V3 Question 3. (32 points) Consider a particle in a box with inﬁnitely high walls. Let us 55 V‘th '
use dimensionless units where h =1, m=1, for convenience Let’s choose the box length to be 1, with the left boundary at the origin. An ensemble of systems is prepared in the following normalized state at time 0: (I) = sin(7rx) — sin(27cx) a) (7 points) Write the Hamiltonian in dimensionless units for the particle in the box,
and give the boundary conditions at the edges of the box. Does the above function satisfy the boundary conditions? f ohms‘tm‘u 3 Mﬂ‘l‘b
2
K .. .t'\1 A1 : “l a 4 \j
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M VM ~ 2 66’) em b) (9 points) Establish whether or not the above function is an eigenfunction of total energy.
I} 'rl is avx ~\‘i Musk 503653
35% > Ea; 2
fi§ : ~Aiégz<gmmw§m21tlx W, wnm’r (Mb 0. kaawl" Smelt ’i‘wd’
’Wxs Mush E§ , Sc 34"“ '2} (“A c) (9 points) If a large number of systems were prepared in the above state, and the
value of position (x) were measured for each of them at time zero, would the
particle be more likely to be found in the left or right side of the box? Give a
simple but convincing reason for your answer (hint: work out, or visualize the probability distribution). SR ’\ W $1“ 21m s‘m W» {mm This, M Wm wit ‘32 mm libel{3+3
‘01 M m M ltjblt Me ad {>0 d) (7 points) Suppose all the systems prepared in the above state at time zero were
allowed to evolve (if they want to) for a time t. If position were then measured at
time t, would you expect the average value to be the same as at time 0? Explain
why, without doing a whole lot of algebra. TLL 'RW'Q' a 33¢ 6M Palfvamllﬁk 73 1M ‘51,
a JAR/kw c2553“ . as
LEIt a —E\M—W¢e 53v I '6' 941136 + SM“ (7“) :9; t'
e L {3‘0 ' § : SEW 'WV” éu’tW
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Question 4. (22 points) Sup ose som em as energy eve s’ given by the expression:
E1 = (n +%)w, . n = 0,1,2.. These are the energy eigenvalues for this system. Now suppose that the following state of
the system is prepared: lw>z3l¢0>+5l¢1>_4l¢3> Here [(1)0 >, $3) are normalized energy eigenstates corresponding to n=0,l and 3. $1), (a) (ll points) Is the state 'l/l> normalized? If not, work out its normalized form. Nolvwllkét :5 E lCnl1= 1
4% 37+S‘+%1=e1. so M 12* «meta. w. M a Cowskm’r A, such ’Wr: lAl7<zlsl>JJ>= 1. » W(emus<¢t\—q<¢3l)(st¢m+s(«ma(«53»)
wmkcm = E0) “t” 1' wan
— — — __\__
:7 \Mﬂawgumrl :; \AVGQA. » AVE;
COK'de WW9, (b) (11 points) The energy of the system in the state ‘1]! > is measured. What are the possible values of the energy that can be obtained and their probabilities? EN: 0%) w ‘ s 1— q E0” "\ZW) V: “Aw5'
Ml»: El : (Hélw : ’32:“) (9: KY. ' g :
“635 " E3 : (9%“ 2 13w. ?: \Cs\2= lg? ...
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 Spring '11
 headgordon

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