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**Unformatted text preview: **J 9
[4] Step #4: “Eigenstateleigenvalues” -) “Spectroscopic Phenomena” (total = 20 pts)
After we have worked so hard to solve the wavefunction W(x,t), it’s time to extract valuable information from it based on 5 QM postulates, in particular, Born’s interpretation (postulate 1). [4](a) Born conditions (4 Pts). The statistical interpretation of the wavefunction asks that a “well-behaved”
wavefunction Ll” must satisfy the following Born conditions: LlJ must be continuous, single-valued, finite, and normalizable. ls llJ(x) = exp(-x) a “well-behaved” wavefunction over the interval (0, +00)? If not, please list the Born
condition(s) it violates. Please briefly explain your reasoning. normhtukic A; q «stilt "9" mm“ °' :‘uM‘C’
is «10+ LN“, Wuhabi‘tﬂ 9‘ Hz) lokam. $825.;qu / l
,r, [4](b) Ex ectation values 6 ts . Based on your results in [3](a) and [3](b), please calculate the expectation value
of linear momentum, <p>, and the expectation value of position, <x>, for the ground—state eigenfunction of a
particle in a 1D box. Please show algebraic details. Note again: The box is located between X: -L to x=O. (Hint: You may ﬁnd the integral table listed in the formula sheet helpful)
5 3:. Sm ( “ix. )
v (M " ‘- L~ mm 3..“ = W“
«ML;- I Suﬁ— (an) Stnlnl‘jlf) Ax. ‘ L: 5m “grit )3; [s‘ql'¥)] Mk: By dc; / éféiclxq 10 [4](c) Measurement outcomes + probabilities (10 Qtsz. 5 P435 Suppose that the wave: function fora system can be ' h. ’ written as What are the pOSSible values that you could ()btain in
measuring the kinetic energy on idtznticaiiy prepared 5““ x 8 * *9
. 1 i 3+V’2; 53mm»-
1x=~ -’-—— x+ x . .. . .
M ') 2 dab}? 4 4}“ ) 4 (M ) c. What is the pmbabzhty of measurmg each of these
. ‘ ‘3
and that (#10:)! (53 (xx and (1:3 (ﬂare nomaahzed mgcmamcs‘
eigeni'unmians of the operator Emm with eigenvalues El, d- What is the average "33% 0f Ekiymiv that You WOUM OWL!“
3E and 713 respectively. 0° dc) from a large number of meaxuremcms'?
~{. 2‘ i’ A?“ ‘ ) i 3. Verify that {1:0} is normalized. 10 UWarvn-m! éc/l Mm [5] QM at works, as illustrated by Feynman’s ouble t Experiments of Electrons (total=24 pts)
Richard Feynman (1965 Nobel prize in physics): “the entire mystery of quantum mechanics is in the double-slit experiment of electrons”. He also said: “I think I can safely say that nobod understands quantum mechanics”.
After 4 weeks in Chem113A W‘OT, let‘s see how much you understand qténtum chemistry now. [5|(a) Wavefunction of the electrons (4 pts) Two steps in solving quantum-chemistry problems: (i) Use Postulate #5 (Schrodinger equation) to ﬁnd the spatial
distribution and time evolution of quantum wavefunction of the quantum system (here the quantum system is the electron passing the double slits); (ii) apply Postulates #1-4 (in particular Postulate #1 Max Bom‘s statistical interpretation) to the calculated quantum wavefunctions. Please roughly sketch the wavefunction, I‘FI, of a single
electron when it arrives at the screen for the following 2 cases: (i) Both slits are open. (Hint: we have seen the anima’gon a few times in lectures) —? I“?‘,‘L‘|’a‘lcc pudkn, 11 [5|(b1 Superposition Principle (4 pts) Your result in [5](a)(i) should reveal a big secret in modern chemistry—the Superposition Principle: if the electron
can be in states described respectively by wavefunction LF1(x,t) (e.g.. passing slit #1) and ‘P2(x,t) (e.g., passing slit
#2). then the electron can also choose to be in a “superposition of states” that is described by the wavefunction
‘P(x.t) ==‘I’1(>:.t)+‘i’2(x,t). Please prove the Superposition Principle by algebra. (Hint: show that. if =‘P1(x,t) and LP,-=(x.t)
are solutions to the Schrodinger equation, then ‘I’(x.t) =‘P1(x.t)+'i'2(x,t) is also a solution to the Schrodinger equation) [Remarkt this principle serves as the theoretical foundation for quantum teleportation. Schrodinger‘s cat,
traveling faster than light, etc.] r. “L‘[email protected] solutions on tum: V. Kit qr
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c}+ I/ f] / 5 c Fe nman double-slit ex eriment of electrons: extremel lowintensi beam 4 ts Please draw the pattern appearing on the screen when there are 100000 electrons ﬁred one at a time. Both slits
are open. Please clearly explain your arguments based on your results in [5] (a). 12 We 5C: \nk/“CH “(Q 9mm Am +0 wwv Pmttck dhu‘l“,-wc \HH 'i Cuhrvdk‘ 5“ a band (MW‘W MM“ Hint, PW“ “VAC \{riqcs qr: Hunk/“hm din. 1 +6 JacthHooHic [Constioofwc web. fcf'PCCﬂvck/_ 5° 01% W‘W‘ls‘ i/k. o‘er/Hon Ethych him“. at Wang It \ ’ s .
Lowe) he Jun‘qu pakﬂglcj V‘M's Lot“ Suﬁ-5 “3. a waw’ in“)! kqsscwus as u rwhdc mm lcqug rm 3‘”: WV?” |5|(d) Effect of observation (4 pts) Assume that l have observed 100000 electrons pass the slits one by one and that I can tell which slit the single electrons pass. Please draw the pattern appearing on the screen. Both slits are open. Please clearly explain your
arguments based on your results in [5](a). \hc 30¢ \>uu:’f [31+er out 10 bbscrvrdiwx. ﬂ“: abﬁmhw} ‘WC Xe, o [Ml'ci POt‘H-(Vn Prof. Lin why 40 We sec or belief Pv‘ﬁtrn? Nobody imam. wk: an ’9‘23Crvsd-tah an claw” the results Hg
“15.) “\L SkOc‘cmy Pkc'mmcnon (“NMUH nagckunlg brmj)_ 13 / 5 e Fe nman double-slit ex eriment of electrons: one sin le electron ts Please draw the pattern appearing on the screen when there ' only one electro d. Both slits are open. Please clearly explain your arguments based on your result’ . KG “04 Sm] 0%: c‘cchon WC
III I. ad. a ‘Hhc in 5o“ \s a .mau: NH" "7 l/ we“, 9". c‘U-Vch btlluwb '45 (f H- £1415 O“, p a u
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f who do My} kmw _ chs/AV S‘hgq’ QM \5 “At? 4... 4K dchw) V0 saw. it "P \S bod/i a “4"?- “NJ WNW“?! LUNA Wavc‘loarﬂc'b Jhlwy'; [5|(f) Feynman double-slit experiment of bullets (4 pts) [Remarkz Since quantum mechanics is a more general, complete, and accurate theory than classical mechanics,
if you like, bullets can be described perfectly by quantum mechanics (i.e., as wavefunction). de Broglie matter
wave relation also confirms that bullets indeed have wave property. So, how come there is no interference pattern in the Feynman double-slit experiment of bullets? (No need to answer) Let’s answer this question through the
following exercise] Very roughly speaking, in order to observe an interference pattern in a double-slit experiment, the de Broglie
wavelength of the particles being fired one by one (e.g., bullets, electrons) must be about the same order of
magnitude as the slit separation length. An AK—47's bullet with mass 59 can travel up to 2000 m/s. Do you expect to observe an interference pattern in the double-slit experiment of AK—47’s bullets? Please support your answer
by calculation. . r )‘3 11‘ :;L _n
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‘ ‘ I. ' ‘ . . I . (a w [)dPC (I To t ‘L SQPQI‘Q‘KG': ‘C’ﬁ’k . [Extra Credit] Uncertainty Principle. as illustrated by the zero-point energy of FIB-1 EV/(total=5 pts) ’ Richard Feynman: “every statement in quantum mechanics is a restatement of Heisenberg‘s Uncertainty
Principle”. If your derivation in [3] is correct. you should ﬁnd out that the ground-state (n=‘l) energy for PlB—1 D is
larger than zero. Classical mechanics predicts that the ground-state energy for PIE-1D is zero. Please explain
why the statement “the ground state energy of a particle in a 10 box is zero” violates Heisenberg’s Uncertainty
Principle ﬁxAp 2 W2. Please clearly show your reasoning within the space provided. (Remark: This is another
failure for classical mechanics. Experiments confirm that PlB-‘l D and a lot of other quantum systems indeed possess zero-point energy.) l E “1 ht L‘ L > o
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- Winter '07
- Lin
- Quantum Chemistry