Problems And Solutions on Quantum Mechanics (Major American Universities Ph.D. Qualifying Questions and Solutions)

385 Pages·1998·30.19 MB·English

Problems and Solutions
on Quantum Mechanics
Major American Universities Ph. D.
~ualifyingQ uestions and Solutions
- .
and
Problems Solutions
on
Quantum Mechanics
Compiled by:
The Physics Coaching Class
University of Science and
Technology of China
Edited by:
Yung-Kuo Lim
World Scientific
Singapore New Jemey London Hang Kong
PREFACE
Solving problems in school work is exercise of the mind and enhances
understanding of the principles. In genera1 amhation questions wudy
pardel such problems, Thus workrng out problems forms an essential and
important p& of the study of physics.
Majw Arnm'can University Ph.D. Qptdifging Questiow and Sol~tlons
is a series of seven volumes. The subjects d each volume and the respective
refem (in parenthem) are as foUows:
1. Mechanics (Qiang Yuan-qi, Gu En-pu, Cheng Ji&fu, Li Ze-hua, Yang
D*tian)
2. Electromagnetism (Zhao Shu-ping, You Jun-ban, Zhu Jun-jie)
3. Optia (Bai Gui-ru, Guo Guang-can)
4. Atomic, Nudear and Particle Physia (Jin Huai-cheng, Yang Bao-
xhwg, Fan Yang-mei)
5. Thermadynamim and Stat isticd Fhysim (Zheng Jiu-ren)
6. Quantum Mechanics (Zhang Yong-de, Zhu Dong-pei, Fan Hong-yi)
7. Solid State Physim, Wtivity and Miscellaneous Topics (Zhang Jk-
lu, Zhou You-yuan, Zhmg SM-ling]
This series covers almost all aspects of University Physics and contains
2550 problems, mast of which are solved in detail.
The problems have been carsfully &asen from a collection of.3100 prob-
lems, of which some m e h r n the China-U.S.A. Physics Examination
and Application (CUSPEA) Program, some were selected from the PBD,
Qudifyiag Examination on Experimental High Ener$y Physics sponsored
by Chao Chong Ting. The rest came h m t he graduate preliminary or
qualiiing mamination questiom of mven world-renowned American uni-
versities: Columbia University, University of California at Berkeley, Mas-
s~,&usett s Institute of Technology, University of Wisconsin, University of
Chicago, Princeton University and State University af New York, Buffalo.
Generally speaking, examination problems in physic8 in American uni-
versities do not involve too much mathematics. Rather, thy me to a large
extant characterized by the fallowing three aspects. Some problems hmlv-
ing variaus hntisr subjects and overIapping damains of science are selected
by professon d i t l y from their own research work and thus have an %up
t&ten flavor. Some probIms invoIve broad Eel& and require a quick
vii

I. BASIC PRINCIPLES AND ONE-DWNSIONAL MOTIONS
Quaat- phen~m%na&m ~ WMI@ g&ls in Cite d p s Worl d.
Show this numehdy for the ful~wiryy
(EL] Tbw amnlihde d the z+mwlnft@ t&b for .a pendulum d length
1=1rtiand-m=1kg.
(bj The tmmhg4p robability for .a mbleo f m = S g m&# at s
~ d l O ~ m / w ~ a ~ o ~ o P h ~ $ H = 5 c r n a n d w i d t b
w = l s
( ~ ) ~ $ i ~ t i o n o f a ~ ~ o f ~ m = O . 1 & m ~ & a :
qxd u=0,§ mi* byawhhtlfsk 1 x Ma2.
( WisWwiA)
T~ItHb - pint ~ O Xof aT -pie ~ penddtim is mgLQ$bf,
(b) If we regard the widthand hei& of the rfgld obhcle as the width
d ~ a f @ ~ E a ~ bia&te$,~ thpe ~op ~r o b a M U ~ i s
That Is, the' tunding pwbatr'i br marble is mWly mm.
~ c ) T h & B ~ ~ o f ~ ~ b a l l i s

(*I
~ teffa. ~ c
(b} Bkdcbadg.-*-
(c) hm&Hmb expmimmt.
(d) exptshmk
(el ampton -&zing-
-
Wutfoflr
(a) E f f d
T h h ~ ~ ' & a m i a g i a s o f ~ ~ ~ w ~ o n e ~ ~ a
m e t d u n d t # : ~ ~ ~ftlws&sfw~nd.tb t thema@ku&t
of theahchic cum& tSws producedis proportional ta the oftbe
~ ~ ~ ~ k h e h q ~ o f ~ ~ i s g w a t m t b u a
~ ~ e & a r a e ~ o I ~ ~ w h t l e t b ~ u f t h e ~
dinsnstdepandohthe@t intedy,butonitshquencf. Theeremlts
* o ~ f b o ~ b e ~ b ~ ~ p ~ .
~ i n X g Q S ~ ~ ~ b y a a s r t m i a g ~ , h ~ i n t p r r -
~ n w i t h n t & t t e r , ~ d d f ~ D f ~ ~ , ~ p ~
~ a , ~ ~ ~ # n i ~ ~ r r m & * ~ I t i a t ~ ~ ~
a d t h e e l ~ , * d ~ t ; h e ~ b , ~ a n ~ d w p &
. W & f e i-b b E E I ~ ~ ~ l ~ m & t ~ ~ ~ d k v e s ~ h a h & ~ ~
-
A W b n d y k s a e w M J l ~ * & t b e ~ n ~ ~ n i t , ~ b
~ ~ ~ a n o £ t h e d l a t b n . e m i t ~ ~ n ~ ~ ~ a a b e & v e d
f i x l m t h e ~ h o f ~ b e t m m m a t k a a d r a d i r r t i oThae
w a ~ ~ ~ ~ r m e d ~ h e t e e t r o n r r b y ~ ~ d&~r~~(~i~P&~&~ teare&able anddo&emit&ht. The problem them
Mdeat hem wsrr obteJlred by tm?.bng-m thro*aa&md ~ b ~ a ~ w ~ o d d d ~ t ~ & o m f r o m a m i t ~
putentid. K n o a t i n g t h e ~ d t h e ~ b ~ i t ~ ~ t e t o light. d a e 3 w m a i n l a i t w .
deducean a@m&dPalrn for tbg ehhm wttvelqthtrnd A bdc prhciph of Qmnhm Mdmiw that, *but @ x t ~4Wd -
w e r e i n ~ ~ d w i ttbhed eBm&e~icmA=h /pl '1~- hie ~ ? l , t h e ~ ~ a h a a a E o n r Ti hS iWi ~~ t~b t
P ~ t s ~ m d p i t h e ~ aShfnkatpebr- ~ s n a & m i n a a d ~ ~ t g ( f l a & ~ s E a t e ) w u t d * yon andnot
b t s er are W performed by o t b w ith beams of helium atam and emit light spoa-e. h d t y ,w ,sp nWious &dtbn of gill
~ ~ d ~ ~ t ~ t ~ ~ sxdtedsatQmdtaesocmrrr8adliughtia~ ~ ~ m ~ ~
tode&mm. Aucmdhg to Qwmm mmdym,t he Enhrw4ion d the db
tionMdd~&cbminanatom,w hi&bmtwuqumtum8ystam,
(e) Oompkon Scattering ~~
a krln Of fbe Fihgbphtoa cr&h opera* of, w w do ss not
C0mpt-m o b m dt he wt&n&d X-rays by frPa (or d ybou nd)
~ h ~ i f t h e r e i s n r p ~ ~ Iit~rEtblisite~rm fhat&ahma
amusa ndhund tbe araveIeqg& 0 f t b em~dM~Ofl ~x& tw in aaas taw u tE M, ape- t r d * ~ .
of the h&bt radiation. The difke~mAA nuied as a fm&n of* mgie
8bW~mtkeinddefftand~&e&bm~
~~an~entinwhkhabeamof&t~ol3akdimctda~a
p h t e c o r r t ~ ~ s l i t s , I ~ A w asdeyBO.~ ,dthapl&laasar;een
~ ~ ~ h a n ~ o f d e b e c t o r s w M & e n a b ~ ~ n e ~ ~ ~ w ~
theektmmhit h s m be~ac~hof~ t h e f o ~ ~ & w a m u ~
pa& of tb dative nudma £ incident ehcbns as a f&n d p d t h
along %hem e aa nd giw a b Ma planatth.
(a]SlitAopm,&Bcbd
(b) Nit B o m ,al it A c b d
Xothe~~re~tum~abigtheorebtmlpmbIemwask [c) B&h &%a opea.
%pa an atam from emitting li$ht. &pldn+ Mar Quantum Merddm, Id) 9kn4kIaehv appamtw &adied the s l i i~n su4 a muwar
a bii bmtW pmbhw aa to make atoms in mdt9d stam em& lighE. t M ~ a W - r # i & n s = A / 2 a p ~ t ~ A s a d ~ ~
-lain. w l & ~ ~ ~ & m 8 P r m i t ~ ? ~ t h= -h/2 can pam through B.
(w-1 {a}Ody~Wbhs,=&/2cal3~thrOugEtAmctonly~
Sol*: Sri& am = h/P Can th-h R*
w w t a ~ ~ o f ~ t h e b e a m ~ % y # h t h a t ~ o r r e
h t h e d a J r s h h r e ~ M ~ , ~ t a ~ R u t h m h r d
atomic modd ektmllpl rnwe mutld the d e w ri n ellI@cal orW, Clad ~ i ~ ~ t h m q h t h e a p p a F a t l l s & atrimrey?
(Cobribin)
d & t P o ~ ~ ~ ~ n t o b e a m i t t e d w h a ~ p a a t i a e
d w - . Thus the atom mu& W Eght. TbJs W the htmm
would lclEse wntSdmw?y and ui,tiaataly capturd by the mdms+
~ i o ~ u a l ~ E h e e l s c t r o n s & d f s l J ~ * P n c l s t r s ~
1m
A partide af mas m i~ mbj@ ta a force F(r) = -W(r) auch
t*h-a t the wm WCtjoEt *,6) eLti&es the m v t m wSc hahgm
[ . p B / ~a-v ;r I P ~t,) = *[pt t)W,