Download Problems and Solutions Quantum Mechanics - Y. K. Lim, World Scientific (1998, 751p) PDF

Title Problems and Solutions Quantum Mechanics - Y. K. Lim, World Scientific (1998, 751p) 19.7 MB 385
Contents
I-Basic Principles and One-Dimensional Motions
II-Central Potentials
III-Spin and Angular Momentum
IV-Motion in Electromagnetic Field
V-Perturbation Theory
VI-Scattering Theory & Quantum Transitions
VII-Many-Particle Systems
VIII-Miscellaneous Topics
Index

Document Text Contents
Page 1

Problems and Solutions
on Quantum Mechanics

Page 2

Major American Universities Ph. D.
~uali fying Questions and Solutions

Problems and 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 J e ~ e y London -Hang Kong

Page 192

368 Prob1e.m~ and Solutions on Quanturn Mechnr~ics

Specifically:
(a) What are the expressioiis for the first order corrections to the energy

level? (Do not atternpt to evaluate the radial integrals).
(b) Are there any rernairling dege~ieracies'!
(c) Draw an energy level diagrarli for 11. = 2 which shows the 1evc:ls l~cfore

and after applicat.ior1 of thc electric fieltl. Describe t,hc spet:tral liilcs t,l~a.t
originate from these levels which can 1)r o1)servctl.

( Cl~it:n!lo)

So lu t ion :

Write the Ha111iltolli;~rl of t8hc systc~ri~ as H = Ho + fl', whc1.e

taking tjhc dircc.tjiori of t,hc c!lcctric: fic:lcl E i1.s tlic: z t1irc:c:tioll. I7or ;I wc:a.k
fieltl, H ' << Ho ant1 we (:an trcat IT' ;IS pert,lirl~;~tioii.

Let (O,U), (1 ,0), (1, l ) :iil(l (1,-1) rc:prc!seiit, t,l~(: four (lcg(,ii(:r:~t(: c~ig(?iif'~il~(~-
t ior~s (1,7r1,) o f the stat(, 11 = 2 of t,hc hytlrogc:i1 ;~t,oln.

The matrix r~~)rc:sc?rlt;itioii of HI ill the S I I ~ ) S ~ ) ~ ( : C is

where

being the Bohr radius. Note that (I 'JH'j l) = 0 unless the 1 ' ) i sta.tes have
opposite parties.

Perturbation Theory 369

Solvirig t,he secular equatiorl

I -7u1 (O,OIH'/1,0) 0 0 I

we get four roots

(i,) As w(,'L) = w(I:i) = 0, th(:rc! is st,ill a two-foltl tlrgcr~c:r:~c:y.
((.) Figllrc 5.16 sllows thc: 71. = 2 c:ric:rgy 1cvc:ls. The sc:l(:c:tiori rul(:s for

elect,ric: tlipolc t,ri~llsit,ioiis i ~ r c A1 = 4Z1, = 0, 4Z1, wl~i(:ll giv(: ris(: to two
spect,ral liiic~s:

v L
with apptied E

Fig. 5.16

Page 193

3 70 Problems arid S011~tzons on Quantum Mechan t~s

Consider the n = 2 levcls of a hydrogcri-lik~ atoln. Snpposc the spins of
the orbiting particlc ant1 rlllcleus to he xcro. Neglect all rcllativistic c,ffc~-ts.

(a) Calculate t,o lowcst ort1t.r the ciiergy splitt,ings in thc: prcscncc of a
uniform magnetic ficl(1.

(b) Do the samc for thc! case of a uniforrrl c:lect,ric ficltl.
(c) Do the same for both felcls prcso~lt sirl~~llt,arlco~~sly ;1.11(1 i ~ t right

angles to each othcr.
(Any integral ovcr ratlial wave func:tior~s iic?rtl not t)c c:vitl1iatcd; it (:it11

be replaced by a p;~r;~rncltc~ for t>hc rest of the ( : i ~ l ( : l ~ l ;~ t i~~ i . Thc: same r i ~ i ~ y 1 ) ~ :
done for m y integral over angular wavc: filnction, orlccr you have: asc:crtai~lc:d
that it tlocs rlot vanish.)

( B c ~ r k ~ L : ~ )

Solution:

(a) Take thc: dir.c:c:t;iori of t,llc: llli~gll(:t,i(: fi('l(l iLS t,h(: z (lir(:(:t,io~l. TII(:II t,l~(:
Hairiiltoniari of tllc: syst,orrr is

whcrc V( r ) = -~ \$. Corisi(1cring H' = f2 as ~)(r r t ,~~rt ) i~t , io l i~ tali(! (:ig(:rl-
filnc:tioris for thc: 1l1ij)c:rturl)c~d st,nt,c:s arc.

As (H , 1" ,1) are still corlsc:rvc:d cll~antitics, ( ~ i . t , t n l i ~ ) , I I , ~ ~ I I , ) = ,rri,Ji, i ~ n d thc
energy splitt,ings t,o first ortlcr for 7,. = 2 arc givc.11 t)y

Per~urbatzon Theory 371

(b) The energy lcvel for n = 2 without c'orlsidering spin is four-fold
degenerate. Thc corrcsyondirig energy ant1 s t a t ~ s arc respectively

Supposc a uniforril electric field is apl)liecl alorig the z-axis. Take as

perturbation H' = eE7; = EoVr, w1ic:rc Eo = CEO,", V' = z/a" = rcosB/ao,
2 .

a,(, = + . S l l l ~ ~

H~l,rrL,,r,,lrrL # 0 for o~l ly A1 - &1, Ant = o. Hcnc:c thc: noll-var~ishir~g
elemcr~ts of thc: pc:rturl)ation rlli~t,rix arc

Lrt (H')200,zlo = (H')210,200 = El, i.e., Ho1 = Hlo = E', and solve the
a. lor1 secular c'qu t '

det I H,,, - E(') b,, ( = 0 .

The roots arc E(') = +El, 0,O. Hcncc the energy state n = 2 splits into
three lcvcls:

E2 & E ' , E2 (two-fold dcgcricracy for E2) .

(c,) Assutning that the magnctic field is along the z-axis a.nd the electric
field is along tile z-axis, the perturbation Harlliltorlian of t,he system is

where

Page 384

Index Index

rotational 6044, 7017, 7022
transition between 6049, 7021
vibrational 8009

Motion under linear restoring force 2008
p+e- atom in magnetic field 3048

-- Neutrino oscillation 8028
Neutron oscillation 5083
Occupation probability 5064
Operator

angular momcntum 3004, 3008, 3018
charge conjugation 1067
parity 7017

-
spin 3002

Particle in
circular rnotion 4016
one-dimensional box 1012, 1014, 3074, 5075, 5077, 6045, 6047
three-dimensional box 5055

Paschen-Back effect 5057
Phase shift 6005, 6012
Photon absorption by electron 6060
Photon-counting experiment 8012
Photon emission frorn atom 3022
Polarization vector 8022
Porphylin ring 70 19
Positroniurn 2013

decay 801 1
in magrletic field 5059, 5066, 5067

Potential
central 2017
delta-function 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1044, 6047
imaginary 1057
logarithmic 2010
one-dimensional 1036, 1038, 1041
periodic 1064, 1065
spherical shell 6004, 6008

spherically symmetric 2014, 8008
three-dimensional 2009

Potential well
double 1039, 7005
moving 5082
one-dimensional 1015, 1016, 1019, 1037, 1048, 2019, 2020
spherical 2018

Proton in varying magnetic field 6058
Cluaritization of electromagnetic field 8026
Quantum phenomenon

' experiment on 1004, 1006
historical note on 1005, 8003
in macroscopic world 1001
numerical estimate of 1002, 1003, 8004

Quark in box 2002
Quark-antiquark system 2015, 8019
Ramsauer-Townsend effect 6029
Rotation matrix 3010
Rotator 1013
Scattering

atom-atom 6002
between identical particles 6036
between particles with spin 6031, 6035
by bound particle 6038
by central potential 6003, 6006, 6023, 6025, 6026
by delta-function pot,ential 6004, 6017, 6018
by hard sphere 6001
by potential step 6013, 6016
by potential well 6007, 6022
by regular lattice 8015
by spin-dependent potential 6037
elastic and inelastic 6010
neutron-neutron 6032, 6034
neut ron-proton 6033
of alpha-particle by He 6009
of electron by atom (nucleus) 6011, 6024, 6028
of electron by symmetric charge distribution 6020

Page 385

Index

of nucleon by heavy nucleus 6019
of proton by atom 6027

Scattering amplitude 6014, 6015, 6030
Scattering differential cross section 6021, 6025
Schrodinger equation

.- - in momentum space 1007, 1041
one-dimensional 1008, 1018, 1055

Short questions 8005, 8006
Spin

matrices 3001
of deuteron 3035
of free electron 3005, 3017, 3035

Spin-magnetic field interaction 3043, 3044, 3045, 3046, 3047, 4014, 6059
Spin-orbit interaction 3007, 5058
Spin-spin interaction 3011, 3041
Stern-Gerlach experiment 3037, 3038, 3039, 3040 - System of particles in harmonic oscillator potential 5071
Theorem on eigenvalues 1061
Three-particle system 7008, 7036, 7037
Time reversal 4009
Transition probabilit,~ 6047, 6048, 6050, GO52
Transmission and reflection at

delta-function potential 1053
pot,ential barrier 1046, 1054, 1056
potential step 1049, 1050, 1051, 1052
potential well 1047, 1048

-2- Two-boson system 7007
Two-fermion system 7012, 7013, 7016
Two-hydrogen system 7029, 7032
Two-particle system 2016, 3012, 5062, 7002, 7003, 7004, 7006, 8024

in potential well 7005, 7009, 7010, 7011, 7014, 7015, 7018
in relative and cms coordinates 7001
in time-dependent potential 6061

Two-slit diffraction experiment 4015
Uncertainty principle 1036
Variational method 8016, 8017, 8018, 8020, 8021
Virtual phot,on emission 8035

Index

Wave function
changing with time 1010, 1011, 1031, 1035, 1040, 1042, 1043
under Galilean transformation 1058

WKB approxinlation 2015, 8008
validity of 8007