r.
For example,the wavefirnction for a fiee particle traveling in the positivex directionis r2(;r): p of the particlebyp = hkl2z, Sincee'o: cosa* lsina, Ae^, whereft is relatedto the momentum the free-paiticlewavefunction canbe written V= Ae'o : Acos(k) + iAsin((a). Both the real and imaginaryparts aresinusoidalandboth havethe samewavelength2, relatedto kby k:22d2. The complexconjugateofa complexnumberhasthe samereal part asthe original numberbut its imaginarypart is the negativeof the imaginarypart of the original number.Thus,the complex tl/R- W a\dthe complexconjugateof e'k is e-'k. conjugateof ry: typ+ itylis f: The squareof the magnitudeof a complexnumberis found by multiplying the numberby its complexconjugate:lAt: Vi. Thus,the squareof themagnitudeof qzr is
lv4': (v^+ivl(vn- iwD:vi + ,y?.
It is the sum of squaresofthe real andimaginaryparts.The squareofthe magdtudeofthe freeparticlewavefunctionis 14"'bl2: A"tbA"-ib: A2eo: 42, where we have assumedI is real.
v
If a particle has energyE, its completewave function Y is the product of a coordinate-dependent function and a time-dependent function. The time-dependent firnction has the form ei't, wnere a) is the angular frequency. Since ar 2nf nd.f: Elh. where/is the frequency, the angular frequencyis relatedto the energyby at:2rE/h. Thus, Y: tyei''.V and,r4leadto the same probability density:
= ,y yl ei''e'''t = ,/ =1,//'. lYl' : lrlr i "t''l'
Questionsand Example Problems from Chapter 38 Question1 Ofthe following statementsaboutthe photoelectriceffect,which aretrue andwhich arefalse? (a) The greaterthe frequencyofthe iacident light is, the greateris the stoppingpotential.(b) The greaterthe intensityofthe incident iight is, the greateris the cutoff frequency.(c) The greaterthe work function ofthe targetmaterial is, the greateris the stoppingpotential.(d) The greaterthe work function of the targetmaterial is, the greateris the cutoff frequency.(e) The $eater the frequencyof the incidentlight is, the greateris the maximumkinetic enerry ofthe ejected electrons.(f) The greaterthe energyofthe photonsis, the smalleris the stoppingpotential
(")E-
(!)i+,e-] (.) i-B-!-]
?^r-&f , ta,-la YP,",-"
(r) l}^-](")
d=r*
----),(F,"-,"= hf-f
(f) i:s-_h/
Question2 In the figure be1ow,why are the minima in the values of I ry I' grealet than zeto?
Probabiliq'., densityly(x)l'
^4-eo.toJ Jr.^ra.,{;
ra4af,Aa'n
)*- *.qtUt-*.-$
Ifu rw'rd.o^t
-r rq^,t .&o u-* rl.
r*d q"l
Question3 particlesa1lhavethe samekineticenergy.Rankthemin orderof Thefollowingnonrelativistic greatestfrst: electron,alphaparticle,neutron. their deBrogliewavelengths,
KF= 7lnv"=?h->
P=
!
ur.'.r"I,'.qrg "
,X W-4"
\ = n/f
Q*w"g, a"uar)
{Til'
-^o ";". KE p a,*^" 4o, e-z'.A ) [Ju &,2;,1r, r], J]'o qr-\ ?
fa>p">f.
probrem, I\"riJ[l An ultraviolet lamp emitslight of wavelength400 nm, at the rate (power) of 400 W. An infrared lamp emitslight of wavelength700 nm, alsoat the rate of 400 W. (a) Which lamp emitsphotons atthe greaterrateand(b) what is that greaterrate?
A-*" ot *)S,^ A F}J-."
ot'r: -p,vnfid"
=
,-.ta A,',N"A ntn"'l r^-e*'ftt) -D./t%t'dn \
A = % F--h*=\'/x A = PYn. ( a) qrvrU \
-- g^P-t 4
Qoonq Xrrn^p ,\aa o/t-lA
(u)
B= >
hc-
= (Ltoo5/e)(1oo,^) ( leto.v".-) (r.a, t6'15/"t)
^-*" he = lt.tloeV"nm
Problem2 A spectralemissionline is electromagneticradiationthat is emittedin a wavelengthrange narrow enoughto be takenasa singlewavelength.Onesuchemissionline that is importantin astronomyhas a wavelengthof 21 cm. What is the photon energyin the electromagneticwave at that waveleneth?
F= ( o.at*,)
\-'
E = 9.u7,.rd's5 &'t _,r,r"t",tr"g hc = l?.qO = A.x't-\) A
Problem 6 Considera collision betweenan x-ray photon of initial energy50.0 keV and an electronat rest, in which tki photon is scatteredbackward and the electron is knocked forward. (a) What is the energy of the back-scattered photon? (b) What is the kinetic energy of the electron?
d= rco"
o 11lv-) E= So.ofev o-+ ^
€v^\f\
V*.r
A\=
YE'(
E'
A''13^ld'"rq
h/*..=
(\-r^d)
= (2,-u:'lo'"')(t-o'rlo")
= 4 . 8 l o" l d l r - ,
(q)
A = h^
\ ra ) = f,.9?* 1il'rr",--J
\'=
o.oaq!'n..,= A.'.t1,td"n
h./A = ]3jo:!I9*
------\=
'=
\c/, /^
=
@.oA9J n m
E, = 4.lt".lOq.V _- 4t.8"-V ( b) -Vq
'$ t ,naqft -a
.rr^r-'Xa
KE= 5o-oX"V-t1.8F"y'
problemT
=
( ll alo e V.nr.r
(E"E-E/ 8.2,Kev
Gammaraysof photon energy0.511MeV aredirectedonto an aluminumtargetandare scattered in variousdirectionsby looselyboundelectronsthere.(a) What is the wavelengthof the incident gammarays?(b) What is the wavelengthof gammarays scatteredat 90.0" to the incident beam? (c) rMhat is the photon energy ofthe rays scatteredin this direction?
= o"9ll rlobeV
(u)
n\=
./.r/\
h fieC
h'/g\ =
(") 4:=hf = h.1>*
E = O"5il MeV
>=
(
\1-1'^Q)
"l .),q
O eV "nr.l
\
- a.al3" lo
O"5tr r lDte V
';\ =
1,'r3 p.^
({o.r,3nlo-31T"*')_,
( r_c-o,
- a.KL
e
=
- ?.L
LbJL* -
*
e
-J L'l
errar"i(u"-a
h*
3'rratq(U'-A)
oo tl
"r" (\)"-g) l\
n
1L^
(u"-r) = J,:!*?* ----'E= oo n^') ( 1L') (g
b'(.p"rTI" 3e rI-Ltrt
hc.= laqDe-v'nm
g1,'n*l*(m.ct)
E = G"oov
( Tlrr o at/.n,n)"(ir_a:o9lg.D 3ln* (o.?o",')\ ( o . 5 l l x l o b e V )
A= 5.I.V
i I
