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Unformatted text preview: PHYS 115A HW 5 Solutions 1. First we need expressions for x and p in terms of creation and annihilation ops. A 1 h +
x: (a +a)
me0
ﬁzi‘ hwoghf—a) ,and r2 5 x: (a+a++a+a+aa++aa) 2mm0 Thus for the state 2> we get: <56>2=<25cl2>=1l 2 h <2(a++a)l2>:c0nst*<23>+const*<21>=O
=><5c>3=O Similarly: hwom 2 <2l(a+—a)l2>=const*(2l3>—c0nst*<2ll>=0 <13>2=i =><iy>3=0 For the x2 term we can automatically eliminate the (a+)2 and a2 terms. Thus, £2>22C<2l(a+a+aa+)l2>,where C: h
2mw0
£2>2=¢3C<2la+l1>+6C<2lal3>=ﬁxﬁc<2lz>+J§J§C<2I2>=c(2+3)=5cz2 5”
mwo 7h 2mm0 32>3zc<3l(a+a+aa+)l3>:\/§C<3a+l2>+JEC<3a4>=3C+4C27C= 2) a)
At time t = 0 the state is: Iw>=A(211>+3i2>) <wl=(i<l—3i<2)
<wlw>=1=lAl (4<:I,1>+9<212>)=13A 3A:— £3 Where we have used the orthonormality of the enero c,y eigenstates.
b) By expanding in energy eigenstates we can operate with the time evolution operator: _, , 1 —i l —: 1
IY’(x,t)>=e ”m: —(2 “+"Il> 3,e E’”l2>)
JE
3ﬁw0 5th
where E1: 2—
2, 2 c) Now we can use the same procedure we did in question 1. We must remember that
once we've acted H on an energy eigenstate then we get the corresponding eigenvalue
which is just a number. Thus a+ and a do not operate on them. <x(t)>:<‘I’(x,t)lx‘I’(x, t)> 1 h l I I I —l l —l
(26 E’”<1—3ie“’<2)(a +a)(2e E’”1>+ 3ie Wm) _13 2mm <x(t)>=1—\/§(2e’E"’”< 1I—3ieiE’l/ﬁ<2)(2Jae—iEl'/h2>+3iﬁe_iEZ'/h 1 >) For brevity I‘ve kept only the terms I know will not go to zero. C is deﬁned as in problem
1. l? <x(t)>=—(6ix/Ee_i(Ez_E')I/h<11>—6i6ei(EZ_E‘)l/h<2l2>) 13
12V2C . 12 h <x(t)>: 13 sm(AEt/ﬁ)=E mw 0 sin(AEt/h) For <x2> again the (a+)2 and a2 terms won't contribute as they increase the eigenstate by
two and thus give orthogonal states to our <2 and <].. (x2(t)>=%(2e5'<1I—3ieE2<2l)(a+a+aa+)(2e_E‘1)+3ie_E22>) Where I use E 1 and E 2 to stand for the entire exponent with those respective energies <x2(t)>:%(2e5'<1—3ie52<2)><(2e‘5'1>+121e‘EZ2>+8e‘E'1>+9ie‘522>) (x2(t)>=£(4+18+8+27)=57—C= 57h
13 13 26mw0 3) I think this problem is best done using the functions in the book. Either way you have to
do an integral and so bra—ket notation isn't so useful here as a calculational tool but it
helps set up the process without having to write out all the integrals. If we label the eigenstates of the original potential as In> and the eigenstates of the new
potential as n'>. Then the probability of going from the l> state to the m'> state is <m'1>2. The primed state with energy hw'/2 = hw0/4 is the 0'> state and the state with energy
3hw'/2 = 3hw0/4 is the 1'> state, simply because w' = w0/2. Then we just have to do the
integrals and square them. I'll go over some ways to do Gaussian integrals that you may
not have seen before. I1>=w,(z)=A1H,(c)e‘Cz’2
l0'>:‘l/ '0(CI):AOIHO(C')€_§'2/2:A0'Ho(C/\/E)e_zzl4 since C’z—f:
2 l1'>=w'1(§/l3)=Alwig/6);?” 2 (I) AIAO’ i LH1(§)H0(C/\/E)e_3gll4dz; 711(00— Prbllo.=l<0'll>lz= 2 h °° _2
AIAO' —f2:e “Md: =0 H1600— Since the integrand is an odd function over symmetric bounds. 2 Prblﬂl,:l<1’1>2=AA’ h —TH(?;) H(lj/\/—) e‘3z’4dc mwo 8T2 °° _
AHA' _J‘Cze 3:22/4dg
mwo °° Once way to do this integral is to note: That integral is easy to remember: 00 —0(x2 e—ux7 1 7T
:[oe dx— \/:=>:: x2 dng 5 Thus for our problem we can quickly integrate to get the answer: 81Th
2 X16 \/_
8 5 3mm
AlAl’ " ><i =AfAl'2—O—zl6 2~0.84
3mm0 3 9 27 4) OK, this problem is easily done using bra—ket notation (Dirac notation for those of you
who come to discussion session). 3) Although you don't have to expand in energy eigenstates to do this part and the next part it's a good idea because then you can use the orthonormality of the eigenstates to
simplify our calculations. It turns out we'll need to do this for part (c) anyway so we
might as well do it here. You can expand the given function in energy eigenfunctions in
many different ways. You can do the general process of using Fourier‘s trick to get the
constant coefﬁcients or you can use a trig table to expand and see that it‘s expansion just
happens to be in terms of energy eigenfunctions or you can derive the trig identity
yourself which is what I'll do here. 3 iX+ —iX 1 ix — ix ix —ix 1 ,
cos3(x)= e 26 :§(e3 +e 3 +3(e +e ))=§(2cos(3x)+6cos(x))
cos3(x)=i(cos(3x)+3cos(x))
=>w>=Acos3 ﬂ =Ay/i(l3>+31>)
a 32
2
5AI a 4 _1_ <ww>=1=lAlzi<1+9>= =A= =Iw>= 32 16 [5—61 (3>+31>) J17) b) Since we've already broken down the state into it's energy eigenstate components this
part is easy: Prb.l.=l<2lw>lz=o c) Having the state as a superposition of energy eigenstates makes this trivial: 2 2 2 2
—iF//h —iE//h 977' 71 'IT h
' ) where E3: and E1: 2
Zma 2ma2 ...
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 Spring '03
 Nelson
 mechanics

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