The following lecture notes for PHYS 414, Quantum...
Transcript of The following lecture notes for PHYS 414, Quantum...
The following lecture notes for PHYS 414, Quantum Mechanics II, arethe sole property of Glenn Agnolet, Physics Department, Texas A&MUniversity. They are not to be copied or distributed without the consentof the author.
0 a 0 1 cppv
{-k yt) ) c, cc,v cW%d \/(Y) v C
LL
GO
C 2E 2Z L) e
(x)
r vvi1hJ k,c,oS (LAi /Ly)’)
k= ±jn
\/CX)
+t soIv +L Lc4fic.
CJQ 1jGfl. Ae
ZV7
—
k
+
< i)> 1A LV
S fCcJVl 0 (4L4 to yç € j c.
4j<kI_tx)j’>!
—-— —
U S
L /L) o
J
—
cP
‘j,.?
(z)S-0 44Q. crJj n - vncZ ctvwvvs ocCir
$ - -,
k
e d cr ru 4
SoL
Ci€ b 4o S4tA4c -
E:E
( E _(E:’ -
tj / ±
2
Ev1
+
E± + )2-Wi
I V 4
11 ±) rI1 )2w1
L2-L
+ illvij
-,--
C. Cø
//
3S2-
M +L 1€ -) 7cr)a —
E ±
\I1 \ /
\ (\
0
E’ E/) \b)
4C) ( -q 0
lI
c&e be
un \i t1tr/4. _i e.
—
I ) (OS -v)
;1 2q11 (7rV)
(Con b
.4
±22 r o,tz.
4 d
I tecr1 s7. pdo -
9
6
F
C.
&JAJ1 C1V\ (i () +11 Stft9t 2
eIcc c
I4
3/
rf S
ch oc3 s
2,.J s’ 4
T Ae? todh +0 j
a
() j(-j.
L
c-eL E(k -
so -
___
co O ihf
iOA E+
3
-t1 i’
3e cctJt s
O
do O+
‘!2 L\ V)SJi0-
4L e It c L
4 u (o s
fl
Lo
Is
—-lc/
—lv,
T> D.
u I c c r+r, e
j:
I_J
L.
Io
o +(ec --
J (1‘)
I tçe) =
oct 4J p
j
p
on
OL cc1-f ftf%O
01
21
(; ka)21
w
sko k€reAc
c cA- C C *O Tx:k Cf
-— c4 C je4
C
_1o ioJic fhd
)<o,cL
it_/‘
bOLL
44J
7V
wt
/
3 7
5c,LVrr€ 1Jc:7.
& ETECtV R 2d2
( dS2
thJLi9) Ts
sas acq c 0.çS2. ).
r-ne v4. v 4kL ° d. 4ZkL rzLh
CGV/aç2
50 4 f o-i -LS ui c,V
T*
‘I
:i: sdSd
c or
4/
/
I
‘3
\1Okj; iç -£ aS2.
cS2.
e4 4t o c’4 5 -
1_f or- e a
o- so.wt c
4-{.a. e4zr- -UY?4-
.r[s
-
—
-
Cecr ø oocL.co.
44)
ri) iocSL\IL’’)
b
cc*.SL (43oLr
-ç4j =. EU’)
L t) +
1J11 ( = 31 E.
_3=9
ç nar,t
I
C
q1c tb’)
E( of -k.i aI
-g VL1)29
v1.
ic2k::L2flJ
Lfiv\) f)tw) i-Lu)
-v2(+4 * ‘q)
2.1
c3 q) 9%) E
I
E (4r+4s
+
3 vv,ce Ld l(XA.k
( S
2.
-- V - k4 -
29 LW) S
_
-
:‘ICy’) o.
IPkC
I(3
kID
A
lovtt LJvt
I ‘ c
4 (9 C
S
V 14 - )Lg
1 c.27V)
‘so
+ ) (- kL) ()
‘r5 CJ’ e?ah
3’3
j Jdk s()(kzz
____ ___
eC2c)31
z J
1’/ I t’
Ij
2W)
3Vc) e
Czc)3- ( J(2._k.L
()= j tJ’ ‘) e
-
i’:)’ (i-)
olZ
, zm’) C7,’)
1’ J. Lr-”)
C7cv’) ja
__
C27t (k—k)
j
ç ii) C?,
I)
VC7
3”.’
f WelkoI V2C1‘) KC7‘)
cEj
Jo C7(1’) c7cr.’,,.
0j0 c Ci sfsje t-Q L
o
L3OJtto.ç 4fkL.21. o. diro. ncho oucC-t
r j4 crij{vx so-w. -ç’c) Cc,
r ciV of C7 c ,‘
a.ku - C7c1T’) ck ov
aj[v C7’) i c1’)’)
ka F)
-
jrIC7(1’)) I 4hoj ec&4 So’it fv.).
Ar> °LJ Sô[).. k)
V’M’) i /
A .co t. cc eIop€i
2
V3uv”)
K c 1, h, I 4 ,
& soLh $k 2fcctVfrW,
___
11k
JOVtO.
f
U)3’ ) cLa 3 ( e.
(-I)
“(v ,4I) - ea
- (z)3J
.co
21r 2 c.LD Lr’I (t) 1
)3 j €
or r -( yJ
C)(j1) 2
—0
or3t)
U7t)3
(v’ _q”)
je
L2-T3 J
C.
(A) ç 44U ot(Q
,
(ypt)cQo
cz:c j’
—
2)1
0
CL,
LL)( Sc’nXlv”-r”l J0
C, r’)‘i /
9!VEt
COoLo ‘S
L I k1rr’1osV(Zic)3 j
I I_ ,kiv’-r’Ic05O1
0
CV AI)
C. -
Q
GO
JL kcL
(9”, 9)
(.lYt’)
a- JV1.ijV.
As :LL —tb) 40 ev’JcQ
Co
Yi‘e1 Cc. r’€ v.j -r 4- a,O 2.
A4 Sj t
Co1’v1 _ -
__
x x-LO
H Cs ccwc
- cO fo o o
Yx4.
r’ kj cj-’.
= —0
>
0 Cor r—- ee
e. e
0 ee4.
iidL .
0 0
- 0
che -
cLe-;- -j-.
-(0
&l4_.j 4. (\)j co4- ‘i cst€4. a
V?1O. G. e C Sc 4 —. c.
P- 7ØO e.
GO
1’c4xe (-s)
cO
,
2—
- x
I
J
v..,t Ca.6..
f=
J
-
(OS. _iØ’
C e. e.
Zete*
a
rviç ‘‘
SvyLL& c D On ro
e
crOj acL 1v kiio -k-c
s LA,’Ct -
4f0AAt. £0
00I
a’(e.
-Co
u Cc4. st
tcfn o 4
irFL 4tA a&1 ce Dlèf
de • - o
/2.j‘It.
0
cc b 4
-—
‘(2it)3 (k.-k2i
AOcw\ ft1Le Ck 4{ of 4a - s1Ck+
(r--j k7
0 Lxrj
r2d I I e-e I2
LTk-3 Jo L k -k
Co g -
I aL -e J-Co
Tve
— c2i J-‘
rS
II--I
eou
r5
(
36?
Ic
4s
coSp -lSL19’
e e
-
J1 vkc £o0
oO
( _±(-
(d 4-)(
‘u0
ICA ç,U j
1’
j<z2-
an, v e a. o a
Q
-
CCNAi
ILk
(S
_J
As b +L Sc V*A44s Sfr1c 3L e.
__
CO
So 4 [øJit ci’
t
kc 4 cy_j
R Cc- L civ’ ( 4t r
rCt &
+L.
0
j(.k
S -0+
7
Lit Lo-e )I,
e ç () €
(.Z0L. )
e 2?U4-
r
t{ Xt)1j 4 ch 10 Q Y L) Gk ‘I r
cO 9c) o C
>L5C)
Ori c c
o 2 2
—//1
OxGo
r=
2. —2-
io
r
(2ii
— ±__
_
Soc5
St(c?
4Lio Se(u 0 04 C()
)Z( J)( (e
-c i
*
€ €
s4 —
Z A. f{-t Jo- f e (c, fl ‘J -
L3 cAQ (LJL cxO”. & c*4 1&3 -Lf,
O c0CO
G J
I’
XcIX S’4
-cZ(2Tcij3
r- h) (AJt Co1 (O 4-t-L
tft, pj)tr 1c&-i 4v&‘-I
a I
L i(13 J
tCc kLc A-
ç
d cIo fe1- Y.kEoI- 4t- Lf4P-
-Re
eL
e
kj
AvcI So C-1 (r )L/;c I’1
DtL -tcd 11) icvt 1&
ht4 jojtc
-
4ic !v’)
£0 C7
Orc Cv” o
/
A sptLCL& )CY
o )4 sc4d c&
C1 u’i’) e
Ir
a’
—
—)ttr)
çz_
J’1
2’
L)
#& I. mo.
4K
e. (L’It)‘, )*)
r2’ ‘
A I
1
Icy)
rLjr
0r
qt)
4V Ii— IJe ccMh1l.
Sc
e e So
A
cLdQ 2J)*
i e—
)
ILJL)O
Tltr?4 o 6.
A‘ t1k)k o{
— a1d3
1’
____
v’.c’e
2rnw’) 4
e *
__
liJkc44 a0 4i )coL 44
A
dr rsvn9
4(9’) ck A
:r [4 ‘r J.C 4, T Lktosi9
jrLeI r j
3c-
Jr
r][
-‘LI(OS3 € +
—
___
-
-‘
e (.Y)tk(coIj
r
2
___
.f. IçL)li
)(
j±*s
YkL kJW% < & uIl ot1o.
c.o cci9 i o ‘t9o ]. T
Jcv kcL ç
cW +&
FkLA4Y
JTr
__
M(L
Tk. y s cc
2d<
9
j = k) I
(NA HcL/zrm
S 7.
O(ZS
K ft)( +
L((O’1)-or0-)1 ]CKcosz)) 4 — ]
r
-k4) Vc&h
r [ (Oc1 - coc2J 1.1 j r’72
0
2a z3)
LIr o.
2!
Iaic [ I] - 27r 1f(o,cp)j (J
‘S2
L
4trYyO Ye— (uço. ti. Yr. t)
(o3-j 154
(‘ç t)C) eT• ÷
2.- 1
A 4..f’1) I- f 2
•
*
[ 2
a
Lt a fc1k) (QL)
_
( ) e a + e 4 tCo,)
ckoo kr >2 1 4-L& Y— c/
%JSL ..s’ovt ‘rkL
tk’
e s04-4
- - - -T ç(,)
Coej.1
cI j rdS2. ZJ%
Vol.
J
___
4
1 9
__
e -e)(9-i)
V -C.)
r
___
‘19
2.- z-D I o.
b
.—
‘& Sc LcS
I
_____
f ••
VY) ‘- dfL)
As £ hc *i 4i 4wk flo cv 6 p4 ruc
yeo14
-_i__-
A.cI S°
qicl T1 ..ç(o1) o
y1
QPfl(4L EO’—
___
akJd )c )1
so O t)
—ri flX11CLc
I a cOvAaiQ - io iøc. a
g ccj1VW Y104 eiv -4-Lw*,
k c ci a 4
SccL1kA yyio34 vo 4 4BA. Sc4JLLA Cc3 o
Appz
A-klj E’ e÷
__
rJ
T ‘Icy”) O ( rfldiI1 SCC*IAD)4-t’ 4
e S,kkD );• fr Lk(%)
C.)t
L vrV) L+ .. I— — e.
Jit
__
I F
Ccj Scc&cA.tc. C’t -
soA I
4 )2vr L)
y- jQTc
(,k’)
J
___
— IA
(ILK’l 2kSu14.
1k-k’
I‘
zr_v’c)
Fj 44. css schc cW7’
frd
I-
f
i—or r”-” kCQ S ( 7 hcV s
1v.v.I11 ) k4-4 ce
1
f (ky)2vj
j7I
±
7c
LLT)
tJ CCA
r’cos 3
I r I[r-Vcr) sv’r
1k,D) f(, () ‘2ksw4
ohc
A
5 ec e r i.
L ‘ ( fls ‘
‘1J
‘Jo
6
t’ —t,kr’ ( I tL) (-;L-)
e — e so+e dr e
J LJ0 0
r II -
L— — a 2.
(p( -) (oL.+JL)j (‘1
•zCYkL)
2. 10/ 2.
+
ssiA1 (m/z
(4k÷ 3-)
( Ez-vTh
E÷
ci— — 0 oJ) E — c’ ( VtJ-j ‘k&L. 5 h
LL47çj
2.
12.1,of 4 (J0/df2cR4
(3t ec to* E +L S E
(A tC4Q W1 t
__
wi’- s 4k1
LooAD Gk O VO%4 *‘k*. CAt VLk9 +1c44
4 Q 41
Tv+vvc W c,iicJ 1t*&oeivv\
cT-a 0.
i. L4c- t S f oppvo rc4cwj TZ’ 4rki. Eov
1CAL #ic o 41*. 4-4
j, q(f)
secth oA. TE fO L’Y -,oTiM Q.
(A) CLA.ft. (Irvr CkYO Ir1C4Vh% ) 4 rt%iV 4 44*i.
Jk•4 C 4
(
-) ÷ f—t j Lflt
k•re.
4! zm’k)_k r
J zpc r’ Jwt
+ a 2ovJu’)
I —LIL ‘ (k- k)r”
I — ( e e ]J qç j iç. )
2.
A t)
AAC4Q o
tSt o*n o -vti
Svvc4A ‘,JbeA ‘ft j144.
ce c4j
4jjj
scoj Ice.
P0h avt
b 44 so I
- v 41(%) ÷ /c) iJ) E)zvT1
d9 corc 4 S
T- \C ( ft sfrc \rt[i-?, L1 L?L ° 44
(‘ Q €1344.0 4. ç !_. .
cr) )9j
=
_
co iD.
e3jahc1 fr+ I (r(r): ELr)—
LY7r
& coor E -c
(- ka
Cr o
r) L (K r)
(kr) OA o 4L Jo (cL 4 -CCSY
x
jt -iv7 d hL-
//
/
LL LE
‘II
[ IL.
fl)
>>
—Scrvl ct .;‘ L ck
“e t4 Tn f
(Z’L L )
ck1rQ. P ((os ) ) &i
o( C
we s. -
-e-J 4 a
-Le
kr
_(kr-
e + ) tcoQ4)
____________________
/o
rtD.
4r) k1
I (k-1 _rQ]e.
__
+ zb
i 1oos LJ Q iS r 4L)
e e
r
(k2+( k) CLos)).
±_ /2)
ii
So a o
__
S c CM
//
T e
A +t4L +D4 kj riif ‘ctk
..(kr 4J)1[ e e J Pos)
=
-((kr Q7)
— E Pcost7)
[
____________
-r kk e Q(cost) +
Co
-
0k r
T. Cvv1- o’v 4L of o £pk€AL S-)Lc1
(2
T eM1 +L C— scmc
CcM b Yr-s
SrtLiw k-s- vcIu of
T (co9)
2‘
0
,-14Ec r?orckf
2-
.e
(ba- Qp’)-
2. (Lr-Q))1
= ) (t ajZ tbl)
ioc C ‘“D )= UQi
so Scivc
3D
— Sfr)k c
Fos).
e
-J
-
.. 2L)
‘ e2
(zH) e
co.Lf(
4 (2) (i-ezkr
e coi3)
COr4
L k.-
(2-’) e - Coz) Lc)c c T.
z7c A
±k2
b
() (coca) s[,Tt(cas)I
AOaI
J 52 (cc, ) P SLLt zQ+i
(L4
bu = (*1) ( 1)1<
7
sv dg
r QPT1cAL- 7HO(ZE’
AncL
--
c = = rrv
fzrv w nk z. A+ to
= T b ,
cr epd r cIkj
SccLL.
rro
Th,
T ccj LJ L*•
-° D&oJjti’4. L
LL s0h I f q(
2-
no
c_k1-j
t% rj-) Jçj
/
7,—
2 2.—
I
7 ‘ i Vcr) C7v) =
tt fU
[ r r Qvyr
0
dej ? /I 1 (/ / — 7 a (vLr(7(r3yr>
L 7 rj/âr ‘dr -
/
C7(°) j(o) O
1 (L 7 ‘‘ - drj j& rctr)5r)
0
A14
ir (Vr)
(2
b )
C(r -j/k_(‘ Q%)
- e
os noh
iCr) L e s( -
[C / ) (2+i) k
74r
(Jg
j C7r
50 k4
c)- L(2i) €
0
O( C]v 0k
4r + (k2 t))c7 o c7
C7 tn (r-)
Si d oeo o
Aibf C Li
0Z4 2
zyy1 I\/C) cr) (kr)
‘Jo ‘Jo
o k ICr)
r- C2e
VLY
Jo
Ic) /( ,
L\/
2.—
1’2rr cAL
ItIt
(1+z)
ç)Ac-
+
cs)s,+o5ç
}t
Cso)‘r1>>b>jJ
S\4ç(l+Tz)—
(coIGO
z
Q))S
(i4Z
CJoO4—’A
ç4za
cf)
w:
jct. o‘‘:‘) Y? kL.4 C CJ4
crj A (J(- —kb)
o
ykc4t sf+
—
g0
cr, I !b << ch i4)
E<i1rLL.
0
r (toc Ev )1t
CC1CU -
4
+ (kZ—
- 2V)sIU) cCr)o.
0
4 C +i{ ) < (r O
j<V<oS
A4st
z.
àrz
B> (03k r
V’Lf)o çr r> Sk4 S.
Cb0 c
“4’ s tQ C (a +)
c o(k4 £.
-h’i k1O
(ko (%)k
-k ÷ t4[
1+
_
--
(kc - -i c ciA J)< ?0 kc So.
-- (1 - J_
1.
Z. / -frAmkNLpcc. ( I - I
K2’
3
Qc4A. Sc L(y
I L FrSE:
a.ma cT
+CA,Y ko kc
kO
4- k-fkc.
‘1
2.(tkd s1).
A4 +t*A:,
+(JkA
/