Notes_de_cours Quantique

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Z

eipx/~ −ip2 t/2m~ dp f (p) √ e 2π~

x\|h4e¬V~

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Z

e−ipx/~ f (p) = hp|ϕ(t = 0)i = dx hp|xihx|ϕ(t = 0)i = dx ϕ(x, t = 0) √ 2π~ Z Z −ipx0 /~ ipx/~ e e 2 √ ⇒ ϕ(x, t) = dp dx0 ϕ(x0 , t = 0) √ e−ip t/2m~ 2π~ 2π~ Z Z 0 i(x0 −x)p itp2 ϕ(x , t = 0) = dx0 dp e− 2m~ − ~ 2π~

x\|h4ezV~

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ΠA=

−t m~

Á B=

x0 −x ~

⇒

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2 m~ 2πm~ −iπ/4 −i(x2~02−x) (−t) e e t

Z

ϕ(x, t) =

dx

0

r

x\|h463 ›~

m −iπ/4 i(x0 −x)2 m e e 2~t ϕ(x0 , t = 0) 2π~t

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ˆ −iHt/~

ˆ (t, 0)|ϕ(t = 0)i = ϕ(x, t) = hx|ϕ(t)i = hx|U

–(N5D6n ⇒

hx|Uˆ (t, 0)|x0 i =

Z

ˆ (t, 0)|x0 i = hx|U

dp hx|Uˆ (t, 0)|pihp|x0 i =

r

Z

Z

dx0 hx|Uˆ (t, 0)|x0 i hx0 |ϕ(t = 0)i {z } | ϕ(x0 ,t=0)

0

ipx/~ −ipx /~ ip2 t e e √ dp e− 2m~ √ 2π~ 2π~

m −iπ/4 i(x−x0 )2 m e e 2~t 2π~t

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x\|h463{KV~

0

{|pi}

x\|h4632m›~ …§ : ϕ(p, f (p)e ¾ÈNV8›:E<wW5Á/Z [ W h“A<wWunwn=DF>58WU8<wWu“A<wTun=WU8›¯:ªN:=DFt)>V8 ={|xi} “‡Wu@i: C :E<=Wb:=<=kungBU>V`a“iZFDF‘;@ATUW54 f (p) = ϕ(p, ¯ t = 0)

2

ip t − 2m~

4

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1/4

p0 m

0

2

+∞

2

−x2 /4σ 2

2

−∞

|ψi

1 hψ|ˆ x|ψi = √ 2πσ

Z

+∞

dx xe−x

2 /4σ 2

=0

−∞ Z +∞

1 1 √ 2 2 hψ|ˆ x |ψi = √ 2π dx x2 e−x /4σ = √ 2πσ −∞ 2πσ p ⇒ ∆ˆ x|ψi = hψ|ˆ x2 |ψi − hψ|ˆ x|ψi2 = σ 2

mi4

>Z [ >V8©NQ@i:ED6ZFD6n=TŽZ6WŽv†N5D6:S‘;@AW ¯ ψ(p) =

xe−Ax

2



1 σ2

3/2

WUnw:p@A8AWJv&>V8ABC:EDF>58+DF`-“/N5DF<wW54

= σ2

x | 463{±V~

r  1/4 σ2 2 σ 2 e− ~2 (p0 −p) ≡ f (p) ~ π

 TU`a>V8inw:E<=N:ED6>V8 ”

1 1 1 ¯ √ ψ(p) =√ 1/4 σ (2π) 2π~

Z

+∞

| −∞

dx e−ipx/~ eip0 x/~ e−x √ {z π B 2 /4α2 e = α

2 /4σ 2

}

x | 463}’5~

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Œ"€

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α2 =

1 4σ 2

WU:

B=

i(p0 −p) ~

x\|h4632¬›~

2 1 1 1 √ − σ2 (p0 −p)2 ¯ ~ √ π2σe ⇒ ψ(p) =√ σ (2π)1/4 2π~

hˆ pi = p0

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Z

{|pi}

”

+∞

¯ ∗ pψ(p) ¯ dp ψ(p)

−∞

= p0 = p0

Z

x\|h4632z›~

Z +∞ 2 ¯ ¯ 2 (p − p0 ) dp ψ(p) + dp ψ(p) {z } | −∞ {z }

+∞

| −∞

=1

=0

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0

Z

+∞

eipx/~ − ip2 t ¯ √ dp ψ(p) e 2m~ 2π~ −∞  1/4 Z +∞ √ px p2 t σ2 2 2 1 1 √ = σ dp e− ~2 (p0 −p) +i ~ −i 2m~ π ~ 2π −∞  1/4 Z (p−p0 )x √ p2 t σ2 2 2 1 1 ip0 x/~ +∞ √ e = σ dp e− ~2 (p−p0 ) +i ~ −i 2m~ π ~ 2π | −∞ {z }

ψ(x, t) =

x\|h4LK ›~

I

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I=

Z

+∞

dp e −∞

2

− σ2 p2 +i px −i ~ ~

(p+p0 )2 t 2m~

x\|h4LK 3}~

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2

it 2m~

Œ‡¡

ip0 t m~

ix ~

√




−

ip20 t 2m~

Á/WC:pN5DF8AnwD

π B22 e 4α I =e α ( 2 it α2 = σ~2 + 2m~ 0t B = ix − ip ~ m~ r    σ4 t2 t~ 2 α = + exp i arctan ~4 4m2 ~2 2mσ 2 r    4 i t~ t2 4 σ + exp arctan ⇒ α= ~4 4m2 ~2 2 2mσ 2
2   2 x − pm0 t B2 1 p0 t x = − 2 2i~t =− − 4σ 2 2it 4α2 ~ m~ 4σ + m + m~ ~2 ip2 t

0 − 2m~

N2­5WuB

n=>VDŸ:

x | 4LKVK5~

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 2 1/4

σ 2π

e

»

− 2i arctan (

q 4

σ4 ~4

t~ 2mσ 2

+

ip2 t

0 )− 2m~

–

t2 4m2 ~2

× eip0 x/~ × e

−

(x− pm0 t )

4σ 2 + 2i~t m

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x | 4LK5mV~

2

p0 t m

4"§¨8W1WU:uÁ

"
# p0 t 2 2 x − 1 σ q hψ(x, t)|ˆ x|ψ(x, t)i = dx x exp − 2 m2i~t 4 2 2π 4σ + m σ + 4mt2 ~2 | −∞ ~4 {z } J " "
2 # 2
2 # 2  Z +∞  Z +∞ x − pm0 t x − pm0 t p0 t p0 t J= dx x − dx exp − 2 2i~t exp − 2 2i~t + m m −∞ 4σ + m 4σ + m −∞ | | {z } {z } 

2

1/2

Z

+∞

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|ψ(x,t)i

=1

2

 2  2 1/2 p0 t 1 σ q hψ| xˆ − |ψi = m 2π σ4 + ~4

t2 4m2 ~2

Z

" x2 dx x2 exp − 2 4σ +

+∞ −∞

# 2 2i~t m

x | 4LKV|5~

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α α − a+ib − a−ib

−

2aα a2 +b2

" #   2 1/2  Z +∞ 2 2 σ 1 p0 t 8σ x q h xˆ − i= dx x2 exp − 2 2 4 2 m 2π σ t 16σ 4 + 4~m2t −∞ + 4 2 2 ~ 4m ~  2 1/2 √ σ2 1 σ q 2πA−3/2 A= = 2 2 2π σ4 t2 σ 4 + ~4mt2 + ~4 4m2 ~2  4     2 1/2 √ σ ~2 t 2 t2 σ −3 2 2πσ =σ 1+ 4 2 = + 2π ~4 4m2 ~2 4σ m

N{­5WuB

∆ˆ x|ψ(x,t)i = σ

⇒

r

1+

x\|h4LK5’V~

~2 t 2 4σ 4 m2

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N 

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 W @



−

~2 00 ψ (x) + V (x)ψ(x) = Eψ(x) 2m

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−

~2 00 ψ (x) = (E − V )ψ(x) 2m

ψ(x) = Aeipx/~ + Be−ipx/~

N2­VWUB

p2 =E−V 2m

ÁAnw>VD6:

p=

p 2m(E − V )

x\|h4LK¬›~ x\|h4LKz›~ x\|h4em›~

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N2­VWUB

−

~2 =E−V 2ml2

ÁAnw>VD6:

E=V

ψ(x) = Aex/l + Be−x/l

Œ‡Œ x | 4emi3{~

~ l=p 2m(V − E)

x | 4em›K5~ §¨8;:=<=WI?AWu@
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1

0

2

0

V (x) V2

V1 0

x

x0

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 (x)

x | 4em5™›~

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x0 −

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0

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0

0

0

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⇒ ⇒ ⇒ ⇒

p(x1 −x2 ) ~

p(x1 −x2 )

− e−i ~ =0   p(x2 − x1 ) =0 sin ~ p (x2 − x1 ) = nπ 2m(E − V1 ) ~  2 nπ~ 2m(E − V1 ) = (x2 − x1 ) 2 2 ~π E = V1 + n2 2m(x2 − x1 )2

x | 4emVzV~

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–(N5D6n

x | 4^™ V~

2ipx1 ~

= Aeipx1 /~ eip(x−x1 )/~ − e−ip(x−x1 )/~   p(x − x1 ) ipx1 /~ = 2iAe sin ~

p=

nπ~ x2 −x1

⇒

ψ(x) = 2iAe

ipx1 /~



x − x1 sin nπ x2 − x 1






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x2

dx |ψ(x)|2 = 1

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x\|h4^™Vm›~

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ψ(x)

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~ 2m(V2 −E) −x/l

1

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1

2

2

1

x/l

2

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( ψ x1 : ψ0

: :

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: :

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⇒

−

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+

+ 1 l 1 l

ip(x2 −x1 )/~

⇒ ⇒ ⇒

–(NDFn

⇒


1
ip 1 − eip(x2 −x1 )/~ = 1 + eip(x2 −x1 )/~ ~ l ~ 1 − eip(x2 −x1 )/~ = ipl 1 + eip(x2 −x1 )/~
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Á/>58Nž?A>V8iB

2

p2 + ~l2 = 2m(E − V1 ) + 2m(V2 − E) = 2m(V2 − V1 ) 2m(V2 − V1 )   p(x2 − x1 ) p2 2 1 + tan = 02 2~ p 2 1 p   = 02 ⇒ p cos2 p(x2 −x1 ) 2~

⇒

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4_¾Š>Vnw>V8An

x\|h4L| ›~

p20 =

x\|h4L| 3}~

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1

2~

ip ~ ip ~

1 l 1 l

−

= −eip(x2 −x1 )/~

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~ i   = ipl tan p(x22~−x1 )   p(x2 − x1 ) ~ = − cot ⇒ pl 2~   1 p(x2 − x1 ) p2 2   = 02 = ⇒ 1 + cot 2~ p sin2 p(x2 −x1 ) 2~

⇒

x | 4L|5mV~

  p(x − x ) 2 1 = p sin 2~ p0 cot



p(x2 − x1 ) 2~



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n=D x < 0 n=D x > 0

x | 4L|5±V~

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0

ip ~

C l



–(N5D6n

1+

ipl ~

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ip2 x/~


−ip1 x/~

ψ ψ0

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WC:

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⇒

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j= x>0

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∗ ip1 x/~


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ip2 ∂ψ = |C|2 ∂x ~

⇒ j=



x\|h4e±5±›~

p2 p2 |C|2 = |A|2 T 2 m m

p1 (1 − R2 ) = p2 T 2

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1

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: :

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1
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: :

Eeipx2 /~ = Ce−x2 /l + Dex2 /l
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x | 4L’V~

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]\_M1 ^F ] 1“ >5@A<S?ATU:=Wu<w`aD68AWu<SZ6WunSDF8iBu>V8A8;@AWUn E WC: B ”

x\|h4L’5KV~ > det(S) Wunw:žZ6Wº?ATC:EWu<w`aD68/N58›:Q?AW+ZFN(`aN:E<wDFBuWa<=WU“A<=TUn=Wu8›:EN58›: Z6W nwf nt:Eku`-W©WU: det(S ) Z6W `u`-WŽ?ATU:=Wu<w`aD68/N58›:g?/N58AngZ6Wu‘;@AWuZc>V8(NÉ<wWu`-“AZ•NBuTbZFWUnSBu> W-BuD6Wu8›:Eng?AWJZFNÉD BU>VZF>58A8AWb“/N5<SZ6W `-Wu`É A<wWb?AWJ?A<=>5D6:EW ~y 4‡Ž8NQ?A>V8AB xi =

det(Sxi ) det(S)

xi

−x1 /l e−ipx1 /~ −ex1 /l −Aeipx1 /~ ip −ipx /~ −e 1 −x1 /l 1 − e e − 1l ex1 /l − ip~ Aeipx1 /~ ~ l det(Sxi ) = 0 −e−x2 /l −ex2 /l 0 1 −x2 /l 1 x2 /l 0 e −le 0 l

Ž8?iTU­VWUZF>V“i“1WJ“/N5<S<=N5“A“‡>V<w: 7žZ•Nɓi<=Wu`-DFkU<=WbBU>VZF>V8i8AW5ÁA?c[^> det(Sxi ) = e

−ipx1 /~

=−

4ip A ~l

§…:pZFWbB2NZFBu@iZ?AW



ip ip − Aeipx1 /~ − Aeipx1 /~ ~l ~l

det(S)



x\|h4L’$™;~

0 0 eipx2 /~ ip ipx2 /~ e ~




ip ∆x/l
1 ∆x/l −∆x/l −∆x/l =e e −e − e +e l2 ~l  
1 ∆x/l
ip ip∆x/~ ip ∆x/l −∆x/l −∆x/l + e e −e e +e − ~ ~ l          1 p2 ∆x 2ip ∆x ip∆x/~ × 2 − 2 + 2 cosh × − =e 2 sinh l l ~ l ~l ip∆x/~

 $'&(E)>WI.I@3`Z43



O

x\|h4L’m›~

  ip −ipx1 /~ A ipx1 /~ A ipx1 /~ + e − e − e ~ l l

?A>V8i8AW

−x1 /l e−ipx1 /~ −ex1 /l ip −ipx /~ −e 1 −x /l 1 − e e 1 − 1l ex1 /l l det(S) = ~ 0 −e−x2 /l −ex2 /l 1 −x2 /l 0 e − 1l ex2 /l l = ...





x\|h4L’5|V~

Œ   A!}!#&Ä;!}%('%»ÂŽ/0{% .10}#&% ¥
%;0a_,O.O.O% ¥ >Z [ >V8©N-“‡>VnwT ∆x ≡ x − x 4/¯ŠDF8/N5Z6Wu`-Wu8›:2Áh>V8©>V i:=DFWu8›: 2

⇒



1

− 4ip e−ip∆x/~ A ~l  E=
2 × l12 − ~p2 + 2 cosh 2 sinh ∆x l |E|2 |A|2 16p2 = 22  ~l 4 l12 −

|T |2 =



2 2

p ~2

∆x l




× − 2ip ~l




x | 4L’5±V~

1 sinh2

∆x l




+

16p2 ~2 l 2

cosh2

∆x l




p2 ~2 l 2 |T |2 =   2 2 1 p 4p2 (x2 − x1 ) 2 (x2 − x1 ) − sinh + cosh2 2 2 2 2 l ~ l ~ l l 4

⇒

– u`-Wžn=DWuZ6ZFWžN+@A8AW-Tu8iWu<=j5DFWž“AZF@inH AN5n=nwWa‘;@AW-Z•N+ /N5<w<=D6ku<=Wx E < V ~CÁµ@A8iWa“/N<w:ED6Bu@AZ6WžN ?A>V8ABp@A8AW „V%;!{0{/#†.O%aNJ!}"ÊiÃÈ#†¥\#ƒ0{-'%+¥†+0{!} ‡%;!)U% ! 4 —[ Wunt:_BuWp‘;@[ >V8ºN5“i“1WUZFZFW D F IF^L LTU6P
 PdF^D 4YWC:=:=WQ“A<=>V AN5 ADFZ6D6:=TJ:=Wu8A? 8iT2N58A`->VD68Anp­5Wu<wn  n=DŠZ•N Z6>V8AjV@AWU@A<—?iWZ•Na /N5<=<wDFkU<=W |x − x | :EWU8A?­5Wu<wn +∞ ÁA>5@nwDµZ [ Tu8AWU<=jVD6WŽ:EWu8i?­5Wu<=n V Á"N5@A‘;@AWuZµBuN5n p → 0 4 2

2

1

1

E > V2

 N58AnSBUWbB2N5nUÁiD6Zcn=@
-:g?iWŽ<=WU`a“AZFN5BuWU<_ZFWUn_W h“‡>V8AWU8;:=DFWUZFZFWUn_<=TUWuZ6ZFWunY“/N5<_?AWUnSW h“‡>V8AWu8›:=DFWuZ6ZFWUn

Bu>V`-“AZ6W hWunÉ>5@ Wu8ABU>V<=W ?AW <wWu`-“AZ•N5BUWu< “/N5< p ÁA>V8©:E<=>5@i­VWbœ/8/NZFWu`-Wu8›: p = 2m(E − V ) 1 l

1

1

|T |2 =

(p21

+

2 p22 )

sin

2



ip2 ~

N{­5WuB

p2 =

p 2m(E − V2 )

4p21 p22    p2 (x2 − x1 ) p2 (x2 − x1 ) 2 2 2 + 4p1 p2 cos ~ ~

4¨s D >V8œ1>VnwW

x | 4L’V’5~

¦

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‡   


®   
    



   

   
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x†±i463{~

ˆ2 ˆ = p~ + V (ˆ r) H 2m p r = x2 + y 2 + z 2

>  §¨8º`aTUB2N58iDF‘;@AW—BUZ•N5nwn=D6‘ @iW5Á Z•NJn=>VZ6@i:ED6>V8a?iW—BuW“A<=>5 AZFkU`aWp<=WU“1>VnwW—nw@AV`aWU8›: BuD68ATU:=DF‘;@AWJWunt:BU>V8AnwWu<w­5T541§¨8W cWC:2Á/ZFWb`->V`-Wu8›:SBUDF8ATC:EDF‘;@AW L~ WUnw:p?ATCœ/8ADµ“/N5< ~ = ~r ∧ p~ L ~ dL d~r d~ p ⇒ = ∧ p~ + ~r ∧ dt dt dt p~ = ∧ p~ +~r ∧ F~ m | {z }

x†±i4LK5~

=0

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¦A€

¢¼1,‡% ¶Å% .10a'O/.O) ,O.Ž/0{% .10}#&% ¥Yˆ% .102!$A¥ @     6N
 


    
0 N   W @    M¨[ >V“1TU<EN:=Wu@AV`-Wu8›:gBuD68ATU:=DF‘;@AW L~ˆ = Lˆ , Lˆ , Lˆ  Wunt:?iTUœ/8AD“/N<

 














x

y

z



 yˆpˆz − zˆpˆy ~ˆ = ~rˆ ∧ p~ˆ = zˆpˆx − xˆpˆz  L xˆpˆy − yˆpˆx

†x ±i4^m›~ Ž8y­5Tu<wD6œ/WaN5DFnwTu`-Wu8›:H‘;@AWaZ6WunJ>V“‡Tu<=N:EWU@A<=n n=>58;:IRAWU<=`-D6:ED6‘;@AWunU4§¨:EN5 AZ6DFn=nw>V8AnbZ6Wun <=kUjVZFWUnS?AWJBu>V`-`É@i:EN:ED6>V8+Wu8›:=<=WJZ [ >V“‡Tu<EN$:EWu@i<SLˆ`a,>VLˆ`-,Wu8›Lˆ:gBUDF8ATC:EDF‘;@AWbWC:pZ [ >V“1TU<EN:=Wu@A<S“‡>VnwD6:ED6>V8 ” x

i ˆx, x L ˆ =0 h i ˆ x , yˆ = [ˆ L ypˆz − zˆpˆy , yˆ] = −ˆ z [ˆ py , yˆ] = i~ˆ z

y

z

h

⇒

h

i ˆ i , xˆj = i~ijk xˆk L

x†nw>V`-` N:=DF>V8 n=@A
†x “‡Wu<w`É@i:ªN$:EDF>58An_“/N5D6<=WUnp?iW-3{K5m›~ N2­5WuB  x†“‡Wu<w`É@i:EN:ED6>V8An_D6`a“/NDF<=WUn_?AWž3}K5mV~ ?/N58AngZ6WungN5@i:=<=WUnBuN5n Ž8:=<=>V@h­VWJ?AWJ` N8ADFkU<=WŽnwDF`-DFZ•NDF<=WpBuWUZFZ6WungN2­VWUBIZ\[^>V“‡Tu<=N:EWU@A<SDF`-“A@AZ6n=D6>V8 ” ijk

 123 231 312   =  =  = 1 : 132 = 321 = 213 = −1   ijk  =0 h

i ˆ x , pˆx = 0 L h i ˆ Lx , pˆy = [ˆ ypˆz , pˆy ] = i~ˆ pz ⇒

h

x†±i4 ™;~ x†±i4e|V~

i ˆ Li , pˆj = i~ijk pˆk

sh@i<—Z•Nž /N5nwWI?iWBuWUn—“A<w>V“A<wDFTU:=TunUÁ‡>58(“‡Wu@i:?ATu`->V8›:E<wWu5`a`É@h:EWHN2­VWuBZ [ R/N5`-DFZ6:=>q 8AD6Wu84 h

ˆ z = xˆpˆy − yˆpˆx L i ˆ z , V (ˆ L x, yˆ, zˆ) ϕ = xˆ [ˆ py , V ] ϕ − yˆ [ˆ px , V ] ϕ ~ ∂V ~ ∂V =x ϕ−y ϕ i ∂y i ∂x   ∂V ∂V ~ x ϕ −y = i ∂y ∂x

x†±i4^±›~

F€ •¤žÇ¨;!}i0{% ,!+¢¼1¶%h.‡0©ˆ#†.O›0}#†‹,% %›0 ¥ 
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2

2

¦/¡

2

∂V ∂V y ∂V x ∂V −y =x −y =0 ∂y ∂x ∂r r ∂r r ˆz ⇒ L V. ˆx L ˆy ⇒ L V. ~ˆ 2 = L ˆ 2x + L ˆ 2y + L ˆ 2z ⇒ L V. x

 WH“AZF@AnUÁA>V8N

Bu>V`-`É@i:=WJN2­VWuB Á BU>V`-`É@i:EWU8›:pN{­5WuB Bu>V`-`É@i:=WJN{­5WuB

h i h i 2  ˆ z , pˆx = [ˆ ˆ  L x , p ˆ ] p ˆ = i~ˆ p ⇒ L , p ˆ py pˆx + i~ˆ px pˆy = 2i~ˆ px pˆy x y y z x = i~ˆ       h i h i ˆ z , pˆy = −i~ˆ ˆ z , pˆ2 = −2i~ˆ L px ⇒ px pˆy L y      i h i h   ˆ z , pˆz = 0 ˆ z , pˆ2 = 0  L ⇒ L z h i ˆ z , p~ˆ 2 = 0 ⇒ L

Á Á ÁABu>V`-`É@i:=Wu8›:pN2­VWUB

ˆx L ˆy L ~ˆ 2 ⇒ L

p~ˆ 2

x†±i4L’5~

x†±i4e¬V~

x†±i4ezV~

Á Á WC: L~ˆ Bu>5`a`É@h:EWu8›:pN2­VWUB H.ˆ ¾ÈN5V8AD6Wu8 Un [^W h“A<=D6`aWbnwDF`-“AZFWU`aWU8;:_Wu8©v&>V8ABU:=DF>V8©?AW L~ "4 §¨8W cWC:2Á ⇒

ˆx L ˆy L ˆz L

2

ˆ 2 = (ˆ L ypˆz − zˆpˆy )2 x = yˆ2 pˆ2z + zˆ2 pˆ2y − yˆ pˆz zˆ pˆy − zˆ pˆy yˆ pˆz |{z} |{z} =

yˆ2 pˆ2z

+

zˆpˆz −i~

zˆ2 pˆ2y

yˆpˆy −i~

+ i~ˆ y pˆy + i~ˆ z pˆz − 2ˆ ypˆy zˆpˆz


~ˆ 2 = xˆ2 + yˆ2 pˆ2 + yˆ2 + zˆ2 pˆ2 + zˆ2 + xˆ2 pˆ2 + 2i~ (ˆ ⇒L xpˆx + yˆpˆy + zˆpˆz ) z x y − 2ˆ ypˆy zˆpˆz − 2ˆ xpˆx yˆpˆy − 2ˆ z pˆz xˆpˆx 2 2 2 2 2 2 2 ypˆy zˆpˆz − 2ˆ xpˆx yˆpˆy = ~rˆ p~ˆ − zˆ pˆ − xˆ pˆ − yˆ pˆ2 + 2i~~rˆ · p~ˆ − 2ˆ z

x

y

− 2ˆ z pˆz xˆpˆx  2 = ~rˆ 2 p~ˆ 2 − ~rˆ · p~ˆ + i~~rˆ · p~ˆ

∂2 ~2 ∂ 1 ~ˆ 2 − ~2 2 − 2 ⇒ p~ˆ 2 = 2 L r ∂r r ∂r

⇒

1 ~ˆ 2 ~2 ∂ 2 ~2 ∂ ˆ + H=− − L + V (r) 2m ∂r 2 mr ∂r 2mr 2

x\±h463V~

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@ A~pnwWu<=N5D6:—?AWÉBªRA>VD6n=D6<Z\[^>V“‡Tu<=N:EWU@A< −i~ 4µ–(N5D6n BuWC:p>V“1TU<EN:=Wu@AV8“‡>Vn=W Aˆ = −i~ ÁAD6Zc­ DFWU8;: ” 2 2 x

1 ∂f r ∂θ θ

x

2

∂f ∂r

x

∂f ∂r r

1 ∂f r sin θ ∂ϕ φ 2



2



2

2 r

2

2




r



∂ ∂r

∂ ∂r

hϕ|A† |ψi − hϕ|A|ψi = hψ|A|ϕi∗ − hϕ|A|ψi Z Z ∂ ∂ ∗ = i~ ψ ϕ d~r + i~ ϕ∗ ψd~r ∂r ∂r Z π Z 2π Z +∞ ∂ = i~ sin θdθ dϕ r 2 (ψϕ∗ )dr ∂r 0 0 |0 {z }
        
 

∂ ∂ r = −i~ ∂r + 1r pˆr = −i~ 1r ∂r

x†±i463{KV~

Wunw:pRAWU<=`-D6:=DFWu8c4A§¨8W cWC:2Á

¾ÈNV8›:E<wW5Á/Z [ >V“‡Tu<=N:EWU@A<

hϕ|ˆ p†r |ψi − hϕ|ˆ pr |ψi Z π Z 2π Z = i~ sin θdθ dϕ

MNQ“/N5<w:=DFWb<=N5?AD•NZFWŽnU[^TUBu<=DŸ:pN5ZF>5<=n ”  Z 0

0

+∞

r

0

2



∂ 1 + ∂r r



∗

ψϕ +



∂ 1 + ∂r r

    ∂ ∂ 1 1 ∗ ∗ r + ψϕ + + ϕ ψ dr ∂r r ∂r r 0  Z +∞  2 ∗ ∂ ∗ 2 (ψϕ ) + ψϕ dr = r ∂r r 0 Z +∞ ∂ 2 ∗ (r ψϕ )dr = ∂r 0  +∞ = r 2 ψϕ∗ 0 = 0 rψ → 0 r → +∞ 0



∗

ϕψ



+∞

2

n=D

“1>5@A<

WU: 4

x†±i4632m›~

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~r xˆ yˆ zˆ ∂ = .~ p = pˆx + pˆy + pˆz Aˆ = −i~ ∂r r r r r x ˆ y ˆ z ˆ ⇒ Aˆ+ = pˆx + pˆy + pˆz 6= A r r r

x†±i463u™;~

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1 2

~r r

y

1 2

~r r



~ ~r f ∇. r

 WH“AZF@AnUÁ

z

+



=



~ ~r ∇. r



~r ~ f + .∇f r

1 ∂ ~ ∇.(g(r)~ er ) = 2 (r 2 g(r)) r ∂r

?[ >

~ ~r = 1 ∂ (r 2 ) = 2 ∇. r r 2 ∂r r

]—D68An=D\Á

    1 ~r ~r ~r 1 .~ p + p~. f = .~ p+ 2 r r r r

Ž8WU8?ATU?A@ADŸ:g‘ @iW pˆ WUnw:pn=D6`a“iZFWu`-Wu8›:SZFNÉv&>V<=`-WŽnwf `-TU:=<=DFnwTuWH?AWJZ [ >V“‡Tu<EN$:EWu@i< r

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  2   x = r sin θ cos ϕ  r y = r sin θ sin ϕ ⇔ tan2 θ     tan ϕ z = r cos θ

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= x2 + y 2 + z 2 2 2 = x z+y 2 = xy

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∂ tan2 θ ∂xi

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ˆ+ = L ˆ x + iL ˆy L ˆ− = L ˆ x − iL ˆy L

 ∂ ∂ + i cot θ = ~e ∂θ ∂ϕ   ∂ ∂ −iϕ − + i cot θ = ~e ∂θ ∂ϕ iϕ



~ˆ 2 = L ˆ −L ˆ+ + L ˆ 2 + ~L ˆz L   z   ∂ ∂ ∂ ∂ iϕ 2 −iϕ ˆ −L ˆ+ = ~ e − + i cot θ + i cot θ e L ∂θ ∂ϕ ∂θ ∂ϕ      2 ∂ ∂ 2 −iϕ iϕ ∂ iϕ ∂ iϕ ∂ =~ e −e −e i cot θ + i cot θ e ∂θ 2 ∂θ ∂ϕ ∂ϕ ∂θ   ∂ ∂ eiϕ i cot θ + i cot θ ∂ϕ ∂ϕ  2 ∂2 ∂ ∂ −i ∂ − eiϕ i cot θ + i cot θieiϕ = ~2 e−iϕ −eiϕ 2 − eiϕ 2 ∂θ ∂ϕ∂θ ∂θ sin θ ∂ϕ  2 ∂ ∂2 iϕ ∂ iϕ iϕ + i cot θe + i cot θie i cot θ + i cot θe i cot θ 2 ∂ϕ∂θ ∂ϕ ∂ϕ   2 2 ∂ 1 ∂ ∂ ∂ ∂ = ~2 − 2 + i 2 − cot θ − i cot2 θ − cot2 θ 2 ∂θ ∂θ ∂ϕ ∂ϕ sin θ ∂ϕ   2 2 ∂ ∂ ∂ ∂ − cot θ − cot2 θ 2 = ~2 − 2 + i ∂θ ∂ϕ ∂θ ∂ϕ   2 ∂ ∂2 ∂ ∂ ∂2 ∂ ˆ2 2 2 ~ − cot θ − cot θ 2 − ⇒ L =~ − 2 +i −i ∂θ ∂ϕ ∂θ ∂ϕ ∂ϕ2 ∂ϕ ⇒

  ∂2 1 ∂2 1 ∂ ˆ2 2 ~ L = −~ − 2 + + ∂θ tan θ ∂θ sin2 θ ∂ϕ2

x†±i4L| ›~



FŒÀ% )
A!$¶Å‡.O#\‹,% )aÆNJ !$#†‹,%h) Y>V8AnwDF?ATU<=>V8Ang@A8iWJv&>58ABU:=DF>V8©“A<w>V“A<=WJ?AW

‡¡

”

ˆz L

x\±h4L|h3{~ –(N5D6nS“1>V@iVD6:g@i8AWbv&>V8ABU:=DF>V8©@A8iD6­V> ‘;@AWJ?AWunpBu> >V<w?A>V8A8ATUWunUÁ/DFZcv†N5@h:g‘ @iW ~ ∂ψ(r, θ, ϕ) = ~mψ(r, θ, ϕ) ⇒ ψ(r, θ, ϕ) = χ(r, θ)eimϕ i ∂ϕ

x\±h4L|VK5~

ψ(r, θ, ϕ + 2π) = ψ(r, θ, ϕ) ⇒ ei2πm = 1

Wu8›:ED6Wu< MµWbn=“‡WuBU:=<=WH?A@ 1 BO1 F^PRL@^Q PdY^LTQTSVURF2BE] VQ L _`D Wunw:gœ/8AN5ZFWU`aWU8;:g?i>V8A8ATJ“/N5< ” (¿ÀN5Z6Wu@A<wnS“A<=>5“A<=WUnp?iW ~ˆ : ~ l(l + 1), l Wu8›:=DFWu< x\±h4L|5mV~ L ¿ÀN5Z6Wu@A<wnS“A<=>5“A<=WUnp?iW Lˆ : ~m, m = −l, . . . , l opWU` N<=‘;@AW ” PO>V@An BuWUnY>V“‡Tu<=N:EWU@A<=n 8AW?iTu“‡Wu8A?AWU8›:S‘;@AWŽ?AW θ WC: ϕ 4i—[ TU:EN5D6: “A<wTU­ DFnwDF AZ6Wg­ @ ‘;@AW—Z6WunBu>V`-“‡>VnEN58›:=WunÀ?A@ `->V`-Wu8›:BuD68ATU:=DF‘;@AWgBu>V`-`É@i:=Wu8›:YN{­5WuBZFWS“A<w>h?A@iD6:À“/N5<8[ DF`-“1>5<w:EW ‘;@AWuZ6ZFWÉv&>V8ABC:ED6>V8 ?AW r 4  @yBu>V@A“cÁ>V8·“1WU@i:JBªRAWU<=BªRAWU<ZFWUn—v&>V8ABC:ED6>V8Anb“A<w>V“A<=WUnJBu>V`-`É@A8AWUn[7 ~ˆ WC: L ˆ BU>V`-`aW-v&>V8ABU:=DF>V8inÉ?AW θ WC: ϕ @A8AD6‘;@AWu`-Wu8›:24…MW “A<=> ?A@ADŸ:É?[ @A8AW+:EWUZFZ6W-v&>V8iBU:ED6>V8 L “A<=>5“A<=WJ“/N5<S8c[^D6`a“‡>V<w:=WŽ‘;@AWuZ6ZFWŽv&>V8iBU:ED6>V8©?AW r n=WU<ENžWu8ABU>V<=WJv&>V8ABC:ED6>V8+“A<=>5“A<=WJ?AW L~ˆ WC: Lˆ 4  [   
  
 
   YRAWu<wBªRA>V8AnÈ?ATunw>V<=`aN5DFnZ6Wunv&>V8ABU:=DF>V8inŠ“A<w>V“A<wWunÈBU>V`-`É@A8AWUn7 L~ˆ WU: Lˆ ?iW ­N5ZFWU@A<=nO“A<w>V“A<=WUn WU: 4  [^N5“A<wkunpBuWJ‘;@ADµ“A<=TUBukU?AW5Á">V8“1WU@i:pZFWUnSBªRAWu<wBªRAWu<n=>V@AngZFNv&>V<=`-W ” ~ l(l + 1) ~m x\±h4L|™›~ Y (θ, ϕ) = F (θ)e MNQv&>58ABU:=DF>V8 Y (θ, ϕ) ?A>VD6:gn=N:ED6nwv†N5D6<=W ” ˆ Y =0 L x\±h4L|V|5~   ∂ ⇒

m

2

2

z

2

z

2







z



2

z

2

l

l

m

m l

imϕ

l

+ l

–(N5D6nSZ [ Tu‘;@/N:=DF>V8©?AD cTU<=Wu8›:=DFWuZ6ZFW

NQ“1>V@iVZF@h:EDF>58

⇒

∂θ

l

− l cot θ Fll (θ) = 0

cos θ df =l f dθ sin θ

f (θ) = (sin θ)l

4"§8W1WU:uÁ

df l cos θ = l cos θ(sin θ)l−1 = f dθ sin θ ⇒ Yll (θ, ϕ) ∝ (sin θ)l eilϕ

x\±h4L|5±V~ x\±h4L|V’5~

A³ >5<=`aN5ZFD6nEN:=DF>V8 ” shD Y (θ, ϕ) = c (sin θ) e Á"N5ZF>5<=n

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l

l

l

l ilϕ

2

Z

π

Z

2π

dϕ(sin θ)2l = 1 0 0 Z π 2 sin θdθ(sin θ)2l |cl | × 2π × Il = 1 Il = |cl |

sin θdθ

=n >VDŸ: Ž8v†ND6:SZ6WbBªR/N58AjVWU`aWU8›:?iWJ­N5<wD•N5 iZFWun

N2­5WuB

x†±i4L|¬›~

0

u = cos θ, du = − sin θdθ, Z 1 du(1 − u2 )l ⇒ Il =

(sin θ)2 = 1 − u2

x†±i4L|z›~

WU: I n=WHB2NZFBu@iZFWŽ“/N
l

I0 = 2 Z Il =

1

du(1 − u2 )l−1 (1 − u2 ) −1 Z 1 = Il−1 − du u × u(1 − u2 )l−1 "−1  # 1 Z 1 (1 − u2 )l (1 − u2 )l = Il−1 − − u − du 2l −2l −1 −1

1 = Il−1 − Il  2l  1 = Il−1 ⇒ Il 1 + 2l (2l)(2l − 2) · · · 2 2l Il−1 = × I0 ⇒ Il = 2l + 1 (2l + 1)(2l − 1) · · · 3

wn >VD6: I = [(2l)(2l − 2) · · · 2] (2l + 1)! >V@Wu8ABU>V<=W I = 2 (l!) (2l + 1)! ¯ÈD68/N5ZFWU`aWU8›:2Ái>58?i>VD6:pN2­V>5DF<

x†±i4e±›~

2

l

2l+1

× I0

2

l

|cl |2 =

(2l + 1)! 1 4π 22l (l!)2

MµWJBªRA>VD ºBu>V8›­5Wu8›:ED6>V8A8AWUZŠBU>V8An=D6nw:=W7Q“A<wWu8A?i<=W

(−1)l cl = l 2 l!

r

(2l + 1)! 4π

x†±i4e±h3}~ x†±i4e±VKV~

FŒÀ% )
A!$¶Å‡.O#\‹,% )aÆNJ !$#†‹,%h) ‡Œ MµWun N5@i:E<wWun¨v&>V8ABC:ED6>V8An¨nu[ Wu8a?ATu?i@ADFnwWu8›:Y“AN5<ÀN5“A“AZ6DFB2N$:EDF>58Énwf nw:=Tu`aN:ED6‘;@AWg?AW Lˆ Á›n=N5BªR/N58›: ‘;@AW 

−

§ iWU`a“iZFWun ” 3V4 l = 0 Kh4

N{­5WuB

1 ˆ − Y m (θ, ϕ) Ylm−1 (θ, ϕ) = p L l ~ l(l + 1) − m(m − 1)   ∂ ∂ −iϕ ˆ L− = ~e − + i cot θ ∂θ ∂ϕ

1 Y00 (θ, ϕ) = √ 4π

x\±h4e±VmV~ x\±h4e±5™›~

l=1

r 6 1 Y11 (θ, ϕ) = − sin θeiϕ 2 4π r 3 sin θeiϕ ⇒ Y11 (θ, ϕ) = − 8π r !  1 cos θ 3 iϕ 0 iϕ −iϕ sin θie Y1 (θ, ϕ) = √ e − cos θe + i − 8π sin θ 2 r 3 1 2 cos θ =√ 2 8π r 3 0 ⇒ Y1 (θ, ϕ) = cos θ 4π

mi4

⇒

Y1−1 (θ, ϕ) =

r

x\±h4e±›|5~

3 sin θe−iϕ 8π

l=2 Y2±2 (θ, ϕ) =

r

Y2±1 (θ, ϕ)

=∓

Y20 (θ, ϕ)

r

=

15 sin2 θe±2iϕ 32π

r

15 sin θ cos θe±iϕ 8π


5 3 cos2 θ − 1 16π

x\±h4e±V±V~

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ogWu`aN5<=‘;@AWUn ” 3V4pMµWun R/N5<=`->V8AD6‘;@AWun-n=“ARiTu<=D6‘;@AWun+nw>V8›:º?AWUn v&>V8ABC:EDF>58An “A<w>V“A<wWun ?c[^>V“‡Tu<=N:EWU@A<=n RAWu<w`aD q :=DF‘;@AWunU4"§¨ZFZFWUnSn=>V8›:p?A>V8iBH>5<w:ERi>VjV>V8/NZFWun ” Z Z x†±i4e±V’V~ dθ sin θ (Y ) Y = δ δ dϕ π

2π

l

0

0

m †

m0 l0

l,l0 m,m0

Kh4pMN?ATu“‡Wu8i?/N58ABUWWU8 θ ?AWUnRAN5<=`->V8AD6‘ @iWun+n=“iRATu<wDF‘;@AWunWunt:<=WuZ6DFTUW 7?AWun©v&>V8ABC:EDF>58An “/N<w:ED6Bu@AZ6DFkU<=Wun_N5“A“‡WuZFTUWun >RBEDKJEP
O1 FCc2F^L - BEPd@^LTQhBEPkca[ FIb F < F^P [`]>F 4 mi4Ž8©“1WU@i:pTU:EN5 AZFD6<_Z [ W h“A<wWunwn=DF>58jVTU8ATu<=N5ZFWbnw@AD6­N58›:=W@ ” (−1)l Ylm (θ, ϕ) = l 2 l!

s

2l + 1 (l + m)! imϕ dl−m e (sin θ)−m (sin θ)2l 4π (l − m)! d(cos θ)l−m

™A4Ž8N-WU8“AN5<w:=DFBu@iZFDFWU
x†±i4e±5¬›~ x†±i4e±5z›~

Yl−m (θ, ϕ) = (−1)m (Ylm (θ, ϕ))∗

 
0       
       
6    ¾O>V@A< @i8AW(­N5ZFWU@A<ž?A>V8A8iTuW»?AW l Á >V8 “1WU@i:º?i>V8AB(BªRiWu<=BªRiWu<ºZ6Wun TC:ªN:=nant:ªN:=DF>V8i8/N5DF<wWun-?AW Z [ Tu‘;@/N$:EDF>58+?AWIshBªRA<=« ?AD68AjVWu5:EWU8;:=DFWUZBuWU8;:=<EN5Zµnw>V@AnpZ•Nv&>V<w`aW ” x†±i4L’ ›~ ψ(r, θ, ϕ) = R(r)Y (θ, ϕ)  







l

>

R(r)

m

Wunw:Ànw>VZF@i:=DF>V8I?AWSZ [ Tu‘;@/N$:EDF>58Q‘ @iDAnu[ >V i:=DFWu8›:¨WU8ž<=Wu`-“AZFNM2N58›: L~ˆ “/N5< ~ l(l + 1) Án=>VDŸ: ”   ~ l(l + 1) ~ ∂ ~ ∂ x†±i4L’ 3}~ + − − + V (r) R(r) = ER(r) 2

2

2

2m ∂r 2

2

2

2mr 2

mr ∂r

P#%$'&,(ELN)D5@36>W)H)DK`U3,KdZ4)?3,3dLNI@+`5.3 CE43 5bO &7. 3dI O >WI@&-@1 ( X (E- R >WCE& ^5.5bO  









 










.O

   

2

À% ¢¼‡,1% ¶Å% .10a'/.O)a,O.Ž/0{% .10}#&% ¥SˆI‡,Š¥&‡¶ ÃÈ#&% . 79ZµWunt:BU>V`-`a> ?AWŽ?AWJBªR/N8AjVWuV8ABC:ED6>V8+DF8ABU>V8A8;@AWJWU:p?AWJ“‡>Vn=WU<

c¦

 



1 R(r) = u(r) r 1 1 du dR = − 2 u(r) + dr r r dr 2 2 du 1 d2 u d2 R = u(r) − 2 + dr 2 r 3 r dr r dr 2  ~2 2 2 du 1 d2 u 2 2 du ⇒ − u(r) − 2 + − 3 u(r) + 2 2m r 3 r dr r dr 2 r r dr  2  ~ l(l + 1) u(r) u(r) + + V (r) =E 2 2mr r r

x\±h4L’VK5~

wn >VD6: − ~ d + ~ l(l + 1) + V (r) u(r) = Eu(r) 2m dr 2mr  N58AnHZFWQBuN5nH“/N5<w:=DFBU@AZFD6Wu<Ž?A@“1>:EWu8›:=DFWuZ¨BU>V@AZ6>V`É AD6Wu8·?[ DF8›:EWU<EN5BC:EDF>58yWU8›:E<=W-@A88A>{f›N5@?AW BªR/N5<wjVW +Ze WC:@A8-TuZFWUBU:=<=>V8ž?AWgBªR/N5<wjVW −e Á V (r) = − 4V¾Š>Vnw>V8An α = +Ze ⇒ V (r) = 4‡shD l 6= 0 Ái“1>V@i< r “1WC:ED6:uÁi>V8“‡Wu@i:p8ATUjVZFD6jVWu< − ?AWC­N58›: ÁA?[ > − x\±h4L’5mV~ d l(l + 1) u(r) ' u(r) dr r sh@A“A“‡>Vnw>V8Anp‘ @iW u(r) nw>VD6:g?iWHZFNv&>V<=`-W r ” 2

2

2

2

2

Ze2 r

α r

α r

2

~2 l(l+1) 2mr 2

2

2

2

s

⇒ s(s − 1)r s−2 = l(l + 1)r s−2 ⇒ s(s − 1) = l(l + 1) ⇒s=l+1 s = −l R +∞ 2 2 R(r) r R (r)dr 0 R +∞ 2 u (r)dr 0

>V@

x\±h4L’™›~

(– N5D6nYZ•NJBu>58A?AD6:=DF>V8-?AW8A>V<=`aN5Z6DFnEN$:EDF>58Qn=@A< D6`a“‡>Vn=Wg‘;@AW n=>5D6:ÀBu>V8›­5WuVDŸ:SBu>V8›­5Wu<=j5Wu8›:EW4‡79Z1v†N5@i:g?A>V8iBbBªRA>VDFnwDF<_ZFNÉn=>VZ6@i:ED6>V8 “1>5n=D6:=D6­5W s = l + 1 4 shD l = 0 ÁA>V8>V i:=DFWU8;:g“i<=kung?AWJZ\[^>V<wDFjVD68AW (r ' 0) ” u(r) d2 u(r) ∝ − 2 dr r u(r) = a0 + a1 r + a2 r 2 + · · · d2 ⇒ u(r) = 2a2 + · · · , dr 2

sD

UB WI‘;@ADµD6`a“‡>Vn=W a Y>V8ABUZF@AnwDF>V8 ”  N58ing:=>V@AngZFWUngB2N5nUÁ/>V8©?A>VD6:pN2­5>VDF<S“‡>V@A< r “‡WU:=D6: ” u(r) ∝ r l+1

x\±h4L’V|5~ 0

=0

x\±h4L’5±V~

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0.2

l=0 l=1

0.15

0.1

U(r)

0.05

0

-0.05

-0.1

-0.15

-0.2 0




5

±i463 J <=N5“ARAWb?iWHZFNv&>V8ABC:EDF>58 

10 r

U (r) =

15

l(l+1) r2

−

1 r

“1>V@i<

20

l=0

WU:

l=1

4

Ⱦ N<JN5DFZ6ZFWu@i<=nuÁ"Z6Wɓ1>:EWu8›:=DFWuZ UW1WuBC:EDŸv  U (r) = NºZ [^N5Z6ZF@A<wWI<wWu“A<wTun=WU8›:ETuW-n=@iV@An—Z6WungB2NnuÁ‡>V8(nU[eN:w:EWU8A?»?A>V8iB@7ž:E<=>5@i­VWU<—?AWUn—TU:EN:EnpZFD6Tung@A8AD6‘;@AWu`-Wu8›: “‡>V@A< E < 0 4A¾ZFNMu>58Antq 8A>V@Ang?/N58inpBUWJB2N5nUÁ/WU:gv†N5DFnw>V8AnSZ6WbBªR/N58AjVWU`aWU8›:?iWJ­N5<wD•N5 iZFWun ~2 l(l+1) 2mr 2

λ=

N2­VWUB

r

α r

E r , ρ= EI a0 mα2 ~2 EI = , a = 0 2~2 mα −

x†±i4L’5’V~

À% ¢¼‡,1% ¶Å% .10a'/.O)a,O.Ž/0{% .10}#&% ¥SˆI‡,Š¥&‡¶ ÃÈ#&% . M¨[^TU‘ @AN:ED6>V8©?AWU­ DFWU8›:  

⇒ ⇒

nw>VD6: šŽ@/N58A?



  ~2 l(l + 1) α ~2 1 d 2 + − − u(ρ) = −EI λ2 u(ρ) 2m a20 dρ2 2ma20 ρ2 a0 ρ  2  ~2 l(l + 1) −2ma20 2ma20 α d 2mEI a20 2 + + λ u(ρ) u(ρ) = dρ2 2ma20 ρ2 ~2 ~2 a 0 ρ ~2  2  d l(l + 1) 2 2m2 α2 ~4 2 − + λ u(ρ) u(ρ) = dρ2 ρ2 ρ 2~4 m2 α2 

x\±h4L’5¬V~

 d2 l(l + 1) 2 2 − + − λ u(ρ) = 0 dρ2 ρ2 ρ

ρ → +∞

ÁA>V8NžN5“A“A<w> iD6` N$:ED6­5Wu`-Wu8›: x\±h4L’5zV~

d2 u ' λ2 u ⇒ u(ρ) ' e±λρ dρ2

MNžBU>V8A?ADŸ:ED6>V8+?AWH8A>V<=`aN5Z6DFnEN$:EDF>58aD6`a“‡>Vn=Wb?iWH<wW wWC:EWuVZ6@i:ED6>V8 e 4 YRAWu<wBªRA>V8Anp?A>V8iBHZFNQn=>VZ6@i:ED6>V8+n=>V@inpZFNv&>V<=`-W u(ρ) = e y(ρ) 4 λρ

−λρ

⇒

du = −λe−λρ + e−λρ y 0 dρ d2 u = λ2 e−λρ y − 2λe−λρ y 0 + e−λρ y 00 dρ2  2  d d 2 l(l + 1) − 2λ + − y(ρ) = 0 dρ2 dρ ρ ρ2

x\±h4e¬V~

YRAWu<wBªRA>V8AnZ•NÉnw>VZF@i:=DF>V8+nw>V@AnpZ•Nv&>V<w`aWb?c[^@A8?ATC­VWuZ6>V“A“‡Wu`-Wu8›: ” y(ρ) = ρl+1

+∞ X q=0

cq ρ q =

+∞ X q=0

cq ρq+l+1

x\±h4e¬i3{~

¢¼1,‡% ¶Å% .10a'O/.O) ,O.Ž/0{% .10}#&% ¥Yˆ% .102!$A¥ MµWJ“A<=TCv†N5BU:=Wu@A<S­ DFWU8;:p?AWJZFNÉBu>V8A?iD6:ED6>V8 u(r) ∝ r “1>V@i< r “1WC:EDŸ:24



l+1

+∞

dy X = cq (q + l + 1)ρl+q dρ q=0 +∞

d2 y X = cq (q + l + 1)(q + l)ρl+q−1 dρ2 q=0 ⇒

⇒

+∞ X  cq (q + l + 1)(q + l)ρl+q−1 − 2λcq (q + l + 1)ρl+q + 2cq ρl+q q=0

 −cq l(l + 1)ρl+q−1 = 0

x†±i4e¬VKV~

( c0 (l + 1)lρl−1 − c0 (l + 1)lρl−1 = 0 cq (q + l + 1)(q + l) − 2λcq−1 (q + l) + 2cq−1 − cq l(l + 1) = 0

n=>5D6: c [(q + l + 1)(q + l) − l(l + 1)] = 2c ‘;@AWHZ [ >V8+“1WU@i:p<=TUTuBu<wDF<wW q

q−1

(λ(q + l) − 1)

cq q(q + 2l + 1) = 2cq−1 (λ(q + l) − 1)

]£“A<=D6>V<=D\Áh‘;@/N8A?

⇒ q → +∞

Á

2 (λ(q + l) − 1) cq = cq−1 q(q + 2l + 1)

cq ∝ cq−1

2λ q

1 2λ q! ⇒ y(ρ) ' ρl+1 e2λρ ⇒ cq ∝

⇒ u(ρ) ∝ eλρ

x†±i4e¬5m›~

x†±i4e¬™;~

‘;@AD"8[ Wunw: N5@h:E<=Wg‘;@AWpZ•Nbn=>5ZF@i:=DF>V8Q<wW wWU:=TuWp“A<=TUBuTU?AWu`-`aWU8›:24;79ZAv†N5@i:À?A>V8iBp‘;@AWpZFWS8;@A`-Tu<=N:EWU@A< nu[^N58A8;@AZ6W5Á"N5@h:E<=WU`aWU8›:?iD6:g‘;@[ DFZW iD6nw:=WH@i8WU8;:=DFWU< k ≥ 1 :=WuZµ‘;@AW λ(k + l) − 1 = 0 x†±i4e¬V|V~ 1 ⇒ λ= k+l MµWungTu8iWu<=j5DFWung“‡>Vn=nwDF AZ6Wungn=>58;:p?A>58A8ATuWUn“AN5< E x†±i4e¬5±›~ E =− k = 1, 2, . . . I

k,l

(k + l)2

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dt2

c

c

c

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p~ =

–(N5D6n

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2

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⇒

Kh4

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]—DF8in=D Á

x\±i4Ÿ3V±›~

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z

B



B

z

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c  A6  D
 N   c @ MµWun¨`aN:=<=DFBUWunŠ?AWS¾ÈN@AZFDhn=>V8›:8A>5:EN5`-`aWU8;:O:E<wkun¨@i:=DFZ6Wunȓ‡>V@A<Š?ATuBU<=D6<=W_Z6Wun…<w>5:ªN:=DF>V8inu45Y>V8An=D q ?ATU<=>V8AnYWu8ºW cWC:_ZFWp“A<w>V AZFkU`aWpn=@iD6­N58›: ” n=>VD6Wu8›: nˆ WU: mˆ ?AWu@ ­5WuBU:=Wu@A<wnS@A8ADŸ:ªN5D6<=Wun<wWuZFD6TunÀ“/N5< @A8AWJ<w>5:ªN:=DF>V8 R ?[^N58Aj5ZFW α = k~αk N5@i:E>5@A<S?AWJZ•NQ?AD6<=WuBC:ED6>V8 αˆ = ÁiB5[ Wunw:tq*7$q ?ADF<wW ” x\’h463{’V~ n ˆ = R (m) ˆ Y>V`-`aWU8›:ÉZFWUnTC:ªN:=n“A<w>V“A<=WUn |ˆn ↑i WC: |ˆn ↓i ?AW S~ · nˆ n=>V8›:wq DFZ6nI<wWuZFD6TunIN5
@ ÅTU:EN:Enɓi<=>V“A<wWun WC: ˆ ↓i ?AW S~ · mˆ  |m ˆ ↑i |m ¾¨<=>V“‡>VnwD6:ED6>V8 ”   x\’h4632¬›~ ~ ·n ~ ·m S ˆ=e S ˆ e 





α ~ k~ αk

α ~

α ~

~ α/~ −iS·~

~ α/~ iS·~

¦Ÿ³ ¢£h0{!$#\„V% )Q'%»ÂŽ/,O¥\#À%›0  "02i0}#&1.O)

1Œ

n ˆ α m ˆ

α ˆ




’ 4^™ op>:ªN:=DF>V8+?[ @A8N58AjVZ6W α N5@i:=>V@A<S?AWJZ [^N hW α~ x α~ “1>VD68›:EWbRA>V<wng?AWJZ•Nv&Wu@iDFZFZ6W{~ 4 

 Tu`->V8Ant:E<=N:ED6>V8 ” §¨8BU>V8An=@iZ6:ªN8;:gFZ Nɜ/jV@A<wW5Á/>V8+­V>5D6:pN5D6n=Tu`-Wu8›:g‘;@AW ”

¾Š>V@i<

x ’ 463{zV~

n ˆ = cos α m ˆ + sin α α ˆ∧m ˆ

dα

D68iœ/8ADŸ:ETUn=DF`aN5Z\Á >58(NQ?A>V8iB ”

n ˆ (α + dα) = n ˆ (α) + dα α ˆ∧n ˆ (α) + O(dα2 ) ~ ·n ~ ·n ~ · (α ⇒ S ˆ (α + dα) = S ˆ (α) + dαS ˆ∧n ˆ (α)) + O(dα2 )

x ’ 4LKV~

–(N5D6nSZ•NQ<=WUZ•N:=DF>V8·x\’h4Ÿ3{m›~YDF`-“AZ6DF‘;@AW h

i ~ ~ ~ ~ · (~a ∧ ~b) S · ~a, S · b = i~S

h

i

i ~ · α, ~ ·n ~ ·n ~ ·n ˆ S ˆ (α) + O(dα2 ) ⇒ S ˆ (α + dα) = S ˆ (α) − dα S ~

shDµ>V8©Bu>58An=D6?Aku<wW

~ ·n S ˆ

Bu>V`-`-WŽv&>V8ABU:=DF>V8+?iW α ÁA>V8NQ?A>V8AB ”

i d ~  i h~ ~ ·n S·n ˆ =− S · α, ˆ S ˆ + O(dα2 ) dα ~   ~ ~ ·m S·n ˆ (α = 0) = S ˆ   ~ ~ ~ ·m Aˆ ≡ e−iS·~α/~ S ˆ eiS·~α/~

x ’ 4LKh3{~ x ’ 4LKVK5~

N2­VWuBHZ•NQBU>V8A?ADŸ:EDF>58+DF8ADŸ:ED•NZFW 4  Tu`->V8›:E<w>V8AnI‘ @iW uW nt:Én=>5ZF@i:=DF>V8 x†WU8 n=Wa<EN5“i“1WUZ•N58›:I?AW ZFN ?ATUœA8AD6:=DF>V8 αˆ = ↔ α~ = αˆ ×kαk ≡ αˆ × α~ 4 :E<=DŸ­ D•N5Z ˆ = 0) = S ~ ·m A(α ˆ x ’ 4LK5mV~ i ~ ·α ~ ·α  ˆ iS ˆ ˆ ˆ iS i h~ dAˆ S  b š ¯ A+A ˆ , Aˆ =− =− S·α α ~ k~ αk

dα

~

~

~

  ÆcÇÈ#†. ogWU­5Wu8A>V8in^7(Z•N©‘;@AWunt:EDF>588A>V@AnI8A>V@AnIn=>V`-`aWUn“‡>Vn=TUWºN5@Å?ATu“/N<w:2Á 7(n=N{­5>VDFVV“i<=Wung?AW S~ · nˆ WU:gBuWU
@ ?AW S~ · mˆ 4‡Y>58An=D6?ATu<w>V8AngZ [ TU:EN: e |mˆ ↑i 4 ~ α/~ −iS·~

  ~ ~ ~ ~ ~ ·n ~ ·m S ˆ e−iS·~α/~ |m ˆ ↑i = e−iS·~α/~ S ˆ eiS·~α/~ e−iS·~α/~ |m ˆ ↑i ~

~ · m| = e−iS·~α/~ S ˆ m ˆ ↑i ~ ~ ˆ ↑i = e−iS·~α/~ |m 2

 WJ`u`-W5Á |ˆn ↓i = e

~

⇒ ~ α/~ −iS·~

x\’h4LK$™;~

|m ˆ ↓i

4

|ˆ n ↑i = e−iS·~α/~ |m ˆ ↑i

ogWu`aN5<=‘;@AW ” MNž?ATu`->V8Ant:E<EN$:EDF>58+<=Wu“‡>VnwWHnw@A
\x ’h4eKV|V~ ŽV@A<À8[^D6`a“‡>V5`aWU8;: BuD68ATU:=DF‘;@AW4h§¨8ºW cWC:2Á 7I“/N<w:ED6< ?AWUn<wWuZ•N$:EDF>58An_?AWJBu>5`a`É@h:ªN:=DF>V8 ˆ ,L ˆ ] = i~L ˆ [L >V8+:E<w>V@i­5W h

i ~ ~ ~ ~ · (~a ∧ ~b) S · ~a, S · b = i~S

x

y

z

ˆ x ax + L ˆ y ay + L ˆ z az , L ˆ x bx + L ˆ y by + L ˆ z bz ] [L ˆx, L ˆ y ] + a x b z [L ˆx, L ˆz] = a x b y [L ˆy , L ˆ x ] + a y b z [L ˆy , L ˆz] + a y b x [L ˆz , L ˆ x ] + a z b y [L ˆz , L ˆy] + a z b x [L ˆ z (ax by − ay bx ) + i~L ˆ x (ay bz − az by ) + i~L ˆ y (az bx − ax bz ) = i~L ~ˆ · (~a ∧ ~b). = i~L

]—D68An=D\ÁA>V8NQ?AWbv†NMu>58:=>V@i: 7Év†N5D6:_jVTu8ATU<EN5Z6W ”

  ˆ ~ ~ˆ ~ˆ · m ~ˆ · n ˆ eiL·~α/~ L ˆ = e−iL·~α/~ L

WU:

x\’h4eK5±›~ x\’h4LK5’V~

x ’h4eK5¬›~  N8AnZ6WQB2N5nH?A@Ånw“ADF8 ÁµBuWC:>5“1TU<EN:=Wu@AV<=`-WQ:E<=kUnJn=DF`-“AZ6W54µ§¨8ÅW cWC:2Á>V8y“1WU@i: TuBU<=D6<=W x\’h4LKz›~ e =e ~ˆ

ˆ i. |ˆ nLM i = e−iL·~α/~ |mLM

1 2

~ α/~ −iS·~

−i~ σ ·~ α/2

¦ Ÿ³ ¢£h0{!$#\„V% )Q'%»ÂŽ/,O¥\#À%›0  "02i0}#&1.O)  ¦ > ~σ Wunt:Z6WŽ­VWUBU:EWU@AV8›:pZFWUnSBu>V`-“1>5nEN58›:EWUnSn=>V8›:pZFWUnS` N$:E<=D6BuWUn_?AW ¾ÈN5@iZFD 4/–(N5DFnUÁ    X  −i~σ · α −iα~σ · α ˆ ~ 1 1 x ’ 4em V~ = sum e = +∞

n

−i~ σ ·~ α/2

WU:p“/N5
2

n=0

n

+∞ n=0

n!

2

n!

2

(~σ · α) ˆ = σ0 α ˆ·α ˆ + i~σ · (α ˆ∧α ˆ ) = σ0

?[ > e

−i~ σ ·~ α/2

X  −iα n 1 1 = σ0 +σ·α ˆ n! 2 n! n=0,2,4,... n=1,3,5,... α α ˆ sin = cos σ0 − iσ · α 2 2 X



e−i~σ·~α/2 = cos

⇒

−iα 2

n

x ’ 4emi3{~

x ’ 4em›K5~

α α σ0 − i sin σ · α ˆ 2 2

§ iWU`a“iZFW ” Y>V8AnwDF?ATU<=>V8Ana@A8AW<=>5:EN:ED6>V8?[^N58AjVZ6W − N5@i:=>V@A<ž?AW xˆ 7·“/N5<w:=DF5:EN:ED6>V8AnY“i<=TuBUTu?AWU8›:EWunUÁ S~ · mˆ = S WU: S~ · nˆ = S 4Ash>VD6: 1 Z [ TU:ªN$:_“A<w>V“A<=W—?AW S ?AW ­N5ZFWU@A<S“A<=>5“A<=W 4  [eN5“i<=kunpZ•Nv&>5<=`É@AZ6WŽ‘;@ADµ“A<wTuBukU?AW5Á‡DFZ‡v†N5@i:pN5“i0“AZFD6‘ @iWu<SZ [ >V“‡Tu<=N:EWU@A< π 2

z

y

z

~ 2

√ √   2 2 2 1 i σ0 + i σx = 2 2 2 i 1 √    √   2 1 i 2 1 1 = ⇒ |ˆ n ↑i |{z} = 0 2 i 1 2 i √

Ž
(7.22)

       1 0 −i 1 1 σy = = i i 0 i i √  ! √  ! 2 1 2 1 ~ ⇒ Sy = 2 i 2 2 i

opWU` N<=‘;@AWun ”

x ’ 4emVmV~

uN-`aN5<=BªRiW 

x ’ 4em5™›~



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1 2

−i~ σ ·~ α/2

0

0

nw>VD6: Ž
⇒

( (

a0 b0

= hˆ n ↑ |ψi = hˆ n ↓ |ψi

a0 b0

= ahˆ n ↑ |m ˆ ↑i + bhˆ n ↑ |m ˆ ↓i = ahˆ n ↓ |m ˆ ↑i + bhˆ n ↓ |m ˆ ↓i ~

~

|ˆ n ↑i = e−iS·~α/~ |m ˆ ↑i ⇒ hˆ n ↑ | = hm ˆ ↑ |eiS·~α/~    0   a   0   b



 = a 1 0 ei~σ ·~α/2 

 = a 0 1 ei~σ ·~α/2

   0 a i~ σ ·~ α/2 a =e b b0

!   1 + b 1 0 ei~σ ·~α/2 0 !   1 + b 0 1 ei~σ ·~α/2 0

0 1 0 1

!

!

x\’h4emV|V~

¦FŒ   ¥†% „50{!}‡..‡.!}%h¥&i0}# #†)U0{%    k     c  6 D   8TuZFWUBU:=<=>V88A>V8(<wWuZFN:EDŸ­hD6nw:=Wb“1WU@i:—?i>V8AB C:E<wWI?ATUBu<wD6:“/N5V@i< Z•NJ“1>5n=D6:=DF>V8Q?AN58AnYZ\[^WUn=“/NBuWpWU: @A8Á;?ADFnwBu<wWU:2Á;“‡>V@A<ZFWgnw“ADF84;]…fVN58›:YBªRA>5DFn=D"@A8iW?AD6<=WUBU:ED6>V8-“1>V@i< ‘;@/N58›:EDŸœ/WuV8©“1WU@i:p?A>V8ABJZFWb?iTuBu<wDF<wWb“/N5V8ABC:ED6>V8©?[^>58A?AW54 1 1 x\’h4em5±›~ ψ(x, y, z; σ) ∈ C x, y, z ∈ R σ ∈ {− , } MNɓA<=>V AN5 ADFZ6D6:=T?AWb:=<=>V@i­5Wu<SZ\[^TUZFWuBC:E<w>V8©N5@©“1>5DF8›: x, y, z N2­VWU2BJZ•2NɓA<w> wWuBC:EDF>58 σ nw@AD6­N58›: z “1>5@A<¨ZFWSnw“ADF8žWunt:À?A>V8A8ATUW“AN5< |ψ(x, y, z; σ)| 4;YWUZFZFWS?iWg:=<=>V@h­VWu<Z\[^TUZFWuBC:E<w>V8žN5@ž“1>5DF8›: x, y, z N2­VWuB @A8 n=“iDF8 ‘;@AWUZFBu>58A‘;@AWºWunt: P |ψ(x, WU:ÉBUWuZ6ZFW-?AW Z6W-:=<=>V@i­5WuV y, z; σ)| R ` NDFnSN2­VWUBI@i8nw“ADF8?ATU:=Wu<w`aD68ATŽ­N5@i: d ~r |ψ(x, y, z; σ)| 4 §¨8iœ/8Á1Bu>V`-`aWJZFNa“A<w>V /N5 AD6ZFDŸ:ETb?AWÉZ6WH:=<=>V@h­VWu<ŽN{­5WuBÉ8c[^D6`a“‡>V<w:=WI‘;@AWUZŠnw“ADF8(8[^D6`a“‡>V ­N5@i:H3VÁA>V8©?A>5D6:gDF`-“‡>Vn=WU<_Z•NQ8A>V<w` NZFDFn=N:ED6>V8 ” Z X x\’h4emV’V~ d ~r |ψ(x, y, z; σ)| = 1 







2

2

σ

2

3

2

3

Žš @AWUnw:=DF>V8 ” sh@i“A“1>5n=>V8Anž‘;@[ @A8 TuZFWUBU:=<=>V8nw>VD6:Q?iTuBu<wD6:ž“/N<É@A8AW+v&>V8ABC:ED6>V8?[ >V8A?AW+> ZFN ­N5<=DFN5 AZFW—n=“AD68WUnw:<=WUZ•N:=D6­VW87-@A8iWH?iDF<=WUBU:=DF>V8 mˆ ” x\’h4^mV¬›~ ψ(x, y, z; mσ) ˆ Y>V`-`aWU8;:gnU[^TUBu<wD6:pZ•Nv&>V8iBU:ED6>V8©?[ >V8A?AWHn=DµZ\[^>V8©BªRA>5DFn=DŸ:g@A8AWJ?AD6<=WuBC:ED6>V8 nˆ “1>V@i<S‘ @AN58›:ED6œAWu< ZFWbnw“ADF8  opTU“1>58An=W ” Z•Nv&>58ABU:=DF>V8©?[/>V8i?AW ψ(x, y, z; mσ) Bu>V<w<=Wunw“1>58A?(N@ WU: ˆ XZ x\’h4em5z›~ |ψi = d ~r ψ(x, y, z; mσ)|~ ˆ ri ⊗ |mσi ˆ ¾ÈN558ÁiZFNv&>V8ABU:=DF>V8©?[/>58A?AW ψ(x, y, z; nˆσ ) WUnw:p?A>V8AB ” σ

3

σ

0

(h~r| ⊗ hˆ nσ 0 |)|ψi =

X σ

ψ(x, y, z; mσ)hˆ ˆ nσ 0 |mσi ˆ

= hˆ nσ 0 |(ψ(x, y, z; m ˆ ↑)|m ˆ ↑i + ψ(x, y, z; m ˆ ↓)|m ˆ ↓i)

Ž5:ªN:=DF>V8inuÁABUWungBu>V`-“‡>VnEN58›:=Wungn=>V8›:?A>V8A8iTuWunp“/N5< ”

x ’ 4^™ V~

x ’ 4^™A3{~ YWU:w:EWJ“A<=>5“A<=D6TU:ETJN`akU8AW[7Q“A<=TCv&Tu<=WUV8ABC:EDF>58?c[^>V8i?AW ?[ @A8TUZFWUBU:E<w>V8“/N5<S@A8©­VWUBU:EWU@AVZ6>V8A8AW 

ψ(x, y, z; n ˆ ↑) ψ(x, y, z; n ˆ ↓)



=e

i~ σ ·~ α/2



ψ(x, y, z; m ˆ ↑) ψ(x, y, z; m ˆ ↓)



"

ÆcÇÈ#†. 

ψ(x, y, z; ↑) ψ(x, y, z; ↓)

x\’h4^™›KV~



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→ h~r|Uf Ψi = hf −1 (~r)|Ψi = Ψ(f −1 (~r)).

−1




            

 



   

  

ˆh;=-I19JPGIE6K;=-I1d
,.-B I/6KB I/€
xˆ → xˆ + a

T † (a)ˆ xT (a) = xˆ + a

ƒ c` =@‡

[ˆ x, T (a)] = aT (a)

ƒ ?`s@“A‡

{ -V5UJL/2-A3Q ˆ^M@[PB
(ˆ xT (a) − T (a)ˆ x)|xi xˆT (a)|xi − xT (a)|xi xˆ(T (a)|xi) ⇒ T (a)|xi

= = = =

aT (a)|xi aT (a)|xi (x + a)T (a)|xi |x + ai

ƒ ?`s ]‡ V\,.-;UGI/=/\[P/#6KE1K=/2JP6\-?’N=/2T[L;=(a)WIW` /2V& ;=/2@6 /2K-A37JLI;=/-I6K-IV&/[nJLq37JLV\_]UJP/-]3w 6 /WY;@6937;@-I1dB2/wGIE>@/2[L;=WIW/2V&/2-A3v[PJPV&JL3KEGNMO-I1d[™M56K/[™MO3KJP;=[ˆ x, T (a)] = aT (a) †[ †[ [ˆ x, − iδK] = δ + O(δ ) ƒ ?`s „ ‡ †[ → [ˆ x, K] = i Ψ(x) → hx|T (a)|Ψi = (T (a)† |xi)† |Ψi = hx − a|Ψi = Ψ(x − a)

2

†

†

2

†

2




 



 

,.-/-G?E2GIOM@-A3w

ƒ ?`  A‡ {37JL;=-I-?1m’N -jq|;=- WY/
pˆ ~




a N ) N 1 a = [ − i K + O( 2 )]N n N

T (a) = T (

†[

@M JL1 { -/kyj/k3‚q

lim [1 +

N →∞

lim ln[(1 +

N →∞

1 x + O( 2 )]N = ex N N

x 1 + O( 2 ))N ] = N N

ƒ ?`s“

A‡

x 1 O( 2 )] N →∞ N N 1 x = lim N ( O( 2 )) N →∞ N N = x

JL-I1KJ˜q

lim N ln[1 +



T (a) = exp(−iˆ pa/~)

ƒ ?`f“?† ‡

., -GIJL3d_]@/2Bk37/2@E269JL’N/ T (~a) ƒ ?`f“=“A‡ T (~a)|~ri = |~r + ~ai — /mC… M ;=-/ ŽcWI[PJLB2JL3K/@qcJP[ W/2


(





†

 

B2/k3Kˆh3K/Q;=-?691K;@JP3”G?M E2376KJP;@;@-I-Z1€/2G?-E2V\1K;@E6KB‚V\M@M@-IJPJL1w_]@MO-]3 ƒ ?`f“ =‡ R (~ α)b ~rR(~ α) = R (b ~r) α ~

†

α ~




             





— efM@WI6K81d[P/21‰CD;=6KV5@/26.1KM\CD;=6KV&/w/kŽcWI[LJPB2Jz37/OqB;=-I19JPGIE6K;=-I1F\-I;@=/2M@<Œc<1K=/B3K/2=/2BQ
2

δ~ n

2

i

G eo;=R

[xbi , R(δ~n)] = δ(R(~n ∧ ~r))i + O(δ 2 ) − i[xbi , K] = (~n ∧ ~r)i [xbi , K] = iijk nj xbk

ƒ c`s“ =‡ K1 ;=Jz3 ,.6:-I;==;=-I1€>c<S[L;=6K1QGI/X[˜efEk37@/2V&/2-A3 B/2-A376KM@[h_]
~b · ~n, xbj ] = i~ijk ni xbk [L

;=</2-IB;=6K/

~b · ~n] = i~ijk nj xbk [xbi , L ~b · ~n L K= ~

{ -EB26KJz>OM@-A3 R(~α) = R( ) qN/3v/2-WI6K/-NM@-A3v[™M [PJLV\Jz37/ N → +∞ qNJP[Y>cJL/2-A3w

ƒ c` =‡

α ~ N N

ƒ c`I†}‡ ,V\.-;@V\69//37-]693d;=@-I/&Ek37[LJL/ _U
~b α ~ ·L R(~ α) = exp(−i ) ~




 



 

 

 


ˆeo/2143[˜ef;=WE26KMO37JL;=-Z_]@/26K19/26v[P/m19JPg=-I/mG?/21vB2;U;=6KGI;@-I-IE2/1w

ƒ ?`]=‡

ueo;=WYE67MO3K/2


,.-/2-GIE2GI
† […`/21dCD;=-IB3KJP;=-?1lG ef;@-IGI/w1K/m376KM@-I14CD;=6KV&/2-A3v1KOM@-A3w Π|~ri = | − ~ri

Π2 =

+M 3767MO-I19CD;=69VM 37JP;@- Π B2;=-I19/264>=/Q[P/mWI69;cGI


Z

2/ 193XGI;=-IBOM@-A3w Π

†

|;@=/26Q { 3v/2-/yY/3‚qN;@-M( 

−1

Π2

ˆ Aˆ → ΠAΠ Πb ~rΠ = −b ~r

Πb ~rΠ|~ri = Πb ~r| − ~ri = −b ~rΠ| − ~ri = −b ~r|~ri Πb ~rΠ = −b ~r

— /wV12V&/ q Πbp~Π = −bp~ ` — E2V&;=-A376K;@-I1d[P/mW;=
= = = = =

i~ i~ i~ −i~ −b p

MO6vB;=-A3769/@qN[P/mW?6K;UGI


i j

,.-/-G?E2GIOM@69J™M@-A3m

Πxbi pbj Π = Πxbi ΠΠpbj Π = (−xbi )(−pbj ) = xbi pi ~b = L ~b ΠLΠ

ƒ ?` ]‡

ƒ c` @‡ † […q‰;@-/2- G?E2GI
Z Z d(−~r) ∗ Ψ (~r)Φ(~r)d~r = Φ† (~r)φ(~r)d~r Ψ (−~r)Φ(−~r)d~r = d~r ∗

ƒ c`A“=‡

†

ƒ ?` „ ‡

ƒ ?` A‡ ƒ ?` A‡ ƒ ?` U† ‡

 /21+ >=  /Bk37`/



             




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ab[M@696KJL>@/w1K;=@/2-A3x_U
q˙

;=<_UOM@6KJPM@-A3lWNM@6dB NMO-Ig=/2V&/2-A3vGI<1KJLg=-I/mGI/ p~  Lcl (q, ˙ q) = Lcl (−q, ˙ q)

ƒ ?`!==‡ ƒ ?` O“A‡

— M@-I1vB2/wB2M@12qN19J ~r(t) /214319;=[P
2

1

2

2

2

1

M@JL1

JL-I1KJ˜q 

(

−q˜˙(t) = ∂2 H(˜ q (t), −˜ p(t)) p˜˙(t) = −∂1 H(˜ q (t), −˜ p(t))

∂1 H(˜ q (t), −˜ p(t)) = ∂1 H(˜ q (t), p˜(t)) ∂2 H(˜ q (t), −˜ p(t)) = −∂2 H(˜ q (t), p˜(t)) ( q˜˙(t) = ∂2 H(˜ q (t), p˜(t)) ˜˙p(t) = −∂1 H(˜ q (t), p˜(t))

ˆ —  ,.-/193vGI;=-IBw-NM 37
~rθ = b ~r θ −1b −1b p~ θ p~θ = −b




 




E2_]OM@[P/-]3vF(



 

( θb ~rθ −1 = b ~r −1 θb p~θ = −b p~

u6K;@WY;=19JL3KJP;=- +M5376KM@-I19CD;@6KV\MO37JL;=- θ /2143MO-]3KJP[LJP-IE2M@JP69/@` — EV\;@-I193K67MO3KJP;=-  — efM@WI69821d[P/1d3K67M@-?19CD;=69VMO3KJP;=-?1^GI/ b~r /k3 bp~ q 

θ[xbi , pbj ]θ −1 = = = =

θ xbi pbj θ −1 − θ pbj xbi θ −1 θ xbi θ −1 θ pbj θ −1 − θ pbj θ −1 θ xbi θ −1 xbi (−pbj ) − (−pbj )xbi −[xbi , pbj ]

ˆ^M@[LB2
V\M@JP1lMO
j

hθΦ|θ[xbi , pbj ]|Ψi = hθΦ|θi~δij |Ψi

hθΦ|θ[xbi , pbj ]|Ψi = = = ⇒ θi|Ψi =

hθΦ|θ[xbi , pbj ]θ −1 θ|Ψi hθΦ| − [xbi , pbj ]θ|Ψi −hθΦ|i~δij θ|Ψi −iθ|Ψi

ƒ ?` =0 ‡ ƒ ?`! „ ‡ ƒ c` =‡

— efM@WI69821v[P/.3%?E2;=6982V&/mGI/  JPg=-I/62qI[PM53767MO-I19CD;=69VM 37JP;@- θ /2193vGI;@-IBQM@-A37Jzp =/2Bk37/OM@[P/cJL/2-A3Q Z −1

∗

θ|Ψi =

d~rΨ∗ (~r)|~ri





 

 

     

    /  



 



1K;=Jz3 θ|Ψi = |Ψ i { -?’N- qN-I;O37;=-I1d[PM5[P;=J G?/w3K67M@-?19CD;=69VMO3KJP;=-GI<V&;=V&/2-A3dB2JL-IE3KJP_]
~b = −L ~b θ −1 Lθ

_]
00$ 

0





-,'$$

~b = b L ~r ∧ b p~

`

ƒ c`!" A‡ ƒ c`!"@‡

1x;==;=-I1GIE F\>U<Œ_]@;=[POM@JL3 13K6K/w6K/WI6KE1K/2-A3KE2/:WIM@6d@;=[PUUOM@iI[P/ƒ 1d_Uc=/2Bk37/=-I;=1d[L@B2;=/wJL3._]OM@69J™M@-A3WNM@6.@MO6KJ™MO-]3 WNM@6‰6K;@37MO37JL;=- ` ƒ— /xC… M 2;=-\g=E2-?E267MO[P/@q]B/3K3K/JL-]>OM@69J™M@-?B2/v1K/x3K67M@G?
A‡ K HK = H ⇔ [H, K] = 0 — /.C… M [n;=eo/2-143v37;@M@
− ~i Ht

†

H

H



H







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JL-I1KJ˜q1KJ K|Φi - eo/21435WNM@1QWI69;=WY;@6937JL;=-I-I/[rF |Φi q+[™MX>OM@[P/U/2=/6K19/2V&/2-A3ZGI< 37/2V&WI1qr3K67M@-?1K[™M 37JP;@-I12qt69;@3”M 37JP;@-‡:1K/1KEWNM@6K/-A3X-NMO3KO1ZM@6914TcJ™M@V&-?B2E/.3K6KGIJL/2/w1Z[˜GIe NJLMO1KB2V\698JL[L3K37/2;@1(-IJP//3(-\[LGN/21ZMO-I191dTUV&B2/2E1v3KGI6KJPJz/yj1ZE6KB2/2;=-A-A3K3K1vJP-]B2


  
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OM M@ˆh6KJLV&/1mJ /21K-[P;=/-A1^W#3A19TUTU[P/V&1K1JP_]E1437=19GITU/269/ŒV&1K/E37V\6K37M@69/JP-I/-]191Q3CD;@G?GI6K< =JL;@6K1-Iq?1xM EkŽc3”/M@1vi?Gj[PJ ef;=[PM:69GICD6K;=/w6KV&- q?/m/GI37/2BO1d`L`z`s;=‡kW` E26 6K/2MOgA37/M@



   



  

—B2;@-IGIMO@/26tUJP/-]E3QV\ ;@-I193K67MO3KJP;=- B2;@-I1KJLGIE269;=-I1: O/3K1 |Φ i /3 |Φ i ` — efM@WI6K81€[˜esMO-]3KJzp
2

iα

iα

a

hθΦa |Φb i = = 2iα e = iα e =

b

hθΦb |θ 2 Φa i = eiα hθΦb |Φa i eiα hθΦa |θ 2 Φb i = ei2α hθΦa |Φb i 1 ±1

‘cJ e = −1 qN;@-M#M@[L;=6K1m GIEGI
a

a

b

b

hθΦ|Φi = 0

b

a

ƒ ?`!0@“A‡ ƒ ?` 0 ]‡ ƒ ?`!0 =‡





 

 

     

    /  



 

c‘ Jh[˜esMOB37JL;=- G?/X[™MH14TUV\Ek376KJL/69/2-A>=/6K1K/V\/-A3GI<03K/2V&WI1:/193 3K/2[P[L/\_]=/2V&† […`I/2ab-A[3:- G e Tef@/26K19/kp V\/-A3xG?<3K/2V&WI12q J~ /193d376KM@-I19CD;@6KV&Em/2- −J~`N,m-M GI;=-IB# 2



∗

2

 z z  θJ = −J θ θJ x = −J x θ   y θJ = −J y θ

ƒ c` 0 "=‡ / 3 ‘GIc;=;=Jz-I3 B#|j, mi =/B` 37/=69/2VBmMO[PM:3KJP;=B2-j;@-]qc>@JL[j/2-A>U37JP/2JL;=-A- 3w~ = 1 qU;=-(M — J |j, mi = m|j, mi ƒ ?`!0 0=‡ J θ|j, mi = −θJ |j, mi = −mθ|j, mi GI;=JL3^GI;=-IB 13K6K/wWI69;=WY;@6937JL;=-I-I/[F |j, −mi
`  ;=1K;@-I1 θ|j, mi ƒ ?` 0 „ ‡ θ|j, mi = e |j, −mi ,.-HM#M@[L;=6K1m θJ + = θ(Jx + iJy ) = θJx − iθJy = −Jx θ + iJy θ θJ + = −J − θ θJ − = −J + θ 2

z

z

z

z

iα(j,m)

J + θ|j, mi = eiα(j,m) J + |j, −mi p = eiα(j,m) j(j + 1) − m(m − 1)|j, −m + 1i

M@JL1 J θ = −θJ q †

−

J + θ|j, mi = −θJ − |j, mi p = −θ j(j + 1) − m(m − 1)|j, m − 1i p = −eiα(j,m−1) j(j + 1) − m(m − 1)|j, −m + 1i

., -GI;@JL3lGI;@-IBQM}>@;=JP6 B2/1x69/2[™M 37JP;@-I1^1K;@-]3v/-WNMO6937JLB2
iα(j,m−1)

eiα(j,m) = eiβ(j) (−1)j−m

ƒ ?`!0 A‡ ƒ c` „ =‡ ƒ c` „ †}‡


 

#



 

,.-HM GI;@-IB# θ|j, mi = eiβ(j) (−1)j−m |j, −mi θ 2 |j, mi = θ (θ|j, mi) = e−iβ(j) (−1)j−m eiβ(j) (−1)j+m |j, mi

ƒ ?` „ =‡ ‘B2c;@J -IB2/[L19=Ek/237BxJL_U-I=;=JL6\@[P/MOB6K37J™69MO;=-]-Q35GNWNM@M@-I6 1 6K=W/2;@693K1K/2/-AV\37JL//2-][A3B2;@GIOqM@69/J™-M@-AW?3|6KE2WN19M@/26J~-I69B⇒/2/:-AG >=−/eo=//‚6KV&MOO6KM@E6K12JP` M@|-IM@B/6 WNM@6l69/2-A>=/6K19/2V&/2-A3mGI@E2/O`ˆS~ef/·193 B~     !N` θ 2 |j, mi = (−1)2j |j, mi



2

1 2

B

 

 







u o e = ; Y W  E 7 6 O M K 3 2 / I < ‰ 6 N W @ M 9 6 L J 7 3 E 2 / 4 1 ^ 3 I < \ = ; Y W  E 7 6 O M K 3 2 / I < – 6 P [ L J I ‚ E O M P J K 6 l / ] _ < o e = ; X  W 2 / c < ‰ 3 … C @ M  B P J L [ 2 / & V 2 / A – 3 I G ™ J O M = g = ; N @ M L [ P J 9 1 2 / 2 6 ` Π ˆh;=V&V\/ GI/&[neo/219WNM@ΠB2/&GI=/\~x† […JLqh[Pi19/2/2691Z3w>O_]M@@1Z/219B;=3K-A/2=/B3K/2@E269JL’N/-]3x/2-/yY/3 E3”M 3dWNM@JP6 Π|Φ i = |Φ i ƒ ?` „ =‡ Ek3”MO3vJPV&WNM@JL6 Π|Φ i = −|Φ i ‘[Pc/Jt1[˜Ee I37MOM@37V\1+JLWI[L3K69;=;=-IWIJP6K//-(1+1K/2E219WI3wM@JL6K-AE>@V\MO6K/J™-]MO3|-]GN3M@GN-?M@1 -I[L1/21@[™/2MOV&JP1K/219/^-A3+[neG NeoM@OGNM@M@69J™-IM@1+-A@` /2+-AM^3–37GI67;=MO-I-IBd19CDM‚;=>@69V;=JP6uM 37OM@69J™M@-A3‚q B;=V&V5=/B q|/k3:;=GIB2;U;=;=Jz3–69GIWY;=;=-I=E;=/2JL1d6‰[L1K/2W 1–IGIE269J™M@JP_]g@=/V\/-A3l[P/vΠB‚M@1` { ƒ c` „ "=‡ (r, θ, φ) ⇒ (r, π − θ, φ + π) /1KJ3t;=-5/219W3v/2JL@MOE2JP6962JL` ’N/6t_]
2

+ −

+ −

+

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i ~ α ~ ·L ~

1

2

1

2

−iαT



ƒ ?` „ A‡ G eo;=R W;=@/2-A3^`NabB[j;=>U-AJL37/2/2-A-?3QJP6  U (α = U/2(α) † … [ ` @ M P J 9 1 = ; I l 1 I < ( I G k E = >  / P [ = ; I W  W 2 / & V A x 3 P [ 0) = U (α) α=0 † [ − iαT + O(α ) U (α) = [H, U (α)] = −iα[H, T ] + O(α ) = 0 ƒ ?`  A‡ [H, T ] = 0 ⇒ |;=OM@iI[L/@q?_]=/BQ[˜e IM@V\JL[L3K;=-IJP/- ` xOM@69J™M@-A3–WNM@6–376KM@-I1K[PMO37JL;=- q [H, P ] = 0 `c‘cJ[ne NM@V&JP[L3K;Op -IJP/-/193vJP-A>OM@69J™M@-A3lWNMO6l6K;@37MO37JL;=- q [H, L ] = 0 ` ˆWY/h@MO_]6K< J™MOeo;=-]- 3 GNO=/B qI;=-WY/


2

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i

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ipx/~

x

p~

y

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p~x p~y p~z

x

y

z

z

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z

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+





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ƒ Π = Π ‡GI/ŒG?E37/6KV&JP-NMO-]3(EgAM@[mF +1 q‰/3[P/HWI6K;UGI




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k

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k

i

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k

k


 



 

/ 193lGI;@-IBx>@/2B3K/2OM@[P/OM@[P/
k

j

jk

j

U (b)U (a)|Ψk i = =

X j

U (b)|Ψj iDjk (a)

XX j

M@JL1lB2;=V&V&/

=

X

l

|Ψl i

qN;=-M#M@
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X l

X

Dlj (b)Djk (a)

j

ƒ ?`  „ ‡

|Ψl iDlk (ba)

ƒ ?`  A‡ ;=<(/2-?B2;=69/ ƒ ?`z† =‡ ˆ — D(ba) = D(b)D(a) xWIWI[PJLB‚MO3KJP;=- c‘cJ=/2B[P/21–E2[LE2V&/2-A371‰G eo@/2-A3 13K6K/xB2[™MO1K1KE1h/-CD;=-IBk37JP;@-GI/1^6K/WI6KE1K/2-A37MO37JL;=-I1hJL6K69E2GI=-I/21QBk37/2/-A=/(/B#GIJL[nV\eoJPGI//-I-]1K3KJL;=JL3K- E†  Gt[n=eo
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j











    



 

 

 





 



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2

0

0

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X N

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a



I1 =

|;=19;=-I1 

In =

−~2 d2 1 + mω 2 x2 2 2m dx 2

Z

Z

+∞ −∞ +∞ −∞

hΨ|H|Ψi hΨ|Ψi

dx 1 x pi = √ [arctan( √ )]+∞ ∞ = √ +a a a a dx 2 (x + a)n x2

dIn = −nIn+1 da 1 dIn ⇒ In+1 = − n da dI1 π I2 = = a−3/2 da 2 3π −5/2 a I3 = 8 5π −7/2 a I4 = 16

G ef;@R

 






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Z



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d 1 ( 2 ) dx x + a

2

dx

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1 −~2 (−4)[I3 − aI4 ] + mω 2 [I1 − aI2 ] 2m 2 hΨ|Ψi = I2 ~2 1 1 hΨ|H|Ψi = + mω 2 a = E(a) ⇒ hΨ|Ψi 4m a 2 dE = 0 da ~ ⇔ amin = √ 2mω 1 ⇒ Emin = E(amin ) = √ ~ω 2 ⇒ hΨ|H|Ψi =

W;=
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n

n

0

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n

n

n

n

ab[ >UJP/-]3Q

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n

(H0 + λVˆ )|Ψn i = En |Ψn i (2) (1) (H0 + λVˆ )(|ϕn i + λ|Ψn i + λ2 |Ψn i + . . . ) (1)

(2)

(1)

(2)

= (n + λEn + λ2 En + . . . )(|ϕn i + λ|Ψn i + λ2 |Ψn i + . . . )

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H0 |ϕn i = n |ϕn i (1) (1) ˆ H0 |Ψ(1) n i + V |ϕn i = n |Ψn i + En |ϕn i

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“?`x,.6KGI69/  ƒ ≥ 2‡ 



      
  

 

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(k) H0 |Ψn(k) i + Vˆ |Ψ(k−1) i = n |Ψn(k) i + En(1) |Ψ(k−1) i + · · · + En(k−1) |Ψ(1) n n n i + En |ϕn i

n

n

n

(1) n

0

n

n

(1) (1) ˆ hϕn |H0 |Ψ(1) n i + hϕn |V |ϕn i = n hϕn |Ψn i + En hϕn |ϕn i

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n

(k) n

(k−1) n

n

(k) n

(1) n

(k) n



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(k) n






(k) n

n

m

m

hϕm |H0 |Ψn(k) i + hϕm |Vˆ |Ψ(k−1) i n (k) (1) (k−1) n hϕm |Ψn i + En hϕm |Ψn i + · · · + En(k) hϕm |ϕm i

=

⇒ m hϕm |Ψn(k) i + hϕm |Vˆ |Ψ(k−1) i n = n hϕm |Ψn(k) i + En(1) hϕm |Ψ(k−1) i + · · · + En(k−1) hϕm |Ψ(1) n n i hϕm |Ψn(k) i =

  1 (1) (k−1) (k−1) (1) hϕm |Vˆ |Ψ(k−1) i − E hϕ |Ψ i − · · · − E hϕ |Ψ i m m n n n n n n −  m

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(k) n













(1) n

n

m

m

(1) n

(k−1) n

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n

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En(2) = hϕn |Vˆ |Ψ(1) n i X = hϕm |Vˆ |ϕm ihϕn |Ψ(1) i n

m

=

WI
(1)

hϕn |Ψn i = 0

X

m6=n

q?G ef;@R 

En(2) =

hϕn |Vˆ |ϕm ihϕm |Ψ(1) n i

ƒ † ?`L†0=‡

2 ˆ X hϕm |V |ϕn i

m6=n

n −  m

2/ V\M@6K_]O=/2M@<ŒGIE2g@Ekp 6

n

n



m




1 n −m

|;@
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hϕm |Ψ(2) n i =

=


 

1 n −  m

 &  -,'

X hϕm |Vˆ |ϕp ihϕp |Vˆ |ϕn i p6=n

" ,'$$  (

n −  p

hϕn |Vˆ |ϕn ihϕm |Vˆ |ϕn i − n −  m

 0",

!

=37g /JL/@=-I6K///2Bk146 ppp 




  

    

 





      
  

 

K1 JL;=-WI[POM@-A37/O` 6 /2WNM@6437;=-?1 GI/ C…M@JP19;=-I1l[L/mWI6K;UGIUJP/-A3

(H0 + λVˆ )|Ψn i = En |Ψn i

m



>@/2BQ[™M5-I;@6KV\M@[PJL17MO3KJP;=-

hϕn |Ψn i = 1

qN;=-MZ X

m6=n

|Ψn i = |ϕn i + λ

{ -Jz37E26KM@-A3dB2/k3K37/wE_UUJP/2-A3Q |Ψn i = |ϕn i + λ

X

m6=n

X

|ϕm i

|ϕj i

|ϕm ihϕm |Ψn i

X

m6=n

|ϕm i



/k3

ƒ † ?`z† „ ‡

(En − m )hϕm |Ψn i = λhϕm |Vˆ |Ψn i

|Ψn i = |ϕn i +

+λ2



ƒ † ?`z†A‡

hϕm |Vˆ |Ψn i En −  m

hϕm |Vˆ |Ψn i En −  m

1 1 hϕj |Vˆ |ϕm i hϕm |Vˆ |ϕn i + . . . En −  j En −  m

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n

0

n

n

n

n

n

n

n

n



n

n

n

ƒ † ?`s==‡ E − ˆ67M@[ h-]/
2 ˆ hϕ | V |ϕ i X m n n

m6=n

 



m






   

 

  



 









  









 



  

1K;@-]3d/-(g=E2-IE67M@[jV\/JP[P[L/2=/[P;=WIW/2V&/2-A3 6
 

-,


/0$/  

‘C…M@@/2 @MO[P/2OM@iI[L/#GNM@-I1Q[neoE37MO3wWY/693K= hΨ |O|Ψ i @E/:/2[P;@WNWYM@6/V\hϕ/-]|Ψ3 i = 1 q hΨ |Ψ i 6= 1 `ab[C…M@=;=Jz3d_U
n

n

n

n

n

n

n

(1) n

n

n

2

n

(2) n

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n

k+l

n

(k) n

(l) n

k,l=1

n

n

2

n

2

n

(1) n

(1) n

(1) n

m

(1) n

m

m6=n

hΨn |Ψn i = 1 + λ2


 

  "  

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/0$  



ƒ † ?`  „ ‡

2 ˆ X hϕm |V |ϕn i

m6=n



(n − m

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2

n

 




   

(2) 2 E n











  
   

  

2 X hϕm |Vˆ |ϕn i = (n − m ) m6 = n 2 X hϕm |Vˆ |ϕn i ≤ |n − m | m6=n 2 1 X ˆ ≤ hϕm |V |ϕn i ∆ m6=n 1 = ∆

X m

2 hϕn |Vˆ |ϕm ihϕm |Vˆ |ϕn i − hϕn |Vˆ |ϕn i

 1  2 2 ˆ ˆ = hϕn |V |ϕn i − hϕn |V |ϕn i ∆

JL-I1KJ 

!

(2)

En

 1 hϕ |Vˆ 2 |ϕ i n n ˆ − hϕn |V |ϕn i ⇔∆  hϕn |Vˆ |ϕn i (1)

ƒ † ?`s A‡ —M@WIWI69/r;‚C…Ž?M@JLB2V;=M-Q37g=JP;@E--IE21K6KJjM@[P[P/2/O1hq‚37E26K[P/;@V\-I_]/@/‚M@
(1) n

m

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0

(1) n

ni

nj




ni

0

n

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n



 

   

 

  

  



 





   





 

   

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= (n +

λEn(1)

+λ

2

En(2)

+ . . . )(

i

†=`x,.6KG?6K/ 0  H0

c`x,.6KG?6K/ 1 

X i

H0 |Ψ(1) n i+



X

X

(1) 2 (2) hϕni |Ψ(0) n i|ϕni i + λ|Ψn i + λ |Ψn i + . . . ) (1) 2 (2) hϕni |Ψ(0) n i|ϕni i + λ|Ψn i + λ |Ψn i + . . . )

ƒ † ?`s“c† ‡

hϕni |Ψ(0) n i|ϕni i = n



i

(1) (1) ˆ hϕni |Ψ(0) n iV |ϕni i = n |Ψn i + En

M@JL1K;=-I1l[L/mWI6K;UGIUJP/-]3Q i

,

X hϕni |Ψ(0) n i|ϕni i X i

hϕni |Ψ(0) n i|ϕni i

nj

hϕnj |H0 |Ψ(1) n i +

X i

(1) = n hϕnj ||Ψ(1) n i + En (

G eo;=R

ƒ † ?`s“==‡

ˆ hϕni |Ψ(0) n ihϕnj |V |ϕni i X i

hϕni |Ψ(0) n ihϕnj |ϕni i

ƒ † ?`s“@“A‡ x@/2Bk37/2=/B3K/2@/2[P[L/ iNM@1K/ GI<‹1K;=
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(1) n

nj

ni

n

ni

nj

n1

nk

ni

n

ˆeo/2143GI;@-IBwiIJP/-[PM5iNM@19/€_]
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E±

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m

ˆ hϕni |Ψ(0) n ihϕm |V |ϕni i

X hϕni |Ψ(0) n i hϕm |ϕni i | {z } i

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⇒ Pi→n

i = e− ~ n (t−t0 ) hn|UˆI (t, t0 )|ii 2 ˆ 2 = |cn (t)| = hn|UI (t, t0 )|ii

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t>0

2 Z t i i(E −E )t /~ n 1 i dt1 e = − hn|Vˆ |ii ~

ƒ †=†=`s "A‡ ƒ †=†=`s 0=‡

t0

t

dt1 e

i(En −Ei )t1 /~

=

t0



~ei(En −Ei )t1 /~ i(En − Ei )

t

0

~ ei(En −Ei )t1 /~ − 1 = i En − E i

— ;=-IB






   

$/   

Z

,.6




            *  

  

Pi→n

2 ˆ 2 1 − ei(En −Ei )t1 /~ = hn|V |ii En − E i

1 − eix 2 = (1 − eix )(1 − eix )

= (e−ix/2 − eix/2 )(eix/2 − e−ix/2 ) = −2i sin(x/2)2i sin(x/2) = 4 sin2 (x/2)

ƒ †=†=`s „ ‡ (E − E {G eoE3737
ˆ 2 4 hn|V |ii n

4 ω2

i

)2

sin2

(En − Ei )t 2~

2 ωt 2

ˆ 2 E n − E i ) = ~ hn|V |ii f ( ~

,.->@E26KJz’N/QM@JP19E2V&/2-A3d[P/1dWI6K;=W?6KJPEk37E1l1K@MO-]3K/21w 19J ωt = kπ ⇒ ω = 2πk (k 6= 0) f (ω) = 0 Pi→n

2

2

lim f (ω) = t2

ω→0

t






  

 

 



  

 





  





 



  

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4 ω2

2

2

sin (xπ)/(xπ) 1 0.8 0.6 0.4

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0

1

x

2

3

4

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2π~ ∆t 2π~ ∆t

2π~ ∆t ∆E∆t & 2π~ ∆E &

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~ 2

4 ω2 4 f (ω) ≤ ⇒ →0 f (ω = 0) ω 2 t2 f (ω) ≤

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ab[jC…M@



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Z

−∞

Z +∞  x  dx t2 x 1 2 sin (x)ϕ = sin2 (x)ϕ( )dx 2 2 t t t −∞ x −∞ x Z +∞ Z +∞ 2 2 sin (x) x sin (x) lim ϕ( )dx = f (0) dx = πϕ(0) t→∞ −∞ x2 t x2 −∞ +∞

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19J

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t>0

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ƒ †=†=`f“=‡ JP-IG?
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M@JL1l
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1

2

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AB|ϕi = BA|ϕi = Ba|ϕi = aB|ϕi 

rM@@ /2A|ϕi t q  / 3 2 / 9 1 & 3 I @ ; c b p I G  E = g 2 E ? 2 E K 6  E @ / u q O B f e  / 9 1 9 3 b p F b p ? G P J K 6 Z / ] _ < o e P J h [ f e / c Ž P J 9 1 K 3 / = a|ϕi a B3K/2@WIMO6K;=[P/2W?  /  B K 3 2 / I <  6 I W K 6 @ ; I W K 6 / I G / I G Œ / O > @ M P [  / I < 6 A /2[F |ϕi q]1K;@JL3^-]@/2B b E>@/2-A37

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U‘ /2B2;@-IGB2M@1m?ab[j/kŽcJP1437/m=/B3K/2=/B3K/2OM@[P/
a

a

a

a








a



‘cJGI/


  





 

 



  



/21dB2;=V&V5
• [A, B] = − [B, A] • [A + B, C] = [A, C] + [B, C] • [AB, C] = A [B, C] + [A, C] B

— E2V&;=-I193K67MO3KJP;=- 

[AB, C] = ABC − CAB A [B, C] + [A, C] B = ABC − ACB + ACB − CAB = ABC − CAB

•

abG?/2-A37Jz37EwGI/
=M@B2;@iIJu [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 — E2V&;=-I193K67MO3KJP;=- 

(ABC − ACB − BCA + CBA) + (BCA − BAC − CAB + ACB) + (CAB − CBA − ABC + BAC) = 0  

  

 









 





{ -GIJLV\/-I1KJL;=-X’N-IJL/  ;=-(GIE’I-IJL3v[™M:3K67M@B/wG ef
ii

i

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#

 






  



 

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6 (A · · · A ) = X(A · · · A M@JL1l[neoE2[LE2V&/2-A3dGI/wVM 376KJLB2/ (A · · · A ) 1efEB26KJz3w 

(1)

(n)

(1)

(n)

)ii

i

(1)

X

i1 ,··· ,in−1

⇒

6 (A 

(1)

(n)

(1)

ij

(2)

(n−1)

, · · · , A(n) ) =

X i

X

i1 ,··· ,in−1

X

=

(1)

i1 ,··· ,in

=



X

=

6 (A

• • • •







   

66 (AB) =  6 (BA)  (A + B) =  6 (A) +  6 (B)  6 ([A, B]) = 0  6 (ABC) =  6 (CAB) =  6 (BCA) 







(1)

(2)

(n−1)

(n)

Ai,i1 Ai1 ,i2 · · · Ain−2 ,in−1 Ain−1 ,i (2)

(n−1)

(n)

Ain ,i1 Ai1 ,i2 · · · Ain−2 ,in−1 Ain−1 ,in (n)

i1 ,··· ,in



(n)

Ai,i1 Ai1 ,i2 · · · Ain−2 ,in−1 Ain−1 ,j

(1)

(n−1)

Ain−1 ,in Ain ,i1 · · · Ain−2 ,in−1

(n)

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A(1) · · · A(n−1) )



€) ' + (% '
*
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x-#E[PE2V&/2-A3 |ϕi ∈ ` /2143rGIE’I-IJNWNM@6t 2 / B 2 / 9 1 . 3 I G  E N ’ ? J |ϕi = (c , . . . , c , . . .) |ψi = (d , . . . , d , . . .) WNM@6w X ƒ €` =@‡ hψ|ϕi = d c 





 



`(2) c1 , . . . , cn , . . .

3

2

+∞

2

2

n

n=1

1

n

1

n

+∞

∗ n n





n=1

 
 





ƒ :`s@“A‡ ˆef/193x




 

⇒

|hϕ|χi| < +∞



2

2

2

2

2

2

hϕ − λψ|ϕ − λψi = kϕk − λ∗ hψ|ϕi − λhϕ|ψi + |λ| kψk ≥ 0 ∀λ ∈ C

{ -B I;@JP1KJL1K1KM@-A3 λ =

kϕk2 hϕ|ψi

⇒ λ∗ =

kϕk2 hψ|ϕi

qI;=-;=i?37JL/2-A3w

kϕk4 kψk2 ≥0 |hψ|ϕi|2 kϕk4 kψk2 ≥ kϕk2 |hψ|ϕi|2 kϕk2 − kϕk2 − kϕk2 +

‘cJ kϕk = 0 qI[neoJP-IEgAM@[LJL37Ev/2143x>U6KM@JP/O`‘cJ kϕk 6= 0 qN;=-(GIJz>cJL1K/mWNMO6 kϕk ⇒

2

2

 



 
 

   

/32q

M@JL12q

 

2

⇒ |hψ|ϕi| ≤ kϕk kψk

2



  

|ϕi, |χi ∈ `(2)

⇒

|ϕ + λχi ∈ `(2)

hϕ + λψ|ϕ + λψi = kϕk2 + λ2 kψk2 + λ∗ hψ|ϕi + λhϕ|ψi hϕ − λψ|ϕ − λψi ≥ 0

⇒

kϕk2 + λ2 kψk2 ≥ λ∗ hψ|ϕi + λhϕ|ψi


kϕ + λψk2 ≤ 2 kϕk2 + λ2 kψk2 < +∞

ƒ €`  @‡

 

 



 








  



 



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([a, b])

b

2

a





 
 



  



ϕ(x), ψ(x) ∈ L2 ([a, b]) ⇒ ϕ(x) + λψ(x) ∈ L2 ([a, b]))

 

{ [P[L/mGIE2B;=
  

ƒ :`s 0=‡

kϕk2 + λ2 kψk2 ≥ λ∗ hψ|ϕi + λhϕ|ψi

 

‘cJ ϕ(x) q χ(x) ∈ L

 
 



/2143viIJP/-GIEk’N-IJ   

 

B2;@-IGI


(2)

hψ|ϕi ≡

([a, b]) Z

qN[P/mWI69;cGI
b

dx ψ ∗ (x)ϕ(x)

ƒ :`s „ ‡

a

{ -/kyj/k3‚qN[˜efJL-IE2gAMO[PJL3KE:

  

Z

b a

(ϕ∗ − λ∗ χ∗ )(ϕ − λχ) ≥ 0

(∀λ ∈ C)

Iq B;=-IGI
     L ([a, b]) ˆeo/2193&@MO[P[P/ [a, b] 










 

kϕk2 + |λ| kχk2 − λ∗ hχ|λi − λhϕ|χi ≥ 0

λ=

kϕk2 hϕ|χi



2







M‚>@/2B

ϕ(x) =

+∞ X

n=−∞

1 cn = √ b−a

cn √ Z

b

2iπnx 1 e b−a b−a

dx ϕ(x)e−

a

2iπnx b−a

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n

2

√1 e b−a

1 kϕn (x)k = b−a

/3

e





Z

b

dx e−

2iπnx b−a

e

2iπnx b−a

ƒ €`s“ =‡

=1

a

Z b 2iπx(m−n) 1 hϕn |ϕm i = dx e b−a b−a a i 2iπa(m−n) (b − a) h 2iπb(m−n) 1 = e b−a − e b−a b − a 2iπ(m − n)

M@JL1 



  

B2;=-I1437Jz37
2iπnx b−a

2

 



/21rCD;=-IB3KJP;=-?1 ϕ (x) = ` { -/yY/3‚q L ([a, b]) 

 



 

2iπb(m−n) b−a

−e

2iπa(m−n) b−a

=e

2iπa(m−n) b−a

h

e

ƒ €`s“?†}‡ ƒ €`s“==‡

i −1 =0

2iπ(b−a)(m−n) b−a

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⊥





 

⊥



⊥

   
  

⇒ hϕ|ϕ⊥ i = hϕ|ψi − a = 0

/QW?6K; /B37/


P|ϕi

/2143vGIE’N-?JWNM@6

P|ϕi |ψi = a|ϕi = hϕ|ψi|ϕi

ƒ €`s“@‡

   



   

 

   






  



 

       

/ WI6K;UGI@/2Bk37/2OM@69J™M@i?[P/@q ;=- M b ... b iJPI-AJL3K/26K- ;UGI1 I6
 a1 |ϕi = . . . an



∗ 1 1 ∗ 1




∗ n



  



∗ n n



 b1 |ψi = . . . bn 

 



∗

   

      # 





— efM@WI69821vB2/w_]
ƒ €`s“ 0=‡

P|ϕi |ψi = hϕ|ψi|ϕi = |ϕihϕ|ψi P|ϕi = |ϕihϕ|

⇒ 

     

     



‘GI/1x;@WIJL693 ; {|ϕ/2Bk37i,JL;=i-I1l=GI1,/m37. ;@. @/2Bk37/2
  X ƒ €`s“ „ ‡ |ψi = c |ϕ i ˆh;=V&` V\/ef;=[™R M5iNM@19/m/2143x;@69%3 ?;=-I;=69V\E/@q hϕ |ϕ i = δ ` JL-I1KJ˜q P c |ϕ i /193d;=64%3 I;@g=;=-NM@[L/mF — |ϕ i i





 





N

i

i

i=1

i

j

N j6=i j

ij

j

i

P|ϕi i |ψi = ci |ϕi i ⇒

N X

P|ϕi i |ψi =

N X

ci |ϕi i = |ψi

ˆh/3937/wWI69;=WI6KJLE3KEQ1efEB269JL3‚q?/2-H-?;@3”MO3KJP;=-ZG?/ — JP67MOB  † [ = X |ϕ ihϕ | { [P[L/./2143B;=-I-]
i=1

n

n

n

ƒ €`s“ A‡ ƒ €` A‡




              



 

    









  

 



     



 



  





x- ;=WE26KMO37/
mn

 



m

n

 
 



ƒ €`I†}‡ /21tE[PEV\/-]3K1tGI/‰VMO3K6KJLB2/–GI< V\/V5iI69/‰GI/hGI6K;=Jz37/–1K;=-A3uG?;=-I-IE1tWNM@6‰        ! X X ƒ €` ]@‡ hϕ | A |ϕ ihϕ | |ϕ i = A hϕ |ϕ i hϕ |ϕ i = A | {z } | {z } /:V&/2V5iI69/€G?/:GI69;=JL3K/:M&[P/21vV12V&/21vE2[PEV\/-A371G?/:V\MO3769JPB2/w_]
X m,n



 



i



mn

m

n

j

mn

m,n





 

 



Amn |ϕm ihϕn |

i

m,n

 







 







m

δim

n

j

ij

δjn

   
  

ˆh;=-I19JPGIE6K;=-I1dO@B/2;=B3K-I/214OM@/0[L/2[™OM@WI[L69/2;=O1 M@[LG(i) 2 / I < 9 6 1 a i N 1KJv[neoOM@[P/=/B3K/2OM@[L/2=/B37/@MO[P/2
j

i

i

j

i

i ji

i

i

i

i

i

(r) i

i

(r) i

(r 0 ) j

 











  



 

‚B MO6 hϕ |ϕ i = δ ` ,.-(/2-GIE2G?
(r)

(r 0 ) i

(r) i

ˆ • hϕi |A|ϕ i i = ai δrr 0

rr 0

ƒ €`=‡ — /lC…M ;=- B2;@V\VWNMO12BV&37/O/mq=1K;=J -#-I;@3K/21u19` ;==/-A3 Aˆ = P a |ϕ ihϕ | M‚>=/2Bv[™MxB2;=-A>=/-A37JP;@-\_]
G(i) XX r=1

i

hϕi |ϕj i = δij

 

    





(r)

 







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i

i

ai = a j



(r)

ai |ϕi ihϕi |

  

 



 

i

i






  

1v;==;=-I1d>U<(_UOM@iI[L/21lB;=V\V:=67/2MO6h3K[n/2eoO3KJPM@;=[P-/@MO[P/2@/2B Aˆqc/3hGIE219JPg=-I;@-I1^WNM@6 {|ϕ i, n = 1, . . . , N } =E/@q ;@- WY/OM@i?[P/21Q_]=/B3K/2
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n

n

m

n

n

n

n

n

n

m







1

p

i

(1) n















 



(p) n

(i) n

n

n

(p) m

(1) m














† †



  



 

  

†

−1





  





 









  





 
 
    

 





=† `x‘cJ Uˆ /193v=/6K1K/V\/-A3‚q1K@/€[PM -I;=6KV&/@` [P;@6K12q ∀λ ∈ CqI;=-M( †

ˆ (ϕ + λψ)|U ˆ (ϕ + λψ)i = hϕ + λψ|ϕ + λψi hU

M@JP1

ƒ €` =‡

/

hϕ + λψ|ϕ + λψi = hϕ|ϕi + |λ|2 hψ|ψi + λ∗ hψ|ϕi + λhϕ|ψi = hϕ|ϕi + |λ|2 hψ|ψi + 2< (λhϕ|ψi)

/ 3 hUˆ (ϕ + λψ)|Uˆ (ϕ + λψ)i = hUˆ ϕ|Uˆ ϕi + |λ| hUˆ ψ|Uˆ ψi + 2
=‡ uefEgAM@[LJL37E ƒ :` A‡hJLV\W?[PJP_]


ƒ €` =@‡

ˆ ψi = hϕ|ψi hUˆ ϕ|U ˆ † Uˆ |ψi = hϕ|ψi ⇒ hϕ|U ˆ= ⇒ Uˆ † U

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−1

 

 





†





 



Uˆ

  
  

Uˆ

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Uˆ



*



 







     



ˆh;=-I19JPGIE6K;=-I1v
 



 
 

   

 

†



{|ϕn i, n = 1, . . . , N }

/k3v
/21l>=/B3K/2


 

0 n

n



ˆ †U ˆ |ϕi = hϕm |ϕn i = δmn hϕ0m |ϕ0n i = hUˆ ϕm |Uˆ ϕn i = hϕm |U

ƒ €` @“=‡





 






  



 

ba -A>=/6K19/2V&/2-A3‚q|B2;=-I19JPGIE6K;=-?1€GI/
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  0 n

n

0 n

n














 

 

|ψi =

X n

cn |ϕn i

ˆ ψ|Uˆ ψi = ⇒ hU

X

ˆ ψi = |U

⇒

c∗m cn

hϕ0m |ϕ0n i

| {z }

B;=-I1K/69>@/:[™M5-?;=6KV&/@qIJL[ /2143vGI;=-IBw
   

  





  

m,n

=

X n

X n

δmn

cn |ϕ0n i 2

|cn | = hψ|ψi

ƒ €` ]‡

         

c‘ ;@JP/2-A3 {|ϕ i} /3 {|ϕ i} GI/2=/2Bk37/=/V\/-A3#GNM@-I1 B2/1xG?/2
     X ƒ €` =‡ c = Uˆ c 0 n

n

n










0 n

0 n



0 n

  

 

† nm m

m

  

c0n = hϕ0n |ψi = =

X m

X m

=

X m

=



   

X m

 

n

    





   

hϕ0n |ϕm ihϕm |ψi hUˆ ϕn |ϕm ihϕm |ψi ˆ † |ϕm ihϕm |ψi hϕn |U † Uˆnm cm



#  
 

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ij

i

ij

i

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— E219JPg=-?;=-I1mWNM@6 {|ψ i} [PM\iNM@19/ GI/519/21.>=/B3K/2
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i

ij

i

i

i ij

i

ij

i












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i ij

ij



ˆ ji di δij = hψi |A|ψ X ˆ n ihϕn |ψj i = hψi |ϕm ihϕm |A|ϕ m,n

=

X

† Uim Amn Unj



ˆ † AˆUˆ = U

m,n



ƒ €` =‡

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$ 
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n

n

n

n

n

n

n

Aˆ =

an |ϕn ihϕn |

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0

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1

0

ƒ €!` "==‡ 19J x ∈/ x − , x +  ‚M >=/2B 19J x ∈ x − , x +  ƒ ;@-HW/2OM@-A3w δ(x − x0 ) = lim ( a→0 0 ϕx(a) (x) = 0 1

ϕx(a) (x) 0

0

0

a 2 a 2

a 2 a 2

0 0

0

+∞

0

0

0

−∞

0

 



 
 

  

 



Z

  

+∞ −∞

dx |xihx| =

ƒ €`!" ]‡

†[

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Z

+∞

−∞

dx |xihx||ψi =

Z

+∞

−∞





+∞



 





−∞

  

dx hϕ|xi hx|ψi = hϕ|ψi | {z } | {z } ϕ∗ (x)

+∞

−∞



+∞

−∞

ψ(x)




                









 





    



 



  





uef;=WE26KMO37/


p ˆ



pˆϕ(x) = −i~

/21lCD;=-IBk37JP;@-W?6K;=WI69/21dGI/ pˆ 17M 37JP14CD;=-A3

ƒ €`!" „ ‡

∂ϕ ∂x



−i~

WI
2

ipx/~


Z



       

  

⇒

ϕ(x) = Ceipx/~

= −i~ ip~ eipx/~ = peipx/~ +∞ −∞

   



ƒ €`!" =‡

∂ϕ = pϕ(x) ∂x

2

dx |ϕ(x)| = C

2

Z

`xˆh/21CD;=-IBk37JP;@-I1(-I/ 19;=-A3 WNMO1GNMO-I1

+∞

dx

ƒ €` 0

=‡

GIJL>@/269g=/

−∞



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     ƒ €` 0@“=‡ 1 ϕ (x) = √ e 2π~ [™M 6K/[™MO3KJP;=-GI/mCD/26KV&/3K


  

2

+∞

−ikx

−∞

+∞

ikx



−∞










ipx/~

p

+∞

−∞









 





   hx|

Z

 




  



 

  

+∞

dp |pihp||ϕi =

−∞



Z

+∞

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