Quantum Introduction


  



  !#" $%!$ &('' )+*,.- %/02143563 $Id: quantic physics.lyx,v 1.31 2005/01/24 08:18:34 itay Exp $

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n

E

s

†

^

8{½µ¾68:¿D8AÀÁ=5¾ÃÂ Ä zHj`SQ j`z tiJRNiJRNiL(yšj`qw™zIvk€dmzHq‘zIJbaIdPvq%lj`UaINGL(e§kdPJRU N F V F tiJRL(vwutGJMvkdPzIqtiJRNiJRNiL(y»F n F kdPJRU Ngj€tiJRNiJRNiL(y‹L:cŽzHj`JRL(ejp~™tik‘j`L(L(k‘zHj`SQj`zIk F tGJMJbaIU Ni|XJM~›tiJbwDU S‚™kJRvU dPeKF tiJRL(NiJROPJMQTtGJML‡~viL:cŽk€JR“OPJbaIdjpj`N F E ŜÆ]Ç ÈDÉ.ʖ˖˖Ì6͚Î[Ëo˅ÏО˅ÑÌÃÒÓÏgԜÌ6Ô[ÕHÖi×oÌÃҀÕÏžÖ Æ<ØœÆ WYV ”]” „;f€Ù;U kJMj`|XJRd lPUaINiL(etGJRaIL(Q d!zHq€JMj`|Xq•zwDzIq2‚5Sv–j`Q λ = 10 anstron
L‡j`v|žlPkUaIL‡j`e%‚‹j`e‘ke%U d%U j`eÚF V tGJMdPj tiJRdj`UaINGL(e kJML(SqŠNiU;c£RŸ|ÛzHj`U JM~zG£6cXJ6zwDzIqŽL:wDL.kdPJMUNzHj`U JM~DzI‚§k€JMj`L(zÜkaIJRL(Q kÝF n F tGJRaIL(Q d U j`e%kzIq2“S;LTzIJML(dj`JR“Uj`Qj`U QxtiJRdj`UaINGL‡e!ktiU Omk•zIq2“SŠF E zHj`U JM~DzI‚‹kJMj`L‡z}tGJMdPj`UaINiL(e%kL(cˆkJMvU dme%kÝF s UcXe2w E = hυ kJRv–UdPe˜cXJGNiJRNiL(yL:wDLƒF tiJRNiJRNiL(y•L:cޟ|j`e«e%JRkƒkdPJMUNikÜ_`JMJbacXdPJRe‘L:c0_pj`JRSU ~q2q€k§‰ h = 6.63 × 10 (joul × sec) kq€v–jp~DLšF _`q€OiL‡j`QwškJRvU dPe˜e%j`k h L:c»~q2q€k fzIJMzIJMjpj`OGS;dPzrL:cŽe%j`k L(c W JM~w™_pj`UaINiL(e%L kcXj`U~ c0kJMvU dPe!kzIe˜zIU e%zIqLc =wmvr k€~Dj`‚+S5zIJMJR“Nidj`QTcXJzwDzIqƒhL:wDL JROPeKF U U yzcXkL mv = hυ − w ŜÆ]Çß È2ÐXʅԜ̞Õ%àGÔAÒ§ÌÃҀÕÏ Æ<Øbß tiv–j θ zIJmj`OP‚uzHUOPQ zIq‘kdPJMUNik•Uq!j`y‚‹zIS;voj`Q x zIdPJRU N}UcXe!w™F λ = 1WYV ” Z<^—_`vaIdPUfq%£be§kdPJRU Nˆá X zIdPJRU N 3

−34

1 2

0

λ −λ

=



2

h me c



(1 − cos θ)

âW _pj`UaINGL(e%kzI|Xq m f€F _pj`UaINiL(e%k•L(cŽ_pjYaIQ q2j`N«L‡vi¢;Uj`e§e%U NGd   = 0.024A Ucge!w U|XyxNGJMNGL‡yF ~DUe%JML(JR‚TUjp~ w›j`qwTU OPQ zIq2j_pj`UaINGL‡e!‚TSv–j`Q k|XqšU |Xy5NiJRNiL(yk•Wâ_pjYaIQ q2j`N—fŒU ‚Œ|Xk zHjYcXvdPzIkJMdmQ L#k|Xq zHjYcXvdPzIk€kKã¡V6U j`JMe h me c

e

E P

= hυ hυ = c 3

zHjYcXvdPzIk‘U yhe%L E P

0

0

= hυ 0 hυ = c

0

= mc2

Ee P hυ + mc2

~2 P

=

kJMvU dmeKU j`q2JbcŽJROPe 0

= hv +

P~

zGjYcgv–dmzHkk‘U yhe%L›_pj`UaINiL(e%k•L(cXj

p

m 2 c2 + P 2 c2

S;dPzrUj`q€JRc

0 ~ = P~ + P

_YwDL

2  p~ − p~0 0

~ 2 c2 P

= p~2 + p~ 2 − 2~ pp~ !2 !  2   0 0 hv hv hv hv = + −2 cos θ c c c c    0 0 2 = hv − hv + 2 (hv) hv (1 − cos θ)

m2 c 4 + P 2 c 2

=



hυ − hυ

0

2

(1)

k€JRvU dPeƒUj`q€JRc

  0 + m2 c4 + 2 hυ − hυ mc2

(2)

e%“Dj`JzHj`ejpjYcXkƒJMdMcXq

  0 0 (1) , (2) ⇒ mc2 v − v = hvv (1 − cos θ) 1 1 0 − v v

=

0

⇒λ −λ =

h (1 − cos θ) mc2 h (1 − cos θ) mc2

ä ËÓ2å;æ ä ËÒ2ËÒÓç Æ<ØRè éêRëDìhí+îxïhð+ñîï ò…ó¡ôAóRò tiJRU~Dj`|XqŠzGj`L‡j`NGL;j`q%õœtiJRq!jYaIeÞj`‚DcÝcXJR‚+v‚‘zISv–j`Q L‡v4¢‡Uj`eW¡q!£ReXf zIdPJRU NÞL:cÛUj`OPJRQ0WYVZ
sin θ = n



λ 2a

4



j`e

v–U‚™Uj`OPJRQxã¤nUj`JMe

Rê ÷€øùù…î›úžùðhêMú ò…ó¡ôAó¡û JML‡j`U ‚hlPk2~xL‡vg¢;Uj`eƒcXJ P SdPzrL(S;‚™NiJRNiL(yLŠWYVZ:n…E;fJML‡j`U ‚hlPk2~ λ =

p =

a

h p

_YwDL

E = hυ = pc, λ =

c υ

_pjYaHj`Q LTkq€v–jp~DL

hυ h = c λ

S jp~JtiJRUjYcXJRq‘_`JR‚™NGyU q ü ñDùëð.ú6êMýhþ[ð‹ÿDêMøù ü ÷Iî‡í ì ü ù –ê Mò {û ü ð ü þ[ù bùëêMùñ
  



sin θ

=

λ

= = =

⇒V

λ 2a



NiJRQ |Xq‘_YaIN λ ¢‡JMU“™j`kcXq‘zHj`e%U LTL(JR‚ŒcX‚ L(e%JM“dMaHj`Q ‚šÙ;ej`qƒ_pj`U;aINiL(e í êRþ ü é V (volt)

= 1.6 × 10−19 [coul] · V [Joul] √ 2mE h √ 2me 6.6 × 10−34 √ 2 · 0.9 × 10−30 · 1.6 × 10−19 12.3 √ A V 2 (12.3) [Joule]

E = eV p

= n





'

JROPe

m = 10−15 [kg] , v = 10−3

6.6 × 10−34 h = −15 m mv 10 × 10−3 = 6.6 × 10−6 A

λ =

^

m s

Ni‚ŒeƒU JRv–Uv éêRë í+î ïhðhþ[ùñ  



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1 λ

= R







1 1 + n1 n2









"!



vU ‚+~JMUTSj`‚+N R ' 10 A Ucge!w k€dmJMU N}L:cˆzHj`U~|ŽcXJ%Ucge!w U q2L(‚™zIU~| n = 2 W¡eXf _`q2JRL zIU~| n = 1 W¡‚!f _YcgQ5zIU~| n = 3 W¡v{f ü ïhùý™í ú»ÿDùP÷2ìï™í ñDùð …ò ó :óRò zHj`ydPk L(cˆkq2L:cˆkL;j`Q;w‹ej`k‘JRzHJMjpj`OPkS;dPzIk‘j`‚DcŽJRL(v–S;qL‡j`L(|Xq2‚‹S;di_pj`UaINGL(e%kÝF V −3

−1

1 1 1

$#

%

F~ =

h 2π

(n = 1, 2, 3, ..) mvr

= n



h 2π



¡W JRU dj`JR“ aI|ˆ‚Œ“qifF j`zI“e%k‘zHj`U q€LTkdPJMUNraIL;j`Qxj`dPJReK_pj`UaINiL(e j`e«kaIJML(Q ‚.e!aI‚ŒzHq˜k€JRvU dPe%kƒJMj`dPJbcjHUye%L™~ye«JRUcXQ e}L‡j`L(|Xq€q«Ùj`Q NGLxL‡jYwDJX_pj`UaINGL(e%kÝF n F kdPJRU N}L:cŽkS;JML(‚ tiJRJRNiq•JROmwDU q•yjYw‹‚ŒJM‚Œ|«SdmcˆŸ j`v L‡~Dj`q2k e2 r2 E E2 − E 1

mv 2 r e2 1 = − + mv 2 r 2 = hν =

(3) (4) (5) &$' (*),+.-0/1' 24356'$'$78-95$) 3:/<;

6

JROPe (3) →

e2 v e2 v

⇒v r ⇒E R

= mvr   h = n 2π   2πe 1 = h n 2 e h2 = = n2 mv 2 4π 2 me2 2π 2 me4 1 2π 2 me4 1 −4π 2 me4 1 + =− = 2 2 2 2 h n h n h2 n 2 2 4 2π me ' 10−3 A = h3 c

Ucge!w ҀËÒÓç6Ö XÔ[åÖGÎ 6Ö Ó!Í Ô[ËÓÏgÔ Æ<ØbÉ ÿŒêî;ñDëuêRøú bùêRëêMø­ò…ó :óRò Young lPk‚hj I ∼ |E| kq€“Sk§SJRv–qc»k€~ cL:wDLuOme«~DyheˆÙ‡JMUytiJRq2|j`yƒUcge!w«F tiJR“JRU y‘JMdMcÞ¢;U~.U ‚hj`S‡cÞL‡v6_pj`zId zHjYwD‚+e%z _YwDL‡jIW¡zGj`JRU e%JRdPJRL#zHj`ejpjYcXqKtiktGJML(vHL:c y£R~DqGf E = E + E L(‚ŒNidItiJRyj`IzI∼QxtG|EJMNi~+|ˆEJRdm|cŠUj`‚ŒS 2π 2 me4 h2



' 13.2ev =



>



?

@

$A

2

B

1

1

I

2

2

2

2

2

2

2

∼ |E| = |E1 + E2 | = |E1 | + |E2 | + 2< (E1 E2∗ )

tiJRNg~|Xk5~yeš¢;U ~ÃtGJM|XdMwDdmc‘tiJRdj`UaINGL(e%k›zHe‹tGJm~D~Dj`q™j`dmJMJRk5j`L‡JRe‘F tGJMNGJMNGL‡yµtiv–jtiJRdjYaHj`Q—Uj`‚ŒSºkUj`N•U ‚h~k F zIJMNGJMNGL(yk•kzIUj`“LTzIU Oj`yk€e!“Dj`zHk‘kzIJRJMk W¡zIJM|Xe!L‡NzIJRaIdPvq!lj`UaINiL(eukdPJRU NšL:c‘kq2v–jp~‚4JMv–j`L(dPe E Ucge!w€f+W¡L(vzIJMJR“Nidj`Qf ψ (x, t) UJm~vd í þIéêMê €î;øù W¡kdPJRU NG‚™kq2“Dj`S;LTzHJRv–j`L(dPeXfzHj`U ‚+zI|Xkk P = |ψ| U JM~vd kNiJbaIq2zIJRU e lP‚+zI|XkHf¢‡|Xq2‚4_pj`UaINGL(e%k›tgj`NiJRq€LXzGj`U‚+zI|Xkk |ψ | UcXe!w ψ (x, t) yj`zIQÃ~ye™Ni~D|•Uj`‚+S«F V W¡tgj`JR|ˆ‚Œ“q€L#zHj`U zIe ej`“q€LzHj`U ‚ŒzI|Xkk2j ψ = ψ + ψ OmeöF L(vÃzHJRJR“Nidj`Qk€OMcÛU ye%qˆtiJRNg~|JRdmcÛU j`‚+S«F n tGJMNGJMNGL‡yhk |ψ + ψ | kdPNi|Xq F zGj`U‚+zI|Xk‘U e%zIqƒL(vkcŽ_`‚+j`q2‚‹F zIJRL(vikdjYwDzrcXJ%NGJMNGL(yLrF V F P = |ψ| e%JRk•zHj`U ‚ŒzI|Xk€k ← ψ (x, t) L(vk•zIJMJR“NidPj`QˆF n F zHj`JMe%~Œj2e%L‡j2zHj`JMzHj`U ‚ŒzI|Xk‘NiU5_`kNGJMNGL‡yhL zHj`JRaIdPjpj`NŠzHj`JRQ “z F E F kq€“Sxk€~JM~Dq€kq‘zISQcj`qƒk€~Jm~q€kL:cŽke%“Dj`zIkF s F ψ j`e ψ tGJM‚Œ“q€k•~ye§zHeKkU y‚™k€~JM~Dq€kÛF ^ C

2

1

1

2

1

2

1

2

2

2

2

1

DFEHG IHJ
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F _pj`U NiJRSk•L;jp~v‚ éùìñŒùù–ï Mêmì bù ü î ŒíxêRùëêMø–ï‹íúÞïhðhþ[ùñ JMOPeK∆xF zH·jYwD∆p‚Œe%zI>k€‚šh tij`q2JR|XNiq‘_`zHj`dPk a sin θ = nλ kOPe%Q k•cgU Q kL(S›L:wDzH|Xd tij`q2JR|XNiqJMdmcŽ_`JR‚xcXU Qk€k M R

S

x

n ∈ Z; sin θn =

n λa

∆xmax

≈ d sin θn − d sin θn−1   λ ≈ d a

∆xmax

F zHj`e~Djpj`k§JReK_pj`U NiJRSTQD£bS j k€JR“Dj`L‡j`OPUx¢;JRU “utGJMNi~D|Xk•~ye§zIe§~Djp~Dq€LTJM~wxJROPe a 2

∆px

_YwDL

∆θ

∆x

>

2h a

≥ P =

h λ

UcXe2w‹zIJmjpj`OP‚‹zHj`e~DjpjblPJRer_`zHj`d

∆px P

= ∆θ

2λ 2h λ · = a h a

tiej _YwDL

λ > 2 d > ∆xmax a

748 ‹9;BBŒ¾ C™Â6BDC¥½ <8 ½ 5Â6B;À #½ ›B —8:9;B} 9‡8 µÂ CxB TBB IçžÏ»Ê…ÔÔ[Ëoæ GÎÍ Ò€Ë]Ò%Ó çÝÑ2Ô ÜiÖ ÏgÔÔ͚àÖ žÑ %åÖ ßŒØœÆ zIJbaIdPJMNGkkJRvU dPe%k zHJRL(e%JR“dmaHj`QxkJRvU dPe V (x) zIL(L‡jYwDk‘k€JRvU dPe%k E= + V (x) kejpjYcXq€k =T

9U

0V

XW



ZY

\[

,]

fe

_^

a`

bV

1Y

ge

c

VcW

h

p2 2m

P2 2m

i~

∂ψ (x, t) ∂t

= −

~2 ∂ 2 ψ + V (x) ψ 2m ∂x2

zHJM~q2q‘zIL(zIk‘kejpjYcXq€k ~2 2 ∂ψ (x, t) = − ∇ ψ + V (x) ψ ∂t 2m
~2 ∆ = ∇2 ⇒ = − ∆ψ + V (x) ψ 2m i~

E

=

Uj`‚ŒS}UjYwDOPL#kaIJbc

P2 +V 2m ψ = ei(kx−wt)

h λ = ~ω = hν

p = ~k = E

8

yNiJRdj

d

yzIQ d i = − Eψ h

∂ψ ∂t ∂ψ i~ ∂t ∂ψ ∂x ∂2ψ ∂x2 2 ~ ∂2ψ − 2m ∂x2

i pψ ~ p2 = − 2ψ ~ p2 = ψ 2m

=



=

Eψ

= Eψ

p2 +V 2m



j`dPL(‚+JRNrw+£Rk|

ψ

kU Sk

i

L(vizIJRJM“NGdj`Q›JP£bS5L:cXq€L#U e%j`zHqƒJbaIdjpj`N}‚Œ“q•cXJIF ‚DaIJMkU~voj`q‘L‡j`L(|Xqƒ_`JReKNGJMNGL(yLrF V P ã JRL‡U;aINiQ |ŠNgj`U QŠF n c ψ ψ (x, t) = W¡tiJR‚+“q€k~ye!LTk€|XJRU NrcXJ!OPeƒtiJM~Œ~Dj`q‘Ucge!w€f2k|XJRU Nik.aIQcXqÛF E ÖH˅Ñ2Ô[Õ ä рÔ[Õ Î]ÑÌ6Ô ä ËÓ!å žÑ2Ô gç ߌØbß F ‚ŒyU q€kL:wD‚™NiJRNiL(yšej`“q€LTJmjYwDJR|Žj`zHj`eKcXJ%JbwuJRzIJMJRS‚™kO |ψ| = 1 j`dPL(‚ŒJMN e Uj`‚ŒS tiJRL(vizIL(JM‚ŒykdP‚ŒdIL(JM‚Œvk€LµJm~ wuF ‚+yhU q2k•L(wŒj2tiJRdPq2Omk•L:wuJRdPQ›L(STcj`U Q ψ a a a



j

2

f (x)

g (k) =

=

Z

∞ −∞

zIJMUj`NGq2k•k€JR“NidPj`Q L Uj`‚ŒSdGF ∆k = ‚+yj`U kzIe§UJm~vdIJMOPe √2 α

=

Z

Z

∞

mO e |j`e%vX_pj`q€SQ

2

2

e−α(k−k0 ) eikx dk

−∞ ∞

2

kJRUj`Q5tiUj`Q |XdmUa˜L:cŽk€OmOPkk•zIy|j`dP‚™cXq€zcXkL5_`zHJRdkJMk_`JMQj`L(yL

F [f (ax + b)] (ω) =

ω 1 ibω e F (x) |a| a 9

UcXe2w

F [f (x)] (k) = g (k) f (x) = eik0 x δ (x − x0 )

g (k) = e−α(k−k0 )

e−α(k−k0 ) +ikx dk −∞ Z ∞   02 0 0 0 ik0 x e−αk eik x dk k = k0 + k ⇒ = e −∞ Z ∞ “ 0 ”2 0 −α k − ix x2 ik0 x 2α dk e = e e− 4α −∞ r  x2 π − 2α +ik0 x = e α =

i(kx−wt)

g (k) eikx dk

Uj`‚+S5kq2v–jp~LxF tiJRdjYc k lPLTzHj`L‡j`NGcXq§_`zHj`d f (x) L(cŽkJMUj`QxtiUj`Q|XdPUa f (x)



kq€vojp~

OPe 2

2

|f (x)| = |ψ| = |E| =

2

√2 2α

(g (k)) → ∆k =

√ π − x2 e 2α → ∆x = 2 2α α

∆k∆x

L:cŽ‚Œyj`U j`dmL‡‚ŒJRN

= 4

UcXe!w

p = ~k ⇒ ∆p = ~∆k ∆p∆x ∼ ~

kOPe%QkzHj`U JMkq2j e =

vp

Z

f (x, t) =

∞

ω k

w (k) = w (k0 ) + (k − k0 ) f (x, t) =

Z

∞

0

dk e

−αk

02

e



1 + (k − k0 )2 2

−∞

Ome |f (x, t)|2

~2 k 2 2m

=

s

*S

JRcXQ j`yNGJMNGL‡yšUj`‚ŒS›kq2v–jp~L 

d2 ω dk 2

k0 „ i (k−k0 )x− w(k0 )+(k−k0 )( dω dk ) »

 k

Ml

dkg (k) ei(kx−ω(k)t)

−∞

dω dk

tiJMj`|Xq‘L‡v Rð …ýï ŒøXï+í êMý

_`q2Om‚ukSdHe%JRk¢;JReKk€e!U dj2kL(JR‚+yšUj`“JRd

~ω = 

i(kω−ωt)

2β =



k0

+ 21 (k−k0 )2 k0



d2 w dk2

w (k)



k0

“

tiv–j

2

α(x−vg t) π2 − 2 α2 +βt2 ( ) e α2 + β 2 t2

d2 ω dk2

”

k0

« – t

dw dk k0




= vg

UcXe!w

í þ]ï•éêMê €î;øù á t _`q2Om‚hj x tgj`NGq2‚xNiJRNiL(yšej`“q2L zHj`U ‚ŒzI|Xkk zHj`Q JRQ “ P = |ψ| qzHj`JMU;aIdPq2L(e§zGjYcgJMU~ P Sj`‚+JRU ‚™zHj`JRL‡JR‚ŒU vaIdPJMeƒ_`k ψ (x, t) kJR“Nidj`QkaIU Q ‚™_YwDL R P dx = 1 kJM“OmJML(q€Uj`d5F V C

nm

2

∞ −∞

Z

|ψ|2 dx

< ∞

W ψ ∈ ` ‚™U ‚+jp~q€cŽJMLµkq!~d[f ψ ∈ L zHj`e%U NGdij`L(e!w‹zHj`JR“Nidj`Q F _`q€OP‚™kdPzcXq‘kdPJRe}W¡‚+yhU q2k•L(wD‚2fzIJML(L:wDk‘zHj`U ‚+zI|Xkkc kyJbaI‚+q•U vdPJm~UcˆzIejYcXqF n 2

2

K
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IHG DJ
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10




L

tiJRJbcXq2q V yJRdPd™W¡eXf ∂ψ ∂t ∂ψ ∗ −i~ ∂t ∂ψ ⇒ i~ψ ∗ ∂t ∂ψ ∗ −i~ψ ∂t  ∗ ∂ψ ∂ψ i~ ψ ∗ +ψ ∂t ∂t i~

∂ |ψ| ∂t

~2 d 2 ψ +Vψ 2m dx2 ~2 d 2 ψ ∗ − + V ψ∗ 2m dx2 ~2 ∗ d 2 ψ − ψ + V ψψ ∗ 2m dx2 ~2 d 2 ψ ∗ − ψ + V ψ∗ψ 2m dx2  2 ~2 d 2 ψ ∗ ∗d ψ − + V ψ∗ψ − V ψ∗ ψ ψ−ψ 2m dx2 dx2   ~2 ∂ dψ ∗ ∗ dψ − ψ−ψ 2m ∂x dx dx

= −

i~

= = = =

2

=

zHj`U ‚ŒzI|Xk‘tiU OIkU~vk J

~ 2im

=



dψ dψ ∗ ψ − ψ∗ dx dx

∂ ∂ P+ J ∂t ∂x



JROPe

= 0

tGJMJMjYwDJM|XkzHj`Q JM“UxzIejpjYcXqKj`dPL(‚+JRNGcŠkejpjYcXq€L›e%U NGJMd WpnAfkydPkkzIe˜Nijp~‚+d

0=

Z

∞ −∞

J|∞ −∞

 Z ∂ d ∞ ∂ P+ J dx = P (x, t) dx + (J) |∞ −∞ ∂t ∂x dt −∞
d = 0 ⇒ P (x, t) = 0 dt P (x, t) = const



[−a, a] d dt

Z

S‡aIN«Ngjp~‚Œd kq2v–jp~

a −a

P (x, t) dx = J (−a, t) − J (a, t) 6= 0

OPeƒzHJM~q2q‘EkJRS‚Œ‚

kU S;k

∂P ~ · J~ = 0 (~r, t) + ∇ ∂t J~ =

 ~  ∗~ ~ ∗ (~r, t) ψ ∇ψ − ψ ∇ψ 2im V

d dt

Z

P (~r, t) d3 V V

= −

Z

V

~ · Jd ~ 3V = − ∇

11

Z

Jn dσn σ

yhQ dGU j`‚+S

kU S;k

Ome«W R

∂ψ ∂x ∂ψ ∗ ∂x

W P L:cŽzHj`Q JR“U™_`JRe‘zIU yeXf€k€QJM“U ψ F E F ψ zIye˜tGSQ t ‚+j ψ tiJRJMq€SQ xlP‚‹kU JMOmvuF s j`OHkq2v–jp~‚™tidmq!j`eXf ψ = e k€q€v–jp~ =∞ ∂ ∂t

∞ −∞

|ψ|

2

∂ ∂x2

i(kx−ωt)

= ikψ

= −ikψ ∗  ~  2 2 ik |ψ| + ik |ψ| J = 2im ~k 2 = |ψ| m   p |ψ|2 = 1 ⇒ = =v m

F tiUOGkJRk€JbcŠJm~ wxzI‚ŒwDUj`qƒzHj`JRk€L zH‚ŒJRJMy ψ kU S;k éù ëùøgéùêMñDùëêIéù ü ñDþ]ï £bS“Dj`q€q%£GášzIJRQ “z}¢‡U S kU~v–k W x j`eXf hxi Jm£RS›_`q2j`|XJ x L:cŽS;“Dj`q2q ψ (x, t)


hxi

=

Z

∞ −∞

2

x |ψ (x, t)| dx f (x)

hf (x)i =

Z

∞

f (x) P (x, t) dt

−∞

hpi 2

P = |ψ| ; hpi =

i~

∂ψ ∂t

= −

Z

L:cˆS“Dj`q€q p SdPz§cXJ%NiJRNiL(yL kU~vk

pP (x, t) dx

~ ∂ 2 (ψ) +Vψ 2m ∂x2

p2 +V 2m ∂2 ⇒ p2 = −~ 2 ∂x ∂ = = i~ ∂x E

ej`kc NGJMNGL‡y‹L:cŽk€dPjYwDz

=

12

Fp =

UcXe!w ã tGJMU‚+|Xk U vdPJM~UcŠzHe%jYcXq2qF V ~ ∂ i ∂x

JR|Xe%L(NikkS j`dPzIk‘Ngj`yÞF n dx dt

= p

hpi

= m

m

∂ψ ∂t ∂ψ m ∂t d hxi m dt i~

= =

=

~2 ∂ 2 ψ +Vψ 2m ∂x2 ~ ∂2ψ m − + Vψ 2 2i ∂x i~ Z d ∞ ∗ ψ xψdx m dt −∞ Z ∞ ψ ∗ xψdx m −∞  Z ∞ ∗ ∂ψ ∗ ∂ψ m dx xψ + ψ x ∂t ∂t −∞

= = ∞ −∞

∂ ∂x

∂ψ ∂ψ ∗ xψ − ψ ∗ x ∂x ∂x



(6)

(7)

L(‚+NGdjW¡„fplP‚•W † fzHeK‚ŒJM“d

=



U vdPJM~DU;cŽzIejpjYcXq€L›Uj`‚ŒSq

= −

=

Z

d hxi dt

=0⇒ = = hpi =

∞

   2 m m ∂ 2 ψ∗ ∂ ψ ∗ ∗ − V ψ xψ + ψ x − 2 + V ψ dx ∂x2 i~ ∂x i~ −∞  Z ∞ 2 ∗ 2 ∂ ψ ∂ ψ ~ xψ − ψ ∗ x 2 dx 2i −∞ ∂x2 ∂x  ∗   Z ∞ ~ ∂ψ ∗ ∂ ∂ψ ∗ ∂ψ ∗ ∂ψ − dx xψ − ψ x ψ+ψ 2i −∞ ∂x ∂x ∂x ∂x ∂x  Z ∞ ~ ∂ψ ∗ ∂ψ − dx ψ + ψ∗ 2i −∞ ∂x ∂x Z ∞ Z ∂ ~ ∞ ~ ∂ψ ∗ (ψ ψ) dx + dx − 2ψ ∗ 2i −∞ ∂x 2i −∞ ∂x   Z ∞ ~ ∂ ψ∗ ψdx i ∂x −∞ ~ 2i

Z



F kdjYc ke%“Dj`z}L(‚ŒNidS“q€e%‚šj`zHj`e˜tGJRcgdHe!L#tie§_YwDL UjYaIUQj`e˜ej`k zHj`JM“NGdj`Q k‚ŒyU q UjYaIUQj`e p = ψ (x, t) JRU e%JRdPJRL UjYaIUQj`e p = i t R J b J X c 2 q  q ` j ( L ˆ c I z M J  Q  “ H z ‘ k b J D w U  S 2 j b J aIJMq€U k•U jYaIU Qj`e p = hpi ∈ R rq

~ i ~ i ~ i

p ∂ ∂x ∂ ∂x ∂ ∂x

FV Fn FE Fs

zHj`U S;k

;ùPíê .ë ê ûAó¡ûAóRò zHj`JMkLTU~v–j`q‘Ÿ j`L(JRy|XyJ kU~v–k s

Hl

8l

[A, B] = AB − BA

[A, B] = − [B, A]

VœE

tiJRJMNGzHqŸ j`L(JRy|XyJRL

k€USk EHG I
IHG DJ
J
KoL -N),O+.-Qt

Ÿ j`L(JMy.|XyJ%tGJMq€JRJMNGq•tgj`NGJMq€k€j€S;dPzIkݟ j`L(JRyk|XyJ [p, x] =

~ i

zGj`y+wŒj`k

|XyJRk•zIeKNgjp~‚+d5F V [p, x] ψ (x, t)

 ~ ∂ , x ψ (x, t) i ∂x ~ ∂ ~ ∂ (xψ (x, t)) − x ψ (x, t) i ∂x i ∂x ~ ~ ∂x ψ (x, t) = ψ (x, t) i ∂x i



= = =

hpi − hpi

∗

=

Z

∞

∂ψ ψ dx + i ∂x −∞ ∞

∗~

2

Z

∞ −∞

ψ

zHj`JbaIJMq€U kNgjp~‚+d5F n

~ ∂ψ ∗ dx i ∂x

~ ∂ |ψ| = dx −∞ i ∂x i∞ hh 2 |ψ (x)| = i −∞   2 = |ψ (∞)| − |ψ (−∞)|2 = 0 Z

iÎ ÞÓ!Í Ó!å XË–Ë žÒ2ÎÔ[Õ Œß ØRè zHj`U~v–k F S;dPzIk‘L(cˆL(vizIJRJM“NGdj`Qw φ (p, t) _`q2|Xd›F V ~Djp~q2LTzHj`U ‚ŒzI|Xkk |φ (p, t)| F n p L:c kJRJMUj`Q›tGUj`Q |XdPUa˜ej`k φ (p) kU~vk ψ (x) e =

j

.u

2

ψ (x) 

k=

=

p ⇒ = ~ φ (p) = =

φ (p) = = =

Z ∞ 1 √ g (k) eikx dk 2π −∞ Z ∞ 1 i √ φ (p) e ~ px dp 2π~ −∞ Z ∞ 1 √ f (k) e−ipk dk 2π −∞ Z ∞ 1 i √ ψ (x) e− ~ px dx 2π~ −∞

∞ ∞ 1 1 i 0 i √ φ (p) e ~ p x dpe− ~ px dx 2π~ −∞ 2π~ −∞ Z ∞ Z ∞ “ 0” i 1 p−p x dxdp φ (p) e~ 2π~ −∞ −∞ Z ∞  0  1 φ (p) 2π~δ p − p dp = φ (p) 2π~ −∞

√

Z

Z

Vs

L(JMvU z

p

~Djp~q2LTzHj`U ‚ŒzI|Xkk R∞

−∞

Z

Z

φ∗ (p) φ (p) dp =

∞

dp φ∗ √ 2π~ −∞

Z

∞

|φ (p)|

2

Šc yJbwŒj`d kyhwŒj`k cŽke%UdxF V =1 |φ (p)|

2

i

dxψ (x) e− ~ px −∞ Z ∞ Z ∞ i 1 = dxψ (x) √ φ∗ e− ~ px dp 2π~ −∞ −∞ Z ∞ ∗ = ψ (x) ψ (x) dx = 1 −∞

R∞

hpi = Z

∞

−∞

dpφ∗ (p) pφ (p)

ke%U dHU q2j`L(w

hpi

S;“Dj`q2q‘Nijp~D‚Œd5F n

~ ∂ψ i ∂x −∞   Z ∞ Z ∞ ~ ∂ 1 i √ = dxψ ∗ φe ~ px dp i ∂x 2π~ −∞ −∞   Z ∞ Z ∞ i 1 = dpφ (p) p √ dxψ ∗ e ~ px 2π~ −∞ −∞ Z ∞ dpφ (p) pφ∗ (p) =

hpi =

dxψ ∗

−∞

_`q2JR|ˆJM~ w™~S x

ψ (x) = i~

‚ŒyU q€‚š‰ p L#kkOIUjYaIU Qj`e§ej`k x ‰ φ (p) ‚ŒyU q€‚

kU S;k

∂ ~ ∂ =− ∂p i ∂p

hXP i

F zHj`JbaIJMq€U k

ü ùP÷ ü ùìï kU Sk 

bJ wxJRaIJRq2Uk•e%L hXP i UjYaIUQj`eÚF V k€JR“OPJRUaIq€JM|«Jm£RS›zHj`JRaIJRq2Uk•L(‚+NGLT_`zIJMd›F n hXP + P Xi IçžÏ àià XÔË HÓ XÔ]ÎÔ]Ñ gÕ ßŒØ U vdPJM~UcˆzHe%jpjYcXq X, P



i~

∂ψ (x, t) ∂t

= −

c

vw

xev f



"!

~2 ∂ 2 ψ +Vψ 2m ∂x2

p2 ψ + vψ 2m   2 p +v ψ = 2m =

_`JRJML‡jYaIL(JMq€k•e%j`k H = + v UcXe!w _`q€OP‚utGJMJMj`L(zrtidPJRe!cŠyJRdPd ¢‡JML(kzIk V (x) p2 2m

i~

∂ψ (x, t) ∂t

= −

~2 ∂ 2 ψ (x, t) + V (x) ψ (x, t) 2m ∂x2 15

ψ (x, t) = T (t) u (x) i~

∂T (t) u (x) ∂t i~ ∂T∂t(t) T (t)

= −

kUj`“kqK_pj`U zIQ5ej`“q2LTk|XdPd

~2 ∂ 2 u (x) T (t) + V (x) T (t) u (x) 2m ∂x2 ∂ 2 u(x)

~2 ∂x2 = − + V (x) 2m u (x)

zHj`ejpjYcXqKJRdmc j`dmL‡‚ŒJRNrJMOme«W¡kJRvU dPe%kHf E e%L#e!U NidmcˆS j`‚ŒNiLµtiJMjpjYcŽkejpjYcgq2kƒU JR‚+STJRdmcŠkdPNi|Xq dT dy T

= −

iE T ~ i

= e− h Et

UcXe!w™F _`q2Om‚™kJmj`L(z}e%L#zHj`U ‚ŒzI|Xkk‘JMU dPj`JM“ aI|«‚Œ“q2‚‹F zIJRU dj`JR“ aI|ŽkJR“Nidj`Q›zIe%O −

~2 ∂ 2 u + V (x) u (x) = Eu (x) 2 2m ∂x   2 p + v u (x) = Eu (x) 2m

F _`q€OP‚™kJMj`L‡z˜JMzIL(‚™U vdPJM~UcŽzIejpjYcgq§j`dmL‡‚ŒJRN W¡tGJRaIUNi|XJM~xtikkJMvU dPe!k•JbwDU S›U q!j`L:w€fzaaIdjpjpj`Niqƒe%JRk•kJRv–UdPe%k_`e2wcŽzHj`e%U L›_`zIJRd

k U Sk zHj`e%q€v–jp~

kvU~q‘L‡e!JM“dmaHj`QŠF V JMQj`|ˆUj`‚0F n JRQj`|Štgj`|XyqF E zIJMJR“Nidj`Q›L:cˆtiJRL‡e!JM“dmaHj`QŠF s δ tGJMJRUj`OPyq•tiJRL‡e!JM“dmaHj`QŠF ^ F k€“JRQ NGk€q‘NGJMQ|Xq¢j`Ue˜L(vk‘¢;Uj`e§j`‚DcŽ‚Œ“q2L tiJR‚+Uj`NGq•tGk‘j`L(L‡k‘tGJMU‚h~L zHj`dj`UzHQk ïhþ ü ñDð™íìê høP÷€ù Aû ó :óRò L(L(v–‚šJRQj`|0L(e%JR“dmaHj`Q™Uj`‚ŒS.F x = 0 ‚W¡Ÿ;JM“ULx¢;j`Q kd[f!NGJML(ydjHWYVUj`JReXf!tGJMdPjYcXk‘tiJRUj`OPe%‚štGJMU zHj`Q kIa Jbc kvU~q‘L(e%JR“dmaHj`Q™ã¡V6kL(‚Da C

U vdPJM~UcˆzHe%jpjYcXq du dx

= u

0

16




$#

Uj`zIQ d 00

u +

2m (E − V (x)) u = 0 ~2 2mE = k2 > 0 ~2 2m (E − V0 ) = q2 > 0 ~2

00

u1 + k 2 u1 00

2

u1 + q u1

(8)

j`dPL:cˆkejpjYcXq€kKJM~JM“5JRdmc

= 0 = 0

u1 (x)

= eikx + Re−ikx

u2 (x)

= T eiqx

JROPe

it JRS j`‚ŒN T − T ransmition, R − Ref lection UcXe2w F Uj`OPyJIj`kcXqc0_YwDzIJ%e!L#JRw e ‚™JMj`L(z}e%L u kU Sk Y_ wŒj`q€w ψ = T (t) u (x) = e e L(L(v‚‹zIe%O x _pjpj`JRwD‚‹L(vgj`dmL‡‚ŒJRN zIUjYwDOmz −iqk

2

− ~i Et iqx

u1 (x = 0) = u2 (x = 0) 0

0

u1 (x = 0) = u2 (x = 0) k − kR = qT K (1 − R) = qT 1+R = T k−q ⇒R = k+q 2k T = k+q

J~ =

JR

~ 2im

J1

=

J2

=



zHj`U ‚ŒzH|Xk•tiU OI‚DcXyd

∂u∗ u −u ∂x ∂x ∗ ∂u



 ~k  2
1 − R2 m ~k 2 T m

‰ |XdmwDdPk‘zHj`U ‚ŒzI|Xkk§tGU O J ‰ U ‚+S;q§zHj`U ‚ŒzI|Xk P ‰ k€UOPykƒzHj`U ‚+zI|Xk P UJm~vd F U ‚+j`Sk‘zHj`U ‚ŒzH|Xk•tiU O J F U OPyj`q€k‘zHj`U ‚ŒzH|Xkk‘tGU O I

T

R

T

JI

=

JR

=

JT

=

~k m ~k 2 R m ~q 2 T m 17

 2 JR k−q 2 = =R JI k+q q 4kq JT = T2 = = 2 JI k (k + q) = 1

PR PT PR + P T

zHj`U S;k

U vdPJM~DU;cÜzIejpjYcgk}L(c_pj`U zIQ L™Ome tiJRdPzcXqrzH~U Q k –U j`‚+S«F V k JRv–UdPe%‚ukJMj`L(z«_`q2OP‚xkJmj`L(z}JRzHL(‚ucψ£Rqƒ(x,F £RJRU t)dj`JR=“ aIT|ž(t)£HášutG(x)Jmj`|Xq E L‡S;‚‹_pj`U zIQ e U vd~JMUcŠzHj`ejpjYcXqc»¢(w•W¤£RkJRvU dPeI£%UjYaIU Q j`eXf H = − + V (x) = + V (x) F n zy

T (t) =

− ~i Et

p2 2m

~ 2 d2 2m dx2

H (uE (x))

= EuE (x)

Yïhþ ü ñDðhý íìê høP÷€ù DïhðïhøP÷Iî5ïhêMþ ü øìï ü ð+ùPí E < V ü ùý .ïhþ ü ñDð ûAó :ó¡û kvU~kJMUyej2Uq€cXdHL:wDkkvU~q2kJRdPQ L:cˆNiL(yk 



C




4{

*S

$#

00

u + q2u = 0 2m (E − V0 ) q2 = <0 ~2 u = e|q|x + e−|q|x

W¡zHj`U ‚ŒzI|Xk€LŒf2_pj`U zIQxe%L#_YwDL;j€S j`‚ŒJMU‚™JML(JR‚+UvaIdmJMe‘e%L

e|q|x

UcXe!w

u = e−|q|x

F |XQ e!zHq‘tGU OPkzIJbcXq€qkJM“NGdj`Q L#Jbw pR

k − i |q| 2 =1 = k + i |q|

JT = 0

zHj`U ‚ŒzI|Xk€k‘tGU O

Œùý€îxï ü êMýú í +ý ùù–éý™í ùùëhî;ðéùìùùúžð™í ùð Œùý€îTíìê høP÷€ù ý ü þ[øêRñŒú»é€ìùùúžð U vdPJM~UcˆzHe%jpjYcXq S

4S }|





 2m d2 + (E − V ) u (x) dx2 ~2

~2 ∂ 2 d2 − dx2 c2 ∂t2 



ε (x, t)

ε (x, t)  2 2 ~ d − 2 ω 2 E (x) dx2 c

~S

= E (x) e−iωt

18




= 0

= 0

= 0

C

L‡jpj`|XNGq‘zHe%jpjYcXq

L(e%JR“dmaHj`Q5Uj`‚.ã EUj`JRe

Rü ù–ïhøð 2ê ùë5ÿŒùë ð+ùGíìê høP÷€ù ü ùý ûAó :ó¡ô W¡EUj`JReXfWhL‡j`vU zI‚‹U zIQ JRJ…f—L(e%JR“dmaHj`Q5Uj`‚+‚



T

2

|T |

= e−2ika

 

2kq 2kq cos 2qa − i (q 2 + k 2 ) sin 2qa

q

=

k

=

8l

L(‚+kSj`‚+N

C




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2ka

UcXe!w

p

2m (E + V0 ) ~ √ 2mE ~

K

=

(q = ik) ⇒ k

=

JMQj`|Štgj`|Xyhq

√

2mE p~ 2m (V0 − E) ~ 2

(2Kk) 2K 2 cosh2 (2ka) + (K 2 − k 2 ) sinh2 (2ka)

=



|T |2

≈



4kK K 2 + k2

2

2

2

+ý ù ü êî OPe ka  1 U j`‚+S}tij`|Xyq2k‘‚+j`U JRN

Wentzed-Kramers-Brillin WKB

e−4ka  1

JMOPe}W¡L(U vaIdPJRe%L#_`q€JRU5tijYwD|gw€f€zIJRL(q2JR|XJMQdPJMeKj`zHj`e˜vU~dj€j`kcXL:w‹tgj`|XyhqKyhNiJRd |T |

ln |T |

2 2

i

2

|T |2

p

2m (V (x) − E) Z ~ p 2 = − dx 2m (V (x) − E) ~ V (x)>E √ R 2 = e− ~ V (x)>E dx 2m(V (x)−E)

k (xi ) = ln |T |

2

= |T1 | |T2 | |T3 | X 2 = ln |Ti | = −2k (xi ) ∆xi

V[Z

IF€FG I
IHG DJ
J
K‚L -N),O+.-Nƒ

hý ù ü êî;íuéùìðhþ[ùñ JRL‡qcXy.k€~cˆzIL(SQ k‘Jm£RSTzwDzIq2qƒtGJMdj`U;aINiL(eƒL(cˆkaIJML(QŠF V F owlen − N ordheim W KB

|T |

2

2

= e~ = e

Ra√ 0

2m(w−eεx)dx

√ 3 −4 2m 2 3~eε w

cXJi_`JMS;U vk€q§e%“Dj`J α NGJMNGL(yUcXe!wŠF L(e%JR“dmaHj`Q‹Uj`‚Œ‚•tGJM‚Dcj`JGtiJRdj`UaHj`JRdgJMdMcjtiJRdjYaHj`U QšJMdMc α Ngj`U QŠF n L(e%JR“dmaHj`Q ‚šk€~DJRU J%cXJIL(e%JR“dmaHj`QuU j`‚+L›Ù;j`yqƒU q!j`L:wF NiJRNiL(yLT_`JRSU vk‘_`JR‚‹tiJM~v–j`dPq‘tiJRdPS;aIq‘L:cŽk€JRJRy~ e%JRk W KB JRQ L#ke%“Dj`zIkƒF Ùj`yk_`JM‚ŒL Uj`‚Œk‘_`JR‚ tunneling cXJ!U q!j`L:wšF JRQ L „

1 r

2

2

= e− ~

|T |

= e

Rb

R

dx

√

(V (x)−E)2m

2πz1 z2 e2 − ~v

U q!j`L:w b  R U;cXe!w ÊoÔ]Ñ XÕI֎Ó!Í XÔ[˖ӀÓ%æ XÔÎ]Ôæ ÔÃËoÕ!Ô Î[Ë…Ï»Ë àiài× IçÛÑ2Ô ßŒØbÉ ( 0 −a < x < a L(e%JR“dmaHj`Q5Uj`‚š_pj`zId V (x) = ∞ |x| ≥ a U vdPJM~UcˆzHe%jpjYcXq 1 2 2 mv



…

P2 +V 2m ∂ψ (x, t) i~ ∂t

H=

=E

f

= −

z 1 z 2 e2 R







.



~2 d 2 + V (x) 2m dx2

= Hψ (x, t)

tGJMdPzcXq‘zH~U Q k

ψ (x, t) = T (t) u (x)

HuE (x)

JROPe

= EuE (x) i

T (t) = e− ~ Et

ψ (x, t) =

X

JRL(L:w‹_pj`U zIQ5OPe

i

cE e− ~ Et uE (x)

E

F E Uj`‚ŒS›JMU dPj`JM“ aI|Xktg~Niq€k

L#e%U NGdj Uj`‚ŒLTÙj`yq Rù ü é |x| ≥ a i

e− ~ Et



−

~2 00 u + V (x) u = Eu 2m ⇐⇒ u = 0

20

I
EHG I
IHG DJ
J
KML

-N),O+.- †

‚

E =V +T >0

kU~vkkQD£bS5Ome

V = 0 −a < x < a

Uj`‚Œk‘¢j`zI‚

~ 00 = Eu u 2m 00 2mE u + 2 u = 0 ~ −

_pj`U zIQ e!L OPe L‡L(v‚›_YwŒj`qw™W¡zHj`Q JM“UµL(‚+NGLJm~ w€f kQc˜JMe%dmztie S£bSTNiU cgJ!uzI=JMUAeL;j`vU›L(+JM‚+Bej`JRLµtGUjYac«EzI>JMJRS0‚ŒL:c«e%JMkƒ e‘U£M~q‘j`e‘y£R~Dq~Dq€L:cŠu JM(±a) q€LµŸ;|=j`d0U ‚Œ|XkGfF JRUcXQ e W¡tiJRJRL‡JRL:cžlPJMe W¡\S£bS5U j`‚+Shf u = 0 JRL‡e!Jmjpj`UaIk_pj`U zIQ kej`kƒ_pjYcXe!U k‘_pj`U zIQ k zHj`dj`U zIQ kƒU e!c −qx

qx

u = A sin

√

2mE x ~

!

√

+ B cos

WâF tikJRdmcŠzIe˜Ngjp~‚ŒdH_YwDLT‰ tGJML(‚ŒNizIqzHj`dj`U zIQ k‘JRdmc

±a

2mE x ~

!

‚+JR‚Œ|ŠkL:wDJRU~™JMe!dPzI‚™y£M~q‘JRU JRNgj`q€L+f tiJRJMv–j`OlPJRe%k•zHj`dj`UzHQkÛF V

u (±a) = A sin (kx) = 0 ak = nπ ~2 k 2 ~2  π  2 2 n E1,n = = 2m2 2 2m a ~ π n2 = 2ma2

tGJMJRv–j`OPk•zHj`dj`UzHQkÛF n

u (±a) = B cos (kx) = 0   1 ak = n− π 2 ~2 π 2 E2,n = (n − 1)2 2ma2

W¡zIL‡q€Uj`dPqƒzHj`U ‚ŒzI|Xk‘JMe%dmzHL tiJRe%zIq€cÃfzHj`dj`U zIQ kq§~ye§L:wDLTL‡j`q€U dGJRe%dPzKcXJ Z

∞ −∞

|u (x)| dx =

1 = = = ⇒A=B

=

Z

Z

a −a

∞

|u (x)| dx = 1

A2 sin2 x

−∞ Z a

1 A2 x 2 −a

A2 (2a) = A2 a = 1 2 1 √ a

21

_YwDL‡j

kU S;k

tijYcXU dHOPe

l ∈ N; El

=

E2,n

=

E1,n

=

~2 π 2 (l − 1) 2ma2 4 ~2 π 2 l 2 2ma2 4

~2 π 2 2 l 2ma2

_`q2|XdIzHj`yj`di¢;Uj`“L

l = 2n

2

w+£bk|

(E1 = E1,1 , E2 = E2,1 , ...)

F L(aI‚ŒzIq

JML(e%JMjpj`UaIk‘_pj`U zIQ k•tiv–j ü ùýïéùøù ü é 5í úÞéùøù Dé vo£ReKtikzGj`dPjYc zHj`JMvU dme%L#zHjYwDJMJbcXkL(vgzHj`JR“Nidj`QŠF V u=0

o

∀n, m ∈ N;

Z

b

u∗n um dx

δ=

= δn,m

a

(

1 m=n 0 m= 6 n

! cn

Z

a −a

u∗m (x) ψ (x, t) dx

⇒ cm |cm |

1=

Z

a −a

|ψ (x, t)| dx =

Z

=

Z

2

⇒

X n

|cn | = 1

2

a −a a

Z

=

a

u∗m (x)

−a

= cm e Ra Z =

X

−a n

i

cn e− ~ En t un (x) dx

u∗m (x) ψ (x, t) dx

a

−a

i

e − ~ Em t 2 u∗m (x) ψ (x, t) dx

i c∗n e ~ En t u∗m (x)

n

X

tiJRq2~DNGq‚+jYcXJMyÞF n

n

− ~i Em t

−a

=

X


{

2

|cn | |un | dx =

!

X n

X

cl e

éêRíŒî;ê –ê ïéù Œðúžðï zIL(q2Uj`dmq ψ (x, t) F V  

− ~i El t

l


S

!

ul (x) dx

|cn |

kJRv–UdPeƒL(cŽzIJMQ “zr¢;U S«F n hHi =

Z

U vdPJM~UcˆzHe%jpjYcXq Hun (x)

ψ ∗ Hψdx 2

2

~ d H = − 2m dx2 + V (x)

= En un (x) 22

kJMvU dme%kL:cŽU jYaIU Qj`e%k

F S £RS hHi =

Z

=

Z

a

dx −a

dx −a

n

X

⇒ hHi =

n

i c∗n e ~ En t u∗m

X

i ~ En t

!

c∗n e

u∗m (x)

n

2

zHj`JMq€“S›zGj`JR“NidPj`Quj`e%UNiJ u Ucge!w zIJMQ“zHkƒ¢‡U S›zIe§yzIQ d n

X

(x) H

n

a

X

=

X

En

cl e

− ~i El t

ul (x)

l

!

X

cl e

− ~i El t

En ul (x)

l

|cn | En

! !

2

|cn | En

+‚ “q€‚ E kJMvU dme§~Djp~q€LTzGj`U‚+zI|Xkk‘e%JRk |c | kU Sk F tiJRdjYcXk ψ tik u (x) U q2j`L:w‹~Dj`|XJRkJM‚Œ“q€L#zI|XUj`N«e%JMkzwDU Sq€k‘zIe§tGJm~D~Dj`q‘UcXe!w kU S;k ä ËÌ6Î]ÔÔAÒÖ Ñ2Ô gÓr˖Ì6à Xà Ò!Ñ ßŒØ OPe P : x 7→ −x zHj`JMv–j`OPk•UjYaIU Qj`e P =1 ψ (x, t)

n

n

2

n

‡

2

=

ˆ

ce

Š‰

‹

P u (x) = λu (x) u (x) = P 2 (x) = P λu (x) = λP u (x) = λ2 u (x) ⇒ λP = ±1

JML(UaINGQ |Xk•Nij`U JRQ kaIQcgq

ψ (x)

= +

1 [ψ (x) + ψ (−x)] 2 1 [ψ (x) − ψ (−x)] 2

S £RS;L#kq2JRe%zIq‘zIJMv–j`OPk•k€JR“NidPj`Q k u (x) = [ψ (x) + ψ (−x)] UcXe2w S £RSLTkq€JMe!zHq‘zIJRv–j`OlPJMe%kkJM“NGdj`Q k Ÿ j`L(JRy|XyhJtiJRJMNGzHq2jF L:cˆzGj`JRq€“SxzHj`JR“Nidj`Q5tivi_`k§W u (x)uf H(x)L(cˆ=zHj`JMq€[ψ“(x) Sk‘zH−j`JRψ“Ni(−x)] dj`Qk λ=1 λ = −1 P

1 2

+

1 2

−

n

[H, P ] = 0

zHj`JRv–j`OPkUjYaIU Q j`e§zIeKL‡JRSQd ψ (x, t) , i~ PH

= HP

∂ψ ∂t

= Hψ

Uv–dmJm~UcŠzIejpjYcXqKUcXe!w

∂ (P ψ (x, t)) = P Hψ (x, t) ∂t = (P H) ψ (x, t) ~2 ∂ 2 H (−x) = − + V (−x) 2m ∂x2 ~2 ∂ 2 V (x) = V (−x) ⇒ = − + V (x) = H (x) 2m ∂x2 ∂ ⇒ i~ (P ψ (x, t)) = HP ψ (x, t) ∂t i~

zHj`JMU e!JMdPJRL(kqƒF _pj`U zIQ

(1 ± p) ψ

= ψ (x) ± ψ (−x)

P ψ (x, t)

U q2j`L(w

F tiJRJMv–j`OlPJRej€tGJMJRv–j`OzHj`dj`U zIQ D
DG I
IHG DJ
J
KML

noE

-N),O+.-QŒ

F zIU qcXdzHj`JMv–j`OPk._YwDL t L:wD‚›zGj`JRv–j`O%kzHj`e‘k€LcXJ t = 0 Uj`‚ŒS zIq€Jmj`|Xq.zHj`JMv–j`O2cgJ2L(vzIJMJR“NidPj`Q LÃtie kdPNG|Xq zHj`U S;k UjYaIU Q j`e˜Uj`‚ŒS›OPe H = JbcXQj`y.NiJRNiL(y»F V H p2 2m

00

u +

2mE u = 0 ~2 ~2 k 2 E = >0 2m

F k L(cŽŸJR“U5¢;U S›L:w™L(‚ŒNiL kL‡jYwDJIkJMvU dPe!k W¡S;dPzžf p U jYaIU Qj`e˜U j`‚+S«F n



~ ∂ = p ⇒ p eikx = ~k, p e−ikx = −~k i ∂x

F _pjpj`dPq‘e!U NidISdmzrJRwDUS5JRdmc«cXJbc cos (x) = j`qw™kOmwu‚+“q€‚ _pjpj`dmq‚+“q‘ke%UNidIS£bSxj`zHj`eKcXJzHj`JMq€“S›zGj`JR“NidPj`Q.nL:cˆ‚+“q€L kU S;k k€JRvU dPe%‚™_pjpj`dG_`JReƒzHj`Jm~q€q%l~y•L(e%JR“dmaHj`Q5zGj`JRS‚ŒL aIQcXq kzGj`e§tiS u (x) zHj`dj`U zIQ•n kyhwŒj`k E eikx +e−ikx 2

E

2m (−V + E) u1 ~2 00 2m u2 + 2 (V − E) u2 ~ 00 u1 u1

00

u1 +

00

00

u1 u2 − u 2 u1  0 d  0 u1 u2 − u 2 u1 dx 0 0 u1 u2 − u 2 u1   x→∞ u1 , u2 −−−−→ 0 ⇒ 0

u1 u1 ⇒ ln u1 u1

= 0 = 0 00

=

2m u (V − E) = 2 ~2 u2

= 0 = 0 = const = 0 0

u2 u2 = ln u2 + ln c = cu2 =

F tiJRdjYcˆzHj`dj`U zIQue%L#_YwDL;j€zIJMU e!JMdPJRL JMj`L(z}_YwDL‡j SdPzIkU jYaIU Qj`e˜L:cˆzHj`JMq€“SxzHj`JR“Nidj`Q ~ ∂up (x) i ∂x p ∈ R; up (x)

= pup (x) = ce

ipx ~

zGj`JRL(dj`v–j`zIUj`e Z

dxu∗p0 (x) up (x) = |c|

2

Z

i

“

p−p

0

”

x

dxe ~   0 2 = 2π~ |c| δ p − p 24

_YwDL‡j u Z

p

=

p

√ 1 ei ~ x 2π~

L(q2UdPdGJROPe

  0 dxu∗p (x) up (x) = δ p − p

:L c δ p − p  LµkwDQkU NidPj`U NrL:c δ ‰ zHj`JRq2“S›zHj`JM“NGdj`Q›L:cŽŸ;“U5zIq€“Dj`S›L‡STUq!j`L:w€f W¡NiU JM~ zHj`U ‚ŒzH|Xkk zHj`Q JRQ “Ke%JRk |φ (p)| U;cXe!w φ (p) tiJRq!~Niq€k ψ (x) = R dpφ (p) yj`zHJRQ›aIQcXq ~Djp~Dq€L lP‚+zI|Xkk |c | F L(e%JR“dmaHj`Q k›Uj`‚Œ‚ ψ (x, t) = P c e u (x) tiJbaIU NG|XJm~ºtGJRwDpUS6Uj`‚ŒSf F JRaIU NG|XJm~5‚+“q~Œjp~q€LTzHj`U zHj`JMq€“S›zHj`JM“NGdj`Q X lPL#tij`Niq€k•UjYaIU Q j`eÚF E 0

n,m

i

e ~ px √ 2π~

2

n

2

n n

xωx0 (x) ⇒ ωx0 (x) ψ (x)

|ψ (x0 )|

2

− ~i En t

n

= x0 ωx0 (x) = δ (x − x0 ) Z = dx0 ψ (x0 ) δ (x − x0 )

U;cXe!w X UjYaIU Qj`e0L:czHj`JRq2“SzGj`JR“NidPj`Q‘zGj`S;“q2e!‚

L:w§L:cÛyj`zHJRQaIQcXqˆj`dPL(‚ŒJMN F D~ jp~q€LTzGj`U‚+zI|Xkk‘zHj`Q JMQ“ î ü êRñ bùðhêRëDý.ÿDê ü ùP÷î;ùùGÿDêMý ü ð _`q!j`|XJ UjYaINgjpj |ψi zHJRU L(Ni|ˆkL‡Q;wDq‘L‡JRSQq€c L(dj`JR“Nidj`Q k hϕ| zHj`U~v–k OPeF V ψ (x) x0



Ml

:

hϕ|ψi = (hϕ| , |ψi)

Q£Rq€kL:cŽzHj`JbaIJMq€U kÝF n _`q€|Xd›F E hϕ|T |ψi = (hϕ| , T |ψi) zIU~vkÝF s T

hϕ|ψi∗ = hψ|ϕi

†

hϕ|T |ψi = hψ|T |ϕi

∗

JRaIJRq2Uk•UjYaIU Q j`eÚF ^ ∗

hψ|T |ϕi

= hϕ|T |ψi

UjYaIU Qj`e§L(cˆzIJM“JRUaIq‘kv“k•zIzIL›_`zIJMd›F † Tmn

= hun |T |un i DH€FG I
IHG DJ
J
KML

25

-N),O+.- Ž$

|aψi = a |ψi , haϕ| = hϕ| a∗

F„

tiv–j |T ψi = T |ψi F ” FZ (T T ) = T T lItiL:cŽ|XJR|X‚ F Vœ\ {|u i}

hT ψ| = hψ| T †

†

1 2

† † 2 1

n

X n

X

=

|ϕi

=

hϕ|ψi

n

=

X n

0

X n0

= ⇒

X

*

hun0 |ϕi |un0 i ⇒ hϕ| =

∞ −∞

X n

n

0

hψ|un0 i hun0 |

|un i hun | |ψ

+

|XJR|X‚ŒkzHj`q€L(c xˆ UjYaIU Q j`e˜L:cŽtGJM‚ŒJbwDU q2L

|xi hx| dx = 1

hϕ|ψi = =

=

X

hϕ|un i hun |ψi hun |un i

|xi

|ψi

= |ψi

hun |ψi |un i

ϕ|

|un i hun | = 1

Z

_YwDL kq2v–jp~L

X n

= 1

|un i hun |ψi

n

|ψi

|un i hun |

Z Z Z

kU S;k

_YwDL

hϕ|xi hx|ψi dx ϕ∗ (x) ψ (x) dx

∞ −∞

UcXe!w

hx|ψi |xi dx

€êR÷€êRð ü ï ü ùP÷ ü ùì‹í ‹ÿDêb÷ ú6ð F tiJRJbcXq2q•JRaIJRq2Uk•UjYaIU Q j`e§L:c S £RS«F V vo£ReKtiktiJRdjYcŽS £RS;L#tiJRq2JRe%zIq€k•S£Rj F n kyhwŒj`k 

26

A† = A





4S



A |ψi = a |ψi

tGk•S£Rj A = A yhJMdPd›F V †

hAψ|ψi = hψ|A|ψi = hψ|a|ψi

haψ|ψi = a∗ hψ|ψi = hψ|ψi a ⇒ a = a∗

vo£ReKtiktiJRdjYcŽS £RS;L#tiJRq2JRe%zIq€k•S£Rj F n hϕ|A|ψi ⇒ ab hϕ|ψi ⇒ hϕ|ψi

= hη|aψi = hϕ|ψi a = hAϕ|ψi = b hϕ|ψi

= 0 = 0

W¡S£RS;LTS£Rj2JRdmcˆcgJ…f~yeKJMq€“S›UjYaINgjpj`q‘U zHj`J%cXJ a tGJmj`|Xq•JMq€“S›¢;U S;L:c yJRdPd A |ψ1 i = a |ψ1 i A |ψ2 i = a |ψ2 i

_pjpj`d

v…£be§|XJR|X‚uU“JMJRdIOPe

F a S£bSkL:cŽS£Mj€Jm£RSTcXU Q dmk‚+yhU q2kzIv–U ~™j`e§JRU ‚ŒvL‡eƒJMj`‚+JRU›U~v–j`JI_pjpj`JMdPkzIv–U ~ F tGJM‚Œ“q€k•‚ŒyU q€L#|XJM|X‚utiJMjpj`k€q‘S £Mj2F ~DJM~q‘L‡~Œj`viL(wuv“JMJRqJbaIJMq€U k.UjYaIU Qj`e kU Sk ÿDêRê ùPí ê ÿDêRñŒêRñDð ÿDê ü ùP÷ ü ùì W¡L(QwD‚TtiJRQ L(yzIqGf€F tGJm~JM~q™tik [A, B] = AB −BA = 0 Ÿ j`L(JMyhk‹|XyJzIetGJMq€JMNGq€cKtGJMUjYaIUQj`e k€U ~Dvk W¡L(QwD‚‹tiJRQ L(yzIqGf2F tGJMJRQj`L(JMyštGk ⇐⇒ tiJRQ zHjYcXqƒtiJRJRq2“STtGJM‚Œ“q2k.tiJM~Jm~qtGJMU jYaIU Qj`e˜JRdmcXL aIQcgq kyhwŒj`k JMOmeKtGJMQzGjYcgqKtGJMJRq€“STtiJR‚+“qtieKL£b“ŽF V [A, B] _pj`zIdPkU q2j`L:w A (α1 |ψ1 i + α2 |ψ2 i) = a (α1 |ψ1 i + α2 |ψ2 i)



l







A |ψa i = a |ψa i B |ψa i = b |ψa i ⇒ BA |ψa i

⇒ AB |ψa i

= Ba |ψa i = aB |ψa i = ab |ψa i = Ab |ψa i = bA |ψa i = ab |ψa i

∀ψ; (AB − BA) |ψi = 0 ⇒ (AB − BA) = O

27

_YwDL

F tGJMQzGjYcgq‘tiJR‚+“q€k•L;£R“u_pjpj`JMdG_`JRe!c A |ψa i

yJMdmd›F n U yh‚+d

[A, B] = 0

= a |ψa i

OPe

A (B |ψa i) = BA |ψa i = Ba |ψa i = a (B |ψa i)

_pjpj`JRdcXJ%tie

 A ψa1

⇒ B |ψa i = b |ψa i

_pjpj`JRdH_`JMeƒtieƒOPe

 = a ψa1

F yJbwŒj`di_`Q j`e˜j`zHj`e%‚+j UcXQeŽtiJRdjpj`dPq˜tidPJRe«tie |ψ i W¡JRq2“Sš‚Œ“qifIS£bq A cXJbcyhJMdPd ÿDêRñŒêRñDðƒÿŒê ü ùP÷ ü ùìƒíúïhð+í úÚé ü Œð F zHJRL(NiJROPJRQ k•zwDU S;q2kzIe Jm£RS›U e!zHL ŸJR|j`dXtiJRdjpj`JRdXL:cÜU ~D|»L:wDLxw+£bk| F [B, A] = 0 Ÿ|j`dXU jYaIU Qj`eŠtiJRJMNij |ψ i Ÿ|j`dXU jYaIU Qj`e}cX|ψJX_pijpj`JRdgcXJitGe F tGJMJRQj`L(JRy.tGJMU jYaIU Qj`e 748 ™9BBD¾ C™ÂžBDC ½ 8{½ 5Â6B‡À ›B —8:9 ÿDêMøùð+êMë NiU JM~xL:c P_pj`q2JR|XkÝF V U jYaINgjpj |ψi ∈ F JbaIJMq€U k.ej`k F ‚ŒyU q€‚™JRU e%JRdPJML4UjYaIU Q j`e A F n ‚ŒyU q€L |XJR|X‚™tiJMjpj`kqrW¡tiJRq2“STtGJM‚Œ“qifS £RqÛF E F Å ÔàiÔ[Ë iÒ!ÏÐ ä ËÌ6Î]ÔÔAÒÖ žÑ2Ô ÞÓ!Í Ô[ËÓGÒ€Ë <˖ÕHÖ XÔ]çÎ<Ö èDØœÆ tGJML(q€Uj`dPq |ψi , |ai tiJR‚+“q€k•L:w‹F |ψi ∈ F UjYaINijpjIášzIJRL(NiJROPJMQTzwDU S;qƒ‚+“qF V 

o{

S

a

a

a

=T

‘U

XV

0W

Z[

,]

’

“Š

F



@

=

…

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1 = ha|ai = hψ|ψi

‰ JbaIJRq2U kUjYaIU Qj`eKJm£RS5_`zIJMdG£M~JM~q%£%JRL‡NGJMOmJMQTL‡~Dj`v™F n kJM“OmaIdPjpj`N F A L:c a S£RS›NGUx_`k A zH~DJM~q2LTzHj`JRUcXQ e!k‘zGj`e!“Dj`zHkF E A

•

–

A |ai

|ha|ψi|2

= a |ai

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j

2

D—
G I
IHG DJ
J
KML

n”

-N),O+.- Ž˜Ž

|ψi i~

d |ψi dt

H (r, P ) = =

L#_`q2OP‚xzHj`yzIQ zIk˜lHU vdPJm~UcŠzIejpjYcXqF †

= H |ψi

W¡k€JRvU dPe%L JM|XL(NikJmjYaIJR‚ŒkGf_`e!JMdjYaIL(JRq2k H Ucge!w

P2 + V (r) 2m   ~ 2 − ∇ + V (r) 2m

W¡tGJMU jYaIU Qj`eKL(cf€Ÿj`L‡JRy.|XyJ%tiJRJRNizIqÝF „ − [x, P ] = P X − XP = −i~ = [P, x] =

F |ψ i , |ψ i W¡S £RqGfGzHj`JRUcXQ eˆJMdmcÜcXJxF |hψ |ψ i| 1

2

x

s

2

F zHj`e~DjblPJRe}_pj`U NiS ⇐ tGJMQj`L(JRyšJMzIL(‚‹tiJRUjYaIU Qj`e ‚‘NGJMNGL(y§~Djp~q€L.zHj`U ‚ŒzI|Xk ÿŒêî;ñDëêMøúÛêRùëêMøïhðhþ[ùñ x zHj`q2L:cXk‘QD£bS

|ψ1 i hψ1 | + |ψ2 i hψ2 | = 1

|hψx |1|ψs i|

2

~ i

UjYaIQj`e%k‘zIe§¢;JRL(cgdiF UjYaIUQj`erj`dPL(‚ŒJMN

= |hψx |ψ1 i hψ2 |ψs i + hψx |ψ2 i hψ2 |ψs i|

2

F S £Rq€k€qK~ye%LT|j`UNiJ%‚Œ“q2k•F tiJRNiJRNiL(yk•j`U ‚+S›tGk€qcˆtiJRNg~|Xk•zIe§~Œjp~q€dGtGe ÖHË HË AÔ[Õ%рÕ%Ô èDØbß e!JMk .u ™”

Pa (ψ)

= |ha|ψi|

?

2

= |α1 ha|ψ1 i + α2 ha|ψ2 i|2

= α21 ha|ψ1 i2 + α22 ha|ψ2 i2 + 2Reα1 α1 ha|ψ1 i ha|ψ2 i∗

‚ŒJRwDU#cXJ%‰ zIJMUe%JMdmJML4zIq€wDzI|Xqe%LµzHj`U ‚+zI|Xkk§W¡k€~JM~Dq•e%L(L tiJR‚Œ“q•U Q |XqL:cÃfzGjYwD‚Œe%zIk•L:cŠ‚Œ“q2‚xU q!j`L:w zHjYwD‚Œe%zIk 2Reα1 α1 ha|ψ1 i ha|ψ2 i∗

…Ê à Ë…ÔœÓ ÞË iÓ H èDØRè OPeKtiJRdPzcXqzH~U Q k‘kS“‚ŒLT_`zIJRdGF _`q€OP‚ucgUj`Q q€‚‹Jmj`L(zrJRzIL‡‚ H JMkJ g”a

JRL‡L:wT‚Œ“q™cj`U QLµ_`zIJRd2Ome

š

? 

H |En i = En |En i

|ψ (t)i ∈ F

‚ŒyU q€L|XJM|X‚Dc˜Uye%qšU q2j`L(w™F _`q2Om‚TtiJRJmj`L(zJRzIL(‚5tGJM‚Œ“q |ψ (t)i =

X 29

cn (t) |En i

|En i

U q!j`L:w JP£bS

Uv–dmJm~UcˆzIejpjYcXq€‚.‚ŒJR“d i~

X

i~

d hEm |ψ (t)i dt dcm (t) i~ dt ⇒ cn

dcn (t) |En i dt X cn (t) H |En i

d |ψ (t)i = dt H |ψi =

X

dcn (t) |En i dt

=

i~

Wâ_pj`q€JR|ŽL(QwšlIS£bS

= i~

i~

X

_YwDL

cn (t) H |En i

Em = H

JRwxU JbwDOPd[f%Wâ_`q€OP‚uJMj`L(zre%LŒf hE | L(Sx¢;JRL:cXd n

dcm (t) X = cn (t) hEm | H |En i = Em cm (t) dt

= Em cm (t) i

= ce− ~ Em t

Ô]Ï ÔÔ<Öiׅ˜Ï0ʅÔ]ÑÒ èDØ zH~Jm~q€‚‹zGj`e%~Œj_`JRerF tGJMQzGjYcgq§S £Rq‘L:c |XJR|X‚‹tiJRJMN ⇐ tGJMQ j`L‡JRy•tGJMU jYaIU Qj`eKU q!j`L:w [A, B] = 0 yJMdmd • a, b Jm£RSTzHj`e~DjpjblPJMe˜U Jm~vd [A, B] 6= 0 JMkJ • 

q

∆A =

∆A∆B



2

hψ| (A − hψ| A |ψi) |ψi

=

1 |hψ| [A, B] |ψi| 2

e

Wâ_`Niz§zIJRaI|Ãf JROPe tiJbaIJRq2U k

U;cXe!w ï ù–ï UJm~vd

A, B

Mlo{

λ ∈ R; |ψi = (A + iλB) |ψi

JROPe

0 ≤ hψ|ψi

= hψ| (A − iλB) (A + iλB) |ψi



A

2



+ iλ h[A, B]i + λ

2



B

2



hψ| A2 |ψi

= hψ| A2 + iλ [A, B] + λ2 B 2 |ψi ≥ 0

JbaIJRq2U kUjYaIU Qj`eKUj`‚ŒS

= hψ| A · A |ψi ≥ 0 30

"!

JROPe



A2

B

2





h[A, B]i

≥ 0

≥ 0

= hψ| [A, B] |ψi = hψ| BA − AB |ψi

= hψ| BA |ψi − hψ| AB |ψi = hψ| BA |ψi − hψ| BA |ψi∗ = 2Im (hψ| BA |ψi)

zIJMSj`‚+JRU›tgj`dPJRL‡j`QxL:cŽkaIdmdPJMq€U NG|XJm~k.zHeKNgjp~‚Œd

2

(iλ h[A, B]i) − 4 hψ| A |ψi λ hψ| B |ψi

≤ 0

‚+JR“dI_YwDL‡j!Ÿ j`L(JRy|XyJ!tiJRq2JRJMNGq•e%L:cˆtiJbaIJMq€U k

A, B

L:wDL›_pjYwDd

0

A → A = A − hψ| A |ψi

B

0

→ B = B − hψ| B |ψi λ=1

‚+JR“dIJMOmeKkaIdPdmJMq€U Ni|~‚x‚+JR“d

(i h[A, B]i)2 − 4 hψ| A |ψi hψ| B |ψi ≤ 0 D 0E D 0E 1 2 ≥ B ⇒ A (i h[A, B]i) 4 1 ∆A∆B ≥ |h[A, B]i| 2 0

0



U U j`v [x, P ] = tiJbaIdjpj`NikzIUj`z}L:c kq!j`JR|XNie!k•Uj`‚ŒS kdPNi|Xq ~ i

∆x∆P

≥

~ 2

%Ë à»ËÓHÒ2Ë <˖Õ}Ó Ô[åxÓ!Í iÔàiàŽÓ!Í Êoà Ô]ç XÕ ºÖ Dè ØbÉ _`q2OP‚uS“Dj`q€q2kzIU Omv–dmL#JmjYaIJR‚ e%“€q d 



”



›evu

v”a‡

b

š

R %

d hψ (t) |A|ψ (t)i = dt



     d d d ψ (t) |A|ψ (t) + ψ (t) | A|ψ (t) + ψ (t) |A| ψ (t) dt dt dt

U vdPJM~DU;cŽzIejpjYcXqƒQD£bS

d 1 |ψi = H |ψi dt i~  1 d ψ = − hψ| H dt i~ (9) ⇒

(9)

d hψ (t) |A|ψ (t)i = dt

d dt



ψ (t) |

kejpjYcXq€‚šk€UOPy‚ŒJR“d

 1 d A|ψ (t) + hψ (t) | [A, H] |ψ (t)i dt i~ DG I DG DJ
J
KoL -N),O+.- Ž;

31

Hz j`dPNi|Xq Ÿj`L(JMy.|XyJ%tGJMJRNiq A Uj`‚ŒSTUcXe!w F V [A, H] = 0 d hψ|A|ψi dt

⇒

A

= 0

L:c JRq€“ST‚+“qU j`‚+STaIU Q ‚uS j`‚ŒNŠlG‚+“q•L(wD‚ A L(cˆzIJMQ“zHk¢;U S5_YwDL A |ai

= a |ai

ha|A|ai

= a

U j`‚+‚™zHj`JRv–j`OPk P kq2v–jp~ S j`‚ŒNá%f`F a, E Jm£RS.tiJR‚Œ“q2krzIe0_`JRJRQ e%L‹UcXQe F H lPL‡j AlPLštiJRQ zHjYcXqŠS£Rq |a, Ei cXJOPe [A, H] = 0 F n W¡k€Sj`dPz Ë !ÓHÒrʖÕ%Ô]Ï Ôàiæ ä Ëoå ÖGÎ à P ×œÔ x Ó!Í XË–Õ ÞËoæ%Ñ èDØ JRwuk€e!U d [H, P ] = 0 ∞

™

œ

d hxi dt

d hpi = dt





x, P 2

P, xn+1





=

=

>

hP i m

1 h[P, H]i i~

= xp2 − p2 x = [x, p] p + p [x, p] = 2i~p ~ ~ = [p, xxn ] = [p, x] xn + x [p, xn ] = nxn + xn i i ~ n = (n + 1) x i

d hxi dt

H

=

[x, H] = = = =

1 h[x, H]i i~

=

P2 + V (x) 2m 2  P x, + V (x) 2m   P2 + [x, v (x)] x, 2m 1 2i~p + 0 2m ~ip m 32

uˆ

‚e

Š‰

U vdPJm~UcˆzIejpjYcXq€q L(‚Œe

tiv–j (10)

UcXe!w

(10) ⇒

d hxi dt

  1 ~ip i~ m hpi m

= =

tij`NiJRq2‚xS“Dj`q€q2LTS;dPzIkS“Dj`q€q€c _`JM‚xUcXN}cXJj`dmL‡‚ŒJRN}U q!j`L:w d hpi = dt = = = =

1 h[P, H]i i~   P2 1 P, + V (x) i~ 2m    P2 1 P, + [P, V (x)] i~ 2m 1 h0 + [P, V (x)]i i~  d (V (x)) − dx

L(e%JR“dmaHj`Q kzIU OPvdIS“Dj`q€q€LTSdPzIkS;“Dj`q2qƒ_`JR‚™UcXN«j`dmL‡‚ŒJRN}U q!j`L:w _pjYaHj`JRdHL:cˆJMdMcXk•Nij`ykzIe§aIS;q€wuj`dPL(‚+JRN}tivGJMOme

⇒m

d2 hxi dt2



d (V (x)) dx dV (hxi) d hxi

= 6=



;å Ñ GÎ {˜Ï0Ó%Í ÖHå ÖXÖ èDØ JRzHL(‚ H yJRdPd_`q€OP‚ tGJMJMj`L(zJRzIL‡‚ H, P, X tiJRUjYaIU Q j`eƒF _`q2OP‚µtiJRJmj`L(z |ψ (t)i Uv–dmJm~UcKzIv“k™j`d~q€L kq!~NGk lP‚™JMj`L‡z t  ”

u

i

|ψ (t)i = e− ~ H(t−t0 ) |ψ (t0 )i i

Š

JROPe

U (t − t0 ) = e− ~ H(t−t0 )

F JbaIJMq€U k H Uj`‚ŒS›JMUaIJRdj`J2UjYaIU Qj`e L:cˆS£Rq |ψ (t )i yJRdPd H 0

H |En i

= En |En i i

|ψ (t0 )i = e− ~ H(t−t0 ) |En i

vU ‚ŒdPOPJRe‘L(cˆkv“k€k |ψS i = |ψ (t)i

= U (t − t0 ) |ψ (t0 )i

33

F k€v“k‚‹tiJRJMj`L‡z§JMzIL(‚‹zIJMQ“zrJRwDUS;cˆkcgJMU~5cXJgW¡vU ‚+dmOPJMe‘zIv“k‚!f—tGJMU jYaIU Qj`e hψS |AS |ψS i

= hψH |AH |ψH i

L(‚Œe

† hψS |AS |ψS i = ψH U † AS U ψH

⇒ AH

= U † AS U i i = e ~ H(t−t0 ) AS e− ~ H(t−t0 )

dXH dt dPH dT dAH dt d ⇒ AH dt dAH ⇒ dt

= = =

JbwuzHj`e%U L›_`zIJRdIOPe

PH m dV (xH ) = − dXH =

i i i i H(t−t0 ) i i e~ HAS e− ~ H(t−t0 ) − e ~ H(t−t0 ) AS He− ~ H(t−t0 ) ~ ~ i [H, AH ] ~ 1 [AH , H] i~

Uv–dmJm~UcŠzIv“kUj`‚ŒS5zHj`S~Œjpj`k2lPJRe}_pj`U NiJRSTJbw™tivH‚ŒL tiJbcXd ž B



d AS dt



=

1 h[AS , H]i i~

ʅà Ô]Ï ÔÔ<Öiׅ˜Ï0ʅÔ]ÑÒ èDØ F _`q2OP‚5cXUj`Q q€‚šJmj`L(zrJRzIL(‚ (H, P, X) ~JM~q A UjYaIU Qj`e§UcXe!w U vdPJM~UcŽzHj`ejpjYcgq2qKS‚+j`d g”a‡

d hψ|A|ψi dt

1 hψ| [A, H] |ψi i~

=



e

Ÿ

Ÿ j`L(JMy.|XyJtGJMJRNizIq•tGe

d hψ|A|ψi = 0 dt

F U qcXd aIUQ ‚ JML(L:wDkzHj`e~Djpj`k2lJRe}_pj`U NGJMSTOme§Ÿj`L‡|ψiJRy.|X=y|aiJ_`JReKtGe

[A, B] 6= 0 ⇒ ∆A∆B

≥

1 |hψ| [A, B] |ψi| 2 H

∆A∆E

≥ =

tiS›JMQ j`L‡JRy.j`dPJRe!cŽU jYaIU Qj`e A yhNiJRd

1 |hψ| [A, H] |ψi| 2 ~ d hψ| [A, H] |ψi 2 dt

Es

  G I DG DJ
J
KoL -N),O+.- Ž6P

U JM~vd ∆A d hψ| [A, H] |ψi dt ~ 2

∆t = ⇒ ∆t∆E

≥

F zwDU S;q2‚‹JMj`dPJbcXkzIe§_`JMJRQ e%q€k

S;aIN}JRdPQ›L‡S›Ÿ;L;j`yS“Dj`q€q2kƒj`‚Dcˆ_`q2Omk ∆t U ‚+|Xk >
∆A

QT

i~

d |ψi dt

ψ (t) =

aV

¡

= H |ψi

F [X, P ] = i~, X, P zHj`S“q€e%‚._pj`zIdGUjYaIUQj`e H L:c S £RS5~Djp~q2L›~Dj`e%q‘‚+jYcXy H

X n

i

cn e− ~ En t |ψi

bù ü é íuéùP÷€êMú éù…î;êRùñDð.éùP÷€êMú zHJR“JRUaIqkaIJRcÁF V zIJRUjYaIU Qj`eKkaIJRcÁF n zIJML(e%JR“dPU QJm~5k€e%jYcXqƒL(cˆkaIJRcÁF E éùý ü ù…î;ð.éùP÷€êMú _`q€OP‚™zHj`JMj`L(z}JMzIL(‚+k•zGj`S;U Q kk‘zIUj`z F V zGj`JR“e%JRUjpj`k‘zaIJRcÁF n XË G˅ÑÌ6à g̞ËoÍ +ØœÆ JRQj`|ˆtiJR‚+“qUQ |Xq•cXJUcge!w™k‚+jYarzIJM“JRUaIqkaIJbc kU Sk ÿDêmý hðšûÿ .é ü Œð :óRò…óRò zHj`e%q€v–jp~ cgQj`y•zHj`vU~uJRzc0li_pj`U;aINiL(e§L:cŽ_`JMQ|ÁF V ↑↓ tikJRdP‚™SdH~ye§_pj`UaINiL(ej2tiJRdPq2JRq•JRdmcŠL:cŽkL‡j`NiL‡j`qƒU q2j`L:w.lG_`q€JMq_pj`J H F n kJMdPj`q2e%k•zHL‡j`NGL;j`qF E H N tGJM‚Œ“qJRdmcŠtiJRdj`zId kL(e!c 



™u

*C



<S

+ 2

3

H0 |ψ1 i = E1 |ψ1 i H0 |ψ2 i = E2 |ψ2 i 35

o

o{

S

!

#

S jp~JRcŽyJRdPd W e%UNiJbc H lPL#zIQ |j`zrcXJ!zGj`yhdPOPk‘L(L(v–‚ hψ1 |W |ψ1 i = w11 hψ2 |W |ψ2 i = w22

hψ2 |W |ψ1 i = w21 hψ1 |W |ψ2 i = w12 

=

H0



=

W

E1 0

w11 w21

OPe

0 E2



w12 w22



Ome§JbcXq€q ww =‰ ww 12

∗ 21

11

tiv–j€tGJRcgq2q

E1 , E2

tGviyJRdPd

w11 , w22 = 0, w22 = 0

= H0 + W   E1 w12 = w12 E2

H

E±

|XJR|X‚+‚uU q!j`L:w

F tiJbcXq€q 12

Uq!j`L:w

|ψ1 i , |ψ2 i

S £RS5_`q2|XdHF H L(cˆtiJRJRq2“S;ktiJR‚+“q2jzIwDUSq€kkJMvU dPeƒe%“q€d H |ψ± i = E± |ψ± i 

=

H

0 E−

E+ 0

kv“k‚‹UcXe!w



k“JMU;aIq2LzIJRUaIJMdPj`Jk€“JRUaIq‹Jm£RSµ_pj`|gwDL(L4zIdPzIJRd2zIJbaIJMq€U k‹k“JRUaIq‹L:wTL(‚+eU ‚+S;qšzH“JRUaIq‹e%“q€d tiJRdj`|gwDL(e U † HU

= D

Wâ~ye§U JR“k•Ÿ j`NGJRcˆL(L‡jYw™e%LŒfzHJRL(L:wDk‘k€U j`“kU q2j`L(w™v…£be U j`dPL:cŽkU NGq2‚ U

=

U HU

†



=

cos θ − sin θ 

sin θ cos θ



0 E−



E+ 0

_pj`|gwDL(k‘zIcgJMU~ zHj`S“q€e%‚ ej`“q€L#kUaIq2k

θ E1 , E2 , w12       cos θ sin θ E1 w12 cos θ − sin θ E+ 0 = − sin θ cos θ w12 E2 sin θ cos θ 0 E−    cos θ sin θ −E1 sin θ + w12 cos θ ⇒ = 0 − sin θ cos θ −w12 sin θ + E2 cos θ 1 sin 2θ (E2 − E1 ) + w12 cos 2θ = 0 2 2w12 tan 2θ = E1 − E 2 36

w+£Rk| |ψ± i =

det



E1 − E w12

tan 2θ

=

w12 E2 − E



2 E 2 − E (E1 + E2 ) + E1 E2 − w12

E±



cos θ − sin θ 2w12 E1 − E 2

sin θ cos θ



S£RS›e%“q€d

2 = (E1 − E) (E2 − E) − w12 =0

= 0 q 1 1 2 (E1 − E2 ) + 4w12 = (E1 + E2 ) ± 2 2

tiJRJMdPj`“JMNrtGJM‚Œ“q˜nXcXJ |2w12

tan 2θ |θ|  1, cos 2θ

≈ sin 2θ ≈ ≈ 1

|ψ+ i ≈ |ψ− i +

OPe%j E

1

k U Sk U j`‚+S«F V |  |E − E |

θ

|ψ± i =

2

2w12 E1 − E 2

w12 |ψ2 i E1 − E 2

kJMvU dme%‚™JRzHL(yzIkƒ_pjpj`JRdIL(cˆkU Niq€‚™kq€vojp~L |2w

= E2

1

12 |

 |E1 − E2 |

π 4

'

Fn

OPe%j

1 √ (|ψ1 i ± |ψ2 i) 2

_`q€OP‚š_pj`UaINiL(e%k‘zIe§ej`“q2L›zHj`U ‚ŒzI|Xk€kƒkq§‰ |ψ i ej`k‘‚Œ“q2k 1

t=0

_`q2OP‚Dc yJRdPd bð oý•ý hðéù €é é2ï ‚+“q€‚ t |ψ i _pj`U zIQ k ¢ i

 *

*C

l ‚

£

2

|ψ1 i = cos θ |ψ+ i − sin θ |ψ− i

|ψ2 i = cos θ |ψ+ i + sin θ |ψ− i

i

JML(L:wDk‚Œ“q2k

i

|ψ (t)i = a+ e− ~ E+ t |ψ+ i + a− e− ~ En t |ψ− i

|ψ1 (t = 0)i

⇒ |ψ1 (t)i

= cos θ |ψ+ i − sin θ |ψ− i i

i

= cos θe− ~ E+ t |ψ+ i − sin θe− ~ En t |ψ− i

37

k€L(yzIk‘JRe%dPz§UcXe!w ¤ G I DG DJ
J
KhL -N),O+.- Ž$¥

JROPe P1→2 (t) hψ2 |ψ+ i hψ2 |ψ− i P1→2 (t)

−α e ! − e−α2 2

= |hψ2 |ψ (t)i|

2

= sin θ = cos θ i 2 1 i = sin2 2θ e− ~ E+ t − e− ~ E− t 4

L(JMvU z

2 α +α 2 (α! −α2 ) (α1 −α2 ) −i 1 2 2 −i 2 − ei 2 e e

=

2 α1 − α2 2 α1 − α 2 = 1 · 2i sin = 4 sin 2 2

ã t _`q€OP‚

2

P1→2 (t) = sin 2θ sin

2



E+ − E − t 2~

1→2

‚Œ“q€qU ‚ŒSq€L

(Rabi)

zHyh|j`d



˅ÎÔàgÑ%ÖÞрÔ[ÌÓ2Ë Ô]Ï +Øbß êMñDð+ð.ñ .êMøùð ü ï ü ùP÷íêMëùì :ó¡ûAóRò F zIJR|XL(NikzHj`U JM~zHk ω UcXe2w V = mω x ™

!


#

1 2

=

H

2 2

P2 1 + mω 2 x2 2m 2

L;£R“ En

  1 = ~ω n + 2

_`q€OP‚™kJMj`L(zre%L(kU vdPJM~UcŽzIejpjYcgqKzIeKtgjYcXUdg_`zIJMdœf kyhwŒj`k −

~2 d2 uE (x) 1 + mω 2 x2 uE (x) 2m dx2 2

= EuE (x)

W¡zIJML(e%JR“dPU QJm~ukUj`“‚‹kzHj`e˜U j`zHQL;j zIJMU jYaIU Qj`e˜kaIJbcŠJP£bSTUjYaIQ d JRwxtiJRS ~Dj`Jj`dPe [p, x] = −i~

H |Ei = E |Ei

H

=


~ω X2 + P 2 2 38

tijYcXU d

F tGJm~q€qU |Xy X

=

P

=

mω x ~ 1 √ p mω~

£bcX~Dy£U jYaIU Qj`e˜UJm~vdXW¡tiJbaIJMq€U k

_YwDL

X, P

U;cXe!w€f A =



[H, A]

Ÿj`L‡JRyk|XyhJUcXe!w Ÿj`L(JMyk‘|XyhJ%zIe§Nijp~D‚Œd

 ~ω ,A ~ωA† A + 2  †  = ~ω A A, A   = ~ω A† , A A = −~ωA

[H, A] =

H, A†

U Jm~vd

= ~ωA† A +

  A, A† = 1



(X + iP )

_`e%JMdPjYaIL‡JRq€k€k•_YwDL

~ω i [P, X] 2 ~ω ([P, X] = −i) ⇒ = ~ωA† A + 2 H

√1 2

1 √ (X − iP ) 2

=

UcXe!w

r

[P, X] = −i

A†

X, P



= . . . = ~ωA†

lP‚ L:cŽS £RS5~JMU j`q kdPS;a lP‚ HH L(cˆS£bS›k€L(S;q AA k€y+wŒj`k JbwxyJRwŒj`dPj H |Ei = E |Ei yJRdPd›F V ~ω ~ω

H (A |Ei) = (E − ~ω) (A |Ei)

HA |Ei

†

L‡‚Œe

([H, A] = −~ωA ⇒ HA − AH = −~ωA) ⇒ = AH |Ei − ~ωA |Ei = AE |Ei − ~ωA |Ei = (E − ~ω) (A |Ei)

E{Z

JbwuyJbwŒj`d5F n H A† |Ei



H, A

 †

= (E − ~ω) A† |Ei




= ~ωA† ⇒ HA† − A† H = ~ωA


†




L‡‚Œe

HA† |Ei ⇒ = A† H |Ei + ~ωA† |Ei = A† E |Ei + ~ωA† |Ei
= (E + ~ω) A† |Ei

tikqNgjp~‚Œd Šc L(L‡v‚ E ≥ 0 F V £

H

=

1 P2 + mωx2 2m 2

E

F tGJM‚+j`JMyšS £RS5_YwDL‡j2tiJbaIJRq2U ktiJRUjYaIU Qj`eKJRS j`‚ŒJMU›L:cŽtijYwD|ŽU q2j`L:w Uj`SJbcX‚ E ~DJRUj`q A F n ~ω W |Ei = 0 L‡S;qƒ~ye§Uq!j`L:wƒf`F UzGj`JR‚š¢j`q€dH‚+“q A |E i = 0 F E j`kqF s E 0

0

1 H |E0 i = ~ωA† A |E0 i + ~ω |E0 i 2 1 = ~ω |E0 i 2

n ∈ N ∪ {0} ; En

|En i =



=

1 n+ 2



OPe

E0 = 21 ~ω

U q2j`L:w

~ω

L;£R“


n 1 √ A† |E0 i n!

yJbwŒj`d kyhwŒj`k L‡q€U dPL#¢;JRU “Dj |E i|E= bi =A |EA |Ei tGJMSi~Dj`Jj`dPe n

n

n

 hEn |En i = b2n En−1 |AA† |En−1  


 A, A† = 1 ⇒ = b2n En−1 |A† A + 1|En−1 H



1 = ~ω A A + 2

⇒ bn

†

= b2n n = 1 1 = √ n

40



√1 n †

†

n−1

n−1

L(‚Œe L‡j`q2U dmk€qƒ_YwDL

Hz j`U S;k tieKUj`zIQ LTkJMk•_`zIJMd›F V −

r

~2 d 2 1 uE (x) + mω 2 x2 uE (x) 2 2m dx 2



= EuE (x)

F zIJbaHjYaIQ q2JR|XeKzGj`vkdPzIk€j JP£bSTUj`zIQ L›_`zIJMdH_YwŒj`qw F n

A |E0 i = 0

i mω ~ d uE0 (x) = 0 x+ √ 2~ 2mω~ i dx ! r r mω ~ d uE0 (x) x+ 2~ 2mω dx

uE0 (x) = ce− 2~

x2

1 uEn (x) = c √ n!

r

mω

1 = c√ n!

n mω ~ d i mω 2 x− √ e− 2~ x 2~ i dx 2mω~ !n r r mω 2 mω ~ d e− 2~ x x− 2~ 2mω dx

XËÌ6Î]ÔÔAÒ«ÖÒ2Ë <Ë–Õ 0Ë g˅ÔÔ iÎ »Ó!Í ä Ë ià +ØRè F U qcXdGJRzIJmjpj`OHSdmzrOPe V (~r) = V (r) tik‚ŒcŠzHj`JRS‚Œ‚‹~Dj`e%qƒ‚hjYcXy.JRzHJMjpj`O SPd z k U Sk 

”



?

?”ne =

.vu

!

:m

~ = ~r × p L ~

Ÿ j`L(JMy.JR|XyJMktiJRdj`zId

yj`zHJRQ

[x, px ] = i~ [y, py ] = i~ [z, pz ] = i~

W⢇‚hj`|Xq‘U jYaIU Qj`eKej`k L~ _YwDLŒf Ÿj`L(JMy.|XyJ%ke%U d5F V

[Lx , Ly ] = i~Lz [Ly , Lz ] = i~Lx [Lz , Lx ] = i~Ly

kyhwŒj`k [Lx , Ly ]

= [ypz − zpy , zpx − xpz ] = [ypz , zpx ] − [zpy , zpx ] − [ypz , xpz ] + [zpy , xpz ]

= y [pz , z] px − 0 − 0 + [z, pz ] py x = i~ (xpy − ypx ) = i~Lz IH¦
G I DG DJ
J
KML

41

-N),O+.- Ž6p



~2 L  L2 , Lx

p

~ 2 = L ~ L

L:cˆL‡~Œj`vie%“q2d5F n

= L2x + L2y + L2z   = L2x + L2y + L2z , Lx     = L2y , Lx + L2z , Lx = Ly [Ly , Lx ] + [Ly , Lx ] Ly + Lz [Lz , Lx ] + [Lz , Lx ] Lz = −i~Lz Ly − −i~Lz Ly + i~Lz Ly + i~Ly Lz = 0

_`q€|Xd L~ , L L:cŽtGJMJRq2“STtGJM‚Œ“q•tGkJRzHJMjpj`OISdPz˜L(cŽtiJR‚+“q•_YwDLrF E 2

~2

L |λ, mi

Lz |λ, mi

z

= λ~2 |λ, mi

= m~ |λ, mi

m, λ

S£bS›ej`“q2LTk“U d kdPS‡a

o n n ` ∈ l|l = , n ∈ N ∪ {0} ; λ = ` (` + 1) 2 n o n m ∈ l|l = , n ∈ N, −` ≤ l ≤ ` 2

_`q2|Xd kyhwŒj`k

= Lx + iLy = Lx − iLy

L+ L−

kq€e%zIk‚š~DJRUj`q2j2kL(SqƒUjYaIU Q j`e

Lz |λ, mi Lz (L+ |λ, mi) ~ 2 (L+ |λ, mi) L

Lz (L− |λ, mi) ~ 2 (L− |λ, mi) L

= m~ |λ, mi = (m + 1) ~ |λ, mi

= λ~2 (L+ |λ, mi) = (m − 1) ~ |λ, mi

= λ~2 (L− |λ, mi)

[Lz , L+ ] = ~L+ [Lz , L− ] = −~L−

= i~Ly + i (−i~Lx ) ~ 2 , L+ L

i

= ~ (Lx + iLy ) = ~L+ h i ~ 2 , Lx + iLy = 0 = L

[Lz , L− ] = [Lz , Lx − iLy ]

= i~Ly + i (i~Lx )

h

~2

L , L−

i

ke%U d íýhì

[Lz , L+ ] = [Lz , Lx + iLy ]

h

L£b“

= −~ (Lx − iLy ) = −~L− h i ~ 2 , Lx − iLy = 0 = L 42

Jbwu‚ŒL tGJRcXdH_YwŒj`qw L+ L−

= (Lx + iLy ) (Lx − iLy ) = L2x + L2y − i [Lx , Ly ] = L2x + L2y + ~Lz

⇒ L + L−

L− L+

~ 2 − L2z + ~Lz = L ~ 2 − L2z − ~Lz = L

tiJbaIJRq2U k.tiJRUjYaIU Q j`e§L:cˆJMdmcˆU~|Xq‘zIJMSj`‚+JRU›zIJMdP‚Œz˜L:cˆS£bS;c Uye%q ~2 L

éùø D÷
S

λ≥0 •

= L2x + L2y + L2z λ

JP£bS5tij`|Xy

m •

L+ L− = L†− L− ⇒ hλm|L+ L− |λmi ≥ 0 D E ~ 2 − L2z + ~Lz |λm hλm|L+ L− |λmi = λm|L   ⇒ ~2 λ − m 2 + m ≥ 0 ⇒ λ − m (m − 1) ≥ 0 λ − m (m + 1) ≥ 0

`≥0

NGUu_`yh‚+dtHwD|Xk‘F λ = ` (` + 1) U JM~v–d

λ = ` (` + 1) ≥ 0

` (` + 1) ≥ m (m − 1) ` (` + 1) ≥ m (m + 1) ⇒m ≤ ` m ≥ −`

¢j`q€dPk

m2

‰ UzGj`JR‚‹k€j`‚+vk

m1

U j`⇒‚+Sš−`F tij`q2≤JR|XmNi≤q‘cX` JU q2j`L:w•V]lP‚ m k€L(S;q

L(‚Œe F U zHj`JM‚

L+

L+ |m1 i = 0 L− |m2 i = 0

tiv–j Y

ô :óRò–û<óâû µý.ïì ü ï tGvgj`dPL zGj`Sjp~JGF H, L~ , L lL tiJRq2JRe%zIq€c |n, l, mi NGJmjp~q€‚‹j`dPU zIQ bðhêRðÿDùP÷2ì (θ, ϕ) ?§

l,m

2

Z

oC



z

∗ dΩYl,m Yl 0 m0

En

?§§¨#

= δll0 δmm0

= − (13.6ev) 43

1 n2

zIJML(e%JM~U kkejpjYcXq€kq

S jp~JUcge!w‹F Wâ_`q€JMq‘tijYaIe%L#NGUf llP‚™_pjpj`JRdj mlP‚‹_pjpj`JRdHcXJ!UcXe!w

u = rR;



−

Z

∞ 0

Rn,l (r)

= ...

u2n,l dv =

Z

∞ 0

UcXe!w

2 2 Rnl r dr

= 1

vo£Re

 ~2 d2 u ` (` + 1) + u + v (r) = Hr 2µ dr2 2µr2 Hr u = Eu

Z

Z

0 ∞

0

∞

un,l un0 ,l dr

= δnn0

Rnl Rn0 l r2 dr

= δnn0

F vo£ReKzHj`JM“NGdj`QxtGJML(‚ŒNiq•j`dPe

_YwDL‡j2JbaIJMq€U k

l=l

|Yl,m (θ, ϕ)| = |Yl,m (θ)|

|Y0,0 | = const

0

Hr

Rn,l

tie%k

tiv–j

Uj`‚ŒSTNiU5U q!j`L:w

zHj`U ‚ŒzH|Xk

Uj`‚ŒSTL:cXq€L zHJRUjp~ w‹kJRUaIq2JR|ŠU q!j`L:w Uj`‚ŒS

|Y10 | = c |cos θ|

> T¿  8:B(½4C 8;C›½ CxB;=›ÂÀ C™Â6BC Sjp~JMk‘_pj`UzHQk‘JMOPeKF H L:time cˆzHj`−JMvU independent dPe!k2j€tiJR‚+“q€k•perturbation zIe§j`dPU zIQcŽythoery JMdmd =W

M©ªU

«©žU

¬UU

­

0

H0 |ϕn i = En0 |ϕn i

H

= H0 + W

hϕn |H0 |ϕn i

hϕn |W |ϕn i

s]s

=

En0

 En0

F kSUQ k‘zIQ |j`z W lPL#e%U NGd U q2j`L:w‹kcXL(y•kSU Qk€kcŽyJMdmd

tiJM~q2qU |Xy‹S j`‚ŒNrej`k W lj H _`JM‚›cXU Q kkU q!j`L:w

H0

0

L‡~Dj`viU~|X‚

H1

cj`U ~DdIUcge!w

zIe§U j`zHQLTtiJR“Dj`Uxj`dme§U q2j`L:w

= H0 + λH1

H

H |ψn i (H0 + λH1 ) |ψn i

_`q€|Xd _YwDLTF λ  1

W = λH1

= En |ψn i = En |ψn i

ŒÉ ØœÆ F λlP‚‹UjYaw |ψ i l j E zIeKyzIQ d

®"¯"¯"°:±ž²‘³š³“´}¯µX¶·¬¸ˆ¸´¶¯:´ n

∞ X

= En0 + λEn1 + λ2 En2 + . . . =

En (λ)

L‡j`q2UdGtGUj`vw

Enk λk

_`q€|XdItie§‰ |ψ i zIeKyzIQ d

k=0



|ψn (λ)i = N (λ) |ϕn i +

(H0 + λH1 ) |ϕn i + λ

(1)

X

(1)

k6=n



k6=n

X

k6=n

(2) cnk

λ0 ; H0 |ϕn i

(1)

cnk |ϕk i + H1 |ϕn i

|ϕk i + H1

X

(1) cnk

k6=n

n

cnk (λ) |ϕk i

tiJRq!~NGq2kUjYa}_YwDL

(2)

|ϕk i

k6=n

X

k6=n

‚ŒJM“dH_YwDL 

(2)

X

cnk |ϕk i + λ2

En0 + λEn1 + λ2 En2 + . . . |ϕn i + λ

λ1 ; H0 λ2 : H0

X


=

k6=n

N (λ) 

= λcck + λ2 cnk + . . .

cnk (λ)



X

cnk |ϕk i + . . .

(1)

cnk |ϕk i + λ2

X

hϕn | · En0

X

k6=n

(1)



(2)

cnk |ϕk i + . . .

U~|ˆL(wD‚utiJRJMNGzHz§kejpjYcXkc0tiJR“Dj`Uxj`dPe k6=n

= En0 |ϕn i X (1) = En0 cn,k |ϕk i + En1 |ϕn i k6=n

= En0

X

k6=n

(2)

cn,k |ϕk i + En1

X

k6=n

(1)

cn,k |ϕk i + En2 |ϕn i

Rùú6ì üxü ñŒëDýéù ü ïïé ü ù–é :óRò…óRò F kejpjYcXq€‚tGJm~D~“x¢j`Q kd 



n

0



cn,k |ϕk i + En |ϕn i ⇒ ∆En1 = λEn1

45

= En1 = hϕn |W |ϕn i = hϕn |λH1 |ϕn i


S



A

F S;U Qj`qƒJRzHL(‚‹‚Œ“q2‚xkSU Q kk‘L:cˆzHJRQ “zIkƒ¢;USL = _pjYcXe%U5U~|Xq E kJMvU dme%L#_pj`NGJMzIk‘_YwDL _pjYcge%UuU~|Xq‘kJRv–UdPe%L#_pj`NGJMzIk n

= En0 + hϕn |W |ϕn i

En

|ϕk i 

hϕk | · En0

X

(1) cn,k

k6=n



0

|ϕk i + En |ϕn i

L(S›kL:aIk€k•zHeKNgjp~‚Œd

(1)

= Ek0 cn,k + hϕk |H1 |ϕn i (1)

= En0 cn,k

hϕk |λH1 |ϕn i En0 − Ek0 X hϕk |W |ϕn i ⇒ |ψn i = |ϕn i + |ϕk i En0 − Ek0 (1)

⇒ λcn,k

=

k6=n

|hϕk |W |ϕn i| 

0 En − Ek0

= 1 + o λ2

N (λ)

cj`U~dHzGj`S;U Q kkƒzIUj`z«¢;Uj`“L

ej`kL‡j`q€U dPkrW¡zIJM‚uL‡JRvU zžfk€OH‚Œ“q2‚




êMøú ü ñDëðéù ü ïïé ü ù–é :óRò…ó¡û L‡S –¢;JRL:Xc d hϕ |
S



A

n



hϕn | H0

X

k6=n

(2)

cnk |ϕk i + H1

X

k6=n



(1)

cnk |ϕk i

= hψn | H1 

X

k6=n

= hϕn | En0

= ⇒λ

2

En2

=

(1)

cnk |ϕk i

X

k6=n

(2)

cn,k |ϕk i + En1

hϕn | En2 |ϕn i X (1) λcnk hψn | λH1 k6=n

=

:q

X

k6=n



(1)

cn,k |ϕk i + En2 |ϕn i

|ϕk i

X hϕk |λH1 |ϕn i hψn | λH1 |ϕk i En0 − Ek0

k6=n

=

X |hϕk |W |ϕn i|2 En0 − Ek0

k6=n

WE

0 n

U jYaI‚‹Jbw€f€~Dj`|XJRkzIq2U5zIe§k€~JRUj`q‘kSU Q kkÛF V W¤£Mj`OHzHe§j`OHzGj`yjp~™zGj`q€UD£–fµF n F _YaIN}kJMvU dme%kcXUQ kc L:ww™kL‡~vgkdmwcŽkq€UxzIS;QcXkÛF E

zHj`U S;k

− Ek0 < 0

¦
G IHG DJ
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λ2 En2 ≤

 2 2 (∆W ) = ϕn |W 2 |ϕn − (hϕn |W |ϕn i)

(∆W )2n ∆E

F U zHj`JR‚‹k€‚+j`U NGkkJMvU dPe!L NiyhU q2k

λ2 En2

≤ = = =

∆E

L(JRvU z F V Ucge!w

1 X 2 |hϕk |W |ϕn i| ∆E k6=n

1 X hϕn |W |ϕk i hϕk |W |ϕn i − (hϕn |W |ϕn i)2 ∆E k i  1 h

ϕn |W 2 |ϕn − (hϕn |W |ϕn i)2 ∆E 2 (∆W ) ∆E

£pcXL(y£IJRL(q€cgyk€~ cX‚š_`q€JMqtgjYaIe kJRvU dPeƒL(S‚š~yeK‚+“qNGUTcXJ%U q2j`L:w.‰ n = 1, l = m = 0 ~Dj`|XJMkzIq€U ‚ E10

OPe

= −13.6eV 1 = − µc2 α2 2

F _pjpj`JRwD‚‹JRL(q€cXyhkk€~cgk‘L(cŽ_pjpj`JbwDkzIe§U yh‚+d

~ = Ez z E

e2 p~2 − 2µ r W = eEz hz1 i ∼ r1 ∼ 10−8 cm =

H0

_pjYcge%UxU ~D|X‚™~Œj`|XJRkzIJRv–UdPe%LT_pj`NGJMz

∆E11

= λE11 = hϕ1 |W |ϕ1 i

= h1, 0, 0|eEz|1, 0, 0i = eE h1, 0, 0|z|1, 0, 0i Z = eE d3 rϕ∗1,0,0 (~r) zϕ1,0,0 (~r) = ∗

ϕn,l,m (~r) = Rn,l (~r) Yl,m (θ, ϕ)

ϕn,l,m (~r)

kJR“Nidj`Qk•L:c zHj`JRvoj`OmL#‚+LµtiJbcXd

l

ϕn,l,m (−~r) = (−1) ϕn,l,m (~r) ~r → ~r r θ

→ r → π − θ, ϕ → π + ϕ

∗ = 0 47

_YwDL

kq€vojp~

JRdmcŠU~|ŽNgjp~‚Œd ∆E12

E10 − En0

= e2 E 2

Z

Z

Rn =

r

3

∗ drR10

(r) Rn,1 (r)

r

2

∗ drR10

(r) Rn,l (r)



⇒ |h1, 0, 0|z|n, l, mi|2

∗ r3 drR10 (r) Rn,l (r)

Z

∗ dΩY00 Yl,m (r cos θ)

Z

∗ dΩY00 Yl,m cos θ r   Z Z 4π 1 ∗ ∗ Y10 = r3 drR10 (r) Rn,l (r) dΩ √ Yl,m 3 4π Z Z 1 ∗ ∗ dΩYlm Y10 = r3 drR10 (r) Rn,l (r) √ 3 Z 1 ∗ = r3 drR10 (r) Rn,l (r) √ δl,1 δm,0 3 1 = √ Rn δl,1 δm,0 3

(dΩ = sin θdθdϕ) ⇒ =

Z

2

n 6= 1 l, m   1 = −13.6 1 − 2 n     2 e 1 e 2 n2 − 1 = − 1− 2 =− 2r1 n 2r1 n2

h1, 0, 0|z|n, l, mi =



|h1, 0, 0|z|n, l, mi| E10 − En0

X

=

f (n) = ∆E12

X

n 6= 1 l, m

1 2 R = f (n) r12 3 n 2n−5 28 n7 (n − 1) 3 (n + 1)

= −2e2 E 2 r13

2n+5

X

n 6= 1 l, m

n2 f (n) n2 − 1

= 1.125

⇒ ∆E12

= −2.25e2E 2 r13

n2 f (n) n2 − 1

JMUq!j`dI‚+jYcXJMyhq

F JMSj`‚+JRU k•NGUaI|«aINiQ eƒzIe˜j`dmL‡‚ŒJRN zHj`e%U kL _`zIJMdTzIJR‚™L(JMvU z :y

∆E12 ≤ 1 2 r 3
1 3r12 ⇒ 3

e2 E

2

100|Z 2|100 |E10 − E20 |

lP‚‹Jmj`L(zrJRzIL‡‚



‚Œ“q2k•F tiJR‚+“q€k•zHj`q€L:c0JMQL

|100i θ, ϕ

2  2  2  = z = x = y = r12

  G IHG DJ
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U q2j`L:w E

0 1

− E20 =

3 e2 8 r1

Ÿ|j`dP‚™F Y

00

=

√1 4π

Sjp~Jmj

8 3 2 r E 3 1

∆E12 ≤

ÊoÔ]ÔÎ ä Ï XÔ iрÕIÖXÖ Ñ2Ô ÉŒØbß tGJM‚Œ“q2k L:cXq€L _pjpj`JRdPk0|XNg~dPJRe = i _`q€|Xd»F E kJRv–UdPe%‚«_pjpj`dPq |ϕ li=‚Œ0,“q2mk’=F N0, l == 1,4 mã n=‚+0,“q»±1L:c _pjpj`JRdPk0zInv=U~2L‘k€UNid OPe (i = 1, 2, 3, 4) w

0 2

n

?e



=

n

E = En0 ϕ(i) n

E H0 ϕ(i) n

cXJU q2j`L(w™F H tiSj2tGq2“S5_`JR‚+L4tidPJR‚utGJMQ j`L‡JRyktGJMUye§tGJMU jYaIU Qj`e%L#U jYcXNŠ_pjpj`JMdPk•|XNg~dPJRe cŽ¢‡w I tGJMU jYaIU Qj`eKUj`y‚ŒLT_`zIJMdHF hϕ |ϕ [I, H0 ] = 0

n

ϕin |ϕjm

mi

= δnm

_pjpj`JRdGkJRk•e%L UcXe!w

_YwDL S £Mj%j`q€“S‚‹ej`k•zHJRQ “z}¢‡U Skƒj`eKS£bSk‚ŒyU q€‚™UjYaINij2L:w™tGvg_YwDL

H0

Nn X i=1



|ψn i = N 

Nn X i=1

(H0 + λH1 )  =

λ1 : H0

k6=n

Nk X (1)

cnk

i=1

= En0

Nn X i=1

E αi ϕ(i) n

JROPe

i=1

k6=n

Nn X i=1

 Nk E E X (1) X (i)  (i) βi ϕk cnk αi ϕn + λ k6=n



i=1

 Nn Nk E E X X X  (i)  (1) E0 + λEn1 + . . .  αi ϕ(i) +λ c βi ϕ n ⇒ λ 0 : H0

X

= δnm δij

 Nk E E X (1) X (i) (i) αi ϕn + λ cnk βi ϕk + λ2 . . .





E αi ϕ(i) n



Nn X i=1

nk

i=1

k

i=1

k6=n

Nn E E X αi ϕn(i) αi ϕ(i) = E0 n i=1

Nk Nn Nn E E E E X X (1) X X (i) (i) (i) 1 αi ϕ(i) βi ϕk + En cnk = E0 αi ϕn βi ϕk + H1 n i=1

0 + H1

Nn X i=1

D

k6=n

Nn X E (i) ϕ α ϕ(j) W i n n

Nn X i=1

i=1

=

i=1

αi

D

i=1

Nn E E X = En1 αi ϕn(i) αi ϕ(i) n

E (i) ϕ(j) n W ϕn

D

Nn E X (i) 1 ϕ α ϕ(j) λE i n n n

= λEn1 αj

49

i=1

i=1

L(‚+NGdj Dϕ Pl ‚™L(JRQwDd (j) n

_YwDL

Nn × N n

U~|Xq‘k€“JRUaIq€k

E D (i) (j) hji = ϕn W ϕn Nn X

(n)

tGJMdjpj`dmq2ktiJR‚+“q€k•L:cˆ‚+yUq2‚™kS;U Q kk‘zI“JMUaIq

= ∆En1 αj

hji αi

i=1

h i ⇒ h(n) α ~k

1 = ∆En,k α ~k

tiJR‚Œ“q2k˜_YwDLrF _pjpj`dPq€kr‚ŒyU q€‚kSU Q kkrzI“JRUaIq}L:cS £RS;L = _pjYcge%U.U~|Xq}k€JRvU dPe%Lu_pj`NiJRzIkcÜj`dPL(‚ŒJMN»_YwDL F h L:cˆS£Rj`k = k€S;U Q kk‘U ye!L zIdjpj`dmq2k‘zHj`SUQ kk‘zHU j`z tgjYwDJR| D

E (j) ϕ(i) n |W |ϕn   k = 1 . . . k; h(n) (α)k

(n)

= hi,j

0

= ∆En (k) (α)k

det h − ∆En1 (K) I




kdjYcŽ‚ŒJRzIwD‚

= 0

_YwDL#F k€S;U Q kk‘U ye!LT_pjpj`di_`JRe‘w+£RU~‚š‰ N ej`kzHj`dj`U zIQ k‘U Q|Xq |ψn i =

Nn X i=1

n

E αi (k) ϕ(i) n

F ϕ  |XJM|X‚Œ‚uzIJRdj`|gwDL(e H JRzIq‘Ngjp~‚ŒLT¢;JRU “ âW _pjpj`JRdPkL:c CSCO UjYaIU Q j`e I f€F [H , I] = 0 tie˜‰ k‚hjYcgz tiJR‚Œ“q 4 cXJ n = 2 Uj`‚+SKF ‚ŒyU q€‚•JRdPv–j`q!j`k€jIS j`‚ŒNŽJRL(q€cgyKk€~ cX‚•_`q2JRq§tijYaIeŠlžNGUaI| aINGQ e tiJRdjpj`dPq i n

i

1

‡

|0, 0i , |1, 0i , |1, 1i , |1, −1i W

„

k U Sk kq€vojp~

OPe W = eE~~ r kSUQ kU y‚Œd



 h0, 0|W |0, 0i h0, 0|W |1, 0i 0 0  h1, 0|W |0, 0i h1, 0|W |1, 0i  0 0     0 0 h1, 1|W |1, 1i 0 0 0 0 h1, −1|W |1, −1i £

U ye!qr‰ l − l 2

Fm

1

tiSTtiJRQj`L(JRy

[Lz , z] = 0  2  L , z 6= 0

L z , L2

tie%k

e%j`‚•W¡Ngj`L(‚Œ‚!fNiL(y‚uNiU›tiJRdj`|gwDL(e§e!L#OPe§F tGe§NGU5JRQj`L(JRy_YwDL Uj`‚ŒSuNGU hn, l , m|z|n, l , mi 6= 0 cŽL‡‚ŒNid z LJRvoj`OlJReKUjYaIU Qj`erU j`‚+S F (−1) Jm£RSTzIU~v–j`qƒzHj`JMv–j`OPkc k€q€v–jp~‚

= m2 = uneven 1

W

1

z

2

l

h2, 0, 0|z|2, 1, 0i =

Z

∞

3

r drR20 (r) R21 (r)

0

= −3r1

Z

∗ dΩ cos θY0,0 (θ, ϕ) Y (θ, ϕ)

IHJ
G IHG DJ
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w+£Rk| 

tiviL(‚ŒNidIF ∆E

1 n=2

0  −3eEr1   0 0

 0 0   0  0

0 0 0 0

S£bSkq‘tiJRdmcˆF k“JMUaIq€kzIe§_`|gwDL‡dG_pj`NiJRz˜e%j`“q2LTJM~ w

(k = 3, 4) = 0 −∆E21 −3eEr1

3eEr1

‚›k€j`‚+vIkq2Uk

−3eEr1 0 0 0



−3eEr1 = 0 −∆E21

L(‚+NGd F JMUe%JMdmJML(k•NGUaI|«aINiQ eƒkO tiJRq2JRe%zIq2k•tiJRq2“S;k•tiJR‚Œ“q2k

1 ∆En=2 (k = 1, 2) = ±3eEr1

∓1 −1 −1 ∓1



α1 α2



S£bq2k‘_YwDL

1 1 √ (|2, 0, 0i − |2, 1, 0i) , √ (|2, 0, 0i + |2, 1, 0i) , |2, 1, 1i , |2, 1, −1i 2 2 √1 2

F

U j`‚+S›F kJRv–UdPe%‚5_pj`NiJRzK_`JMe |2, 1, 1i , |2, 1, −1iLw+£Rk|«_YwDL ‚™k€~Uj`kƒj`dPL(‚+JRN (|2, 0, 0i + |2, 1, 0i) Uj`‚+Sj 3eEr _`e%JRdjYaIL(JMq€kcXJbcˆyJMdmd ÿDêMý hðšêRøúÜéêRê +ý

(|2, 0, 0i − |2, 1, 0i) 3eEr1

√1 2

1

MC

S

H0 |ϕ1 i = E1 |ϕ1 i

e%j`“q2LTk“U d H = H + W tGcˆyJRdPdHF S£Rq‘JRdmcŠJm£RS›tiJR|XU Q dItiJR‚Œ“q2kcj H0 |ϕ2 i = E2 |ϕ2 i 0

H |ψ+ i = E+ |ψ+ i H |ψ− i = E− |ψ− i |ϕ1 i , |ϕ2 i W

=



w11 w21

w12 w22



j`dPe!“q§F JbcXq€q

‰

|XJR|X‚+‚™‚+j`zwDd

w21 w11 = w22 = 0

j`dPydmk

|ψ+ i = cos θ |ϕ1 i + sin θ |ϕ2 i

|ψ− i = sin θ |ϕ1 i + cos θ |ϕ2 i 2W12 tan 2θ = E1 − E 2 q 1 2 (E1 + E2 ) ± (E1 − E2 )2 + 4w12 E± = 2 sin θ cos θ

w12 E1 − E 2 ≈ 1 ≈

W12 |ϕ2 i E1 − E 2 w12 |ϕ1 i |ψ− i = |ϕ2 i + E1 − E 2

⇒ |ψ+ i = |ϕ1 i +

51

OPeƒkcXL(y•kSU Qk‘yJMdmd

F zHj`SU Q kkƒzIUj`z}zIe§kU Omy•j`dPL(‚ŒJMN OPeKF |XQ e!zHqK_pjYcXe!UxU~|Žj`dPydPkOme w = w = 0 k€yhdPkk 11

E+

= E1 +

22

2 w12

E1 − E 2

47 8AÀ B;9K748 »B9 +ùý€î#êR÷€øþ[ðïhñŒúºý bðhêRðšÿŒùP÷2ì Aó :óRò WpntiJbaIdjpj`N}|XUj`NG‚™e%“q€JMzrzINGJmjp~q€k kyhwŒj`kkHf _`q2JRqtgjYaIe!Lr_`q€JMO%aINGQ e «]

S

QYnV

R

º $§

»

H

=

¹

‹

e2 P~ 2 − 2m r

S j`‚ŒN}JRaIdmv–qk€~cŽ¢j`zI‚štgjYaIe!kcˆyJRdPdHF L~ JRzHJMjpj`OHSdPz˜cgJ_pj`UaINiL(e%L ~ B

= B zˆ

F

kJRv–UdPe%k•U;cXe!w M~ L‡j`Q Jm~uaIdmq!j`q•cgJOPe O£RdPzrj`L cXJ r |j`Jm~U5L‡S›S;d e yJRdPd

~ ·B ~ W = −M L = µrv 1 ~ ML = iπr2 c v i = e 2πr erv ~ M L = 2c e ~ ⇒ M L = − L 2µc ~ ~L = − e L M 2µc

L

W¡‚ŒUj`NGqifF JR|XL‡NGk•UcgNikkOHU q!j`L:w F kJRv–UdPe%LT_pj`NGJMzIk_YwDL

W

= H1 =

eB Lz 2µc

zIJR|XL(Nik•zHj`U JM~zIkrW¡Uj`q2ULTzGj`UJm~z}e%U NGd w f

= ω L Lz

L

ωc

En0 |ϕn i

eB µc

=

JRaIdmv–qk€~cˆJRL‡‚

 1 e2 2r1 n2 = |n, l, mi = −



Uj`‚ŒS›zGj`S;U Q kk‘zIUj`z}tieKNgjp~‚ŒdIOPe

H = H 0 + ω L Lz [H0 , Lz ] = 0 IH¦
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OPe

[Lz , H] = 0

JbwuzHj`S;U Q kkƒzHU j`zHkƒ¢‡Uj`“™_`JRe

H |n, l, mi = (H0 + ωL Lz ) |n, l, mi
= En0 + ~ωL m |n, l, mi

ÊoË–Õ GÖ ŒØœÆ _`JRQ |Xk•yj`zHJRQ WYVZ:nV:f!¢‡L(U vg_`UacÁF V _`q2JRq€ktijYaIe§zHj`q€UuL:cÜWâ_`JM~SfNg~kkdP‚Œq€kF n WYV[Z(nœ^(f!F JRL‡q2j`dPe%k_`q2JROaINiQ eF E F _`JRQ |Xk•L:c yj`|XJMdXWYV[Z(n ” f€JML‡j`e%QŠF s F zIJbaIU Q kzHj`|XyJRk‘zIUj`z}tieƒtiJbaIdjpj`NGk•zIUj`z}zIe˜~ye!q«WYV[Z:nœZ‡f€NGU JM~ˆF ^ ;í ü þ ü ÷2ú0êRùëêMø AóRò…óRò U ‚Dw™j`dPe!“q zIUjYwDOPz 

|

X

~ = ~r × p L ~

tGvgj`dPe%“q

h l

~ 2 , Lz L

i

[rx , px ] = i~ ⇒ [Lx , Ly ] = i~Lz

= 0

UcXe!w

l (l + 1) ~2

‰

¼º

UcXe!w

tik L~ :L cŽS £RS;k‘UcXe!w

3 1 = 0, , 1, , . . . 2 2

F zHj`Q JR“UuJRL;j`NGJRcŠL(L(v‚‹tiJRq2L:cŠe%L(ktiJbwDU S;k‘j`dPq€L‡S;zIk L(cˆzHj`dj`U zIQ kKF −l ≤ m ≤ l UcXe!w m~ L L(cˆS£bS z

~ ∂ Yl,m i ∂ϕ ⇒ Yl,m

⇒ eimϕ

⇒m

= m~Yl,m = Θ (θ) eimϕ = eim(ϕ+2π) ∈

j`‚ŒJMJRy

Z

m = 0, ±1, . . . , ±l

F

∂B ∂z

6= 0

cr¢(w z _pjpj`JbwD‚TJRaIdmv–qxk€~cr¢;U~ x _pjpj`JRwD‚TtGzGj`ej`U U y+cjŸ|gw JMq2jYaIe•j`yhNiL—JMj`|XJMdm‚Ž¢‡L‡UvX_`UacˆJmj`|XJRd JRaIdPvq•aIdmq!j`q•cgJRcˆyJRdPdHF Ù;dPUj`L yjYwš_`JReƒ_YwDL;j2JML(UaIJRdtgjYaIe%k V F~

~ ·B ~ = −M

~ = Mz = −∇V

∂Bz ∂z

j`e%“q«tik}L(‚Œe F yj`L(k«L:w‘L(StGJMq2jYaIe%krzIe ~Djp~q€L.kQ “d6k€~cgk}L(cÝJMdmcgk}~“‚‘yj`LštiJbcXd6tGe cXJ6U q!j`L:wÚF W¡JML(UaIJRdkJMkcÃfGJML:aIJR‚+Uj`e0JRzIJmjpj`OºSdPzIL•U;cXNJRL(‚ F M L:ctiJbwDU S˜nœlPLkJR“OPJRaIdPjpj`N F tGJMq€“S›tiJR‚+“qJRdmcŠUcXQe%q€cÜWâ_`JMQ|ÃfUjYaIU Qj`e z

53

î ñ+ï.ïhøýhðï AóRò…ó¡û _pj`UaINGL(e§L:c ‚+“ q º

|ψi

∈ m (~r) × m (~s)

_pj`UaINiL(e%L

s=

1 2

ψ (~r, sz )

[Sx , Sy ] = i~Sz 1 S = 2 1 ms = ± 2 s=

1 2

_pj`zIdG_pj`UaINiL(eƒL:c kU Niq€‚

~ 2 |s, mi = ~2 s (s + 1) |s, mi S 3 2 ~ |s, mi = 4 sz |s, mi = m~ |s, mi

|s, mi

_YwDL

_`JRQ |X‚uJR‚Œ“q

tGJm~JRyJMkJM‚Œ“q€k•OPe§F s L#U zHj`J|XyJRJMzIdIe%L›‰ ~ye s cXJRcˆU ye!q  m = 1 2  m = − 1 2

= |↑i = |+i = |↓i = |−i

|+i h+| + |−i h−| = I |χi = C+ |+i + C− |−i

σy σz

~ σx 2 ~ σy 2 ~ σz 2

Sx

=

Sy

=

Sz

=

~ S

=

=



0 1 1 0



1 0 0 −1

= =



_`JRQ |ŠL:cŽJRL(L(wu‚+“q JML‡j`e%Q5zHj`“JMU;aIq

OPe

~ ~σ 2

zHj`“JRUaIq2k 

0 −i i 0

54

vo£ReƒtiJR‚Œ“q2k k€~JMyhJMkUjYaIU Qj`e

h±|±i = δ

σx

1 2





[Sx , Sy ] S±

S± |s, mi = ~

p

= i~Sz = Sx ± iSy

s (s + 1) − m (m ± 1) |s, m ± 1i r

1 2 = ~ |−i r 3 S+ |−i = ~ 4 = ~ |+i

S− |+i = ~

S+ S−

UcXe!w

% 

= ~ = ~

·

3 1 1 + · |−i 2 2 2

+





_YwDL

1 1 · |+i 2 2

_YwDL 0 1 0 0 0 0 1 0





êRíùì uéù hê ü ÷€ðéùøù Dé FV [σ , σ ] = 2iσ Fn σ =σ =σ =I zHj`JMQj`L(yJRaIdPeF E σ σ +σ σ =0 JRL(L:w‹_`Qj`e%‚ σ σ + σ σ = 2δ I k‚ŒNiSTzHj`U |Xyj%zHj`JbaIJMq€U k σ • F C L(S5k‚ŒNiS;k•zHj`U |Xy•zHj`“JRUaIq€kL:w™‚ŒyU qzIeKzGjYcgUj`Q • zHj`k€Omk‘k€dPj`zHd • M

C

x

2 x

x y


{

y

2 y

z

2 z

y x

i j

i j

ij

i

2

   ~ ~σ · B ~ ~σ · A = Sx , Sy , Sz hSx i

=

hSy i

=

hSz i

=

zIL‡yj`zHkƒ¢‡U S›zIe§e%“q


~ 2 
∗ ∗ ~ c+ c− 2 
∗ ∗ ~ c+ c− 2

c∗+ c∗−

  ~ · BI ~ + i~σ · A ~×B ~ A

0 1 1 0

|χi = c+



1 0



+ c−



0 1



_pj`zId

L(JMvU z



0 −i i 0

1 0 0 −1


~ c+ = Re c∗+ c− c− 2  
~ c+ = Im c∗+ c− c− 2    ~ c+ 2 2 = |c+ | − |c− | c− 2 I ¤ G IHG DJ
J €ML

55

-N),O+.-,; 

W s tgj`NGq2‚2f xy UjYcXJRq2‚ x U JR“#tiS φ zIJMjpj`O%U “Dj`JRk.UJM“L|XyJR‚ x

± ~2

_`JMQ |§U e%zIq€k.‚Œ“qštgjYcXUd¢;JRe _YwDL

~s = (sx , sy , sz )

sφ

= Sx cos φ + Sy sin φ  ~ 0 = cos φ + i sin φ 2   ~ 0 e−iφ = eiφ 0 2

cos φ − i sin φ 0



S£bSk‘_YwDL

λ2 − 1 = 0

S£Rj`k2j

λ = ±1

λ = 1 : |χiφ

=

λ = −1 : |χiφ

=

 1 √ 2  1 √ 2

L(JMvU z



1 eiφ

1 =√ 2 

eiφ e−iφ



e−iφ eiφ



F φ U JR“™_pjpj`JbwD‚™‚hj`‚ŒJR|ŠU jYaINgj`k‘L(S›L‡JRSQk€L#_`zHJRd™W¡zIJM‚2fL(JMvU z

F X tGe

|χ+ iz π 2

=

+φ

(s)

(s)

i

~

e− ~ ~s·θ

i

zIJmjpj`OHU “Dj`JRk

 xy

UjYcgJMq€‚‹U JR“x‚ŒJM‚Œ| ‚™tiJR‚Œ‚hj`|Xq π 2

ˆ

i

1 √ 2

=

1 0

= e− ~ ~s·θθ

Rθ~

Rθ~





θˆ = (− sin φ, cos φ, 0) θ =  −iφ

1 eiφ

~

−e 1

  θ θ − i ~σ · θ~ sin 2 2

kq2L:cˆzIwDUSqƒtiJMjpj`kq ~ S ~ L,

i

J~ =

|ψi

OPe

zHj`kOGcXJ

= e− 2 ~σ·θ = cos

h

π 2

Sz , ~r

kU Sk

ý ü ðhý Rê ëuý ü ðý ü ùìêmé Ml

M½o 

Ml

= 0

L‡L‡jYw‹zIJMjpj`OGS;dPzrU JM~vd

~ +S ~ L

ã _`JMQ |ž¼h‚ŒyU q•JM‚Œ“qUj`e%JRzILT|XJM|X‚Œk

= |~r, sz i

56

|XJM|X‚Œk•L:cÁzHj`JML(q€Uj`djYaIUj`e D 0 0 E ~r , sz |~r, sz

 0 = δ 3 ~r, ~r δsz ,s0z

|XJM|X‚Œk•zHj`q€L:c0_Ywuj`q€w zHj`q2L:c X Z

sz =± 12

d3 r |~r, sz i h~r, sz | = 1 |ψi =

X Z

d3~r h~r1 sz |ψi |~r, sz i

sz =± 21

tiJRS j`‚ŒNik

ψ± (~r) = h~r, ±|ψi 2×2 [ψ (~r)] = [ψ (~r)]†

hψ|ϕi

=

hψ|O|ϕi

=

=

Z

Z



k“JRUaIqL:cŽkUj`“‚‹U jYaIU Qj`e!wš~‚+j`S5_`JMQ |Xk

ψ+ (~r) ψ− (~r)

† ψ+



† ψ−




P_YwŒj

% 

†

d3 r [ψ (~r)] [ϕ (~r)] †

d3 r [ψ (~r)] O [ϕ (~r)]

F _pj`UaINiL(eKL‡S›k€~Jm~q‘UjYaIUQj`e O UcXe!w tiJRU JR“k•zIeK‚+‚+j`|Xd

ψ (~r) = h~r|ψi
ψ R−1~r = h~r|Rψi

 = R−1~r|ψ h~r|R|ψi =



0

ψ+ (~r) 0 ψ− (~r)



L(JMvU z

JROPe

D 0E 0 ~r|ψ = ψ (r)

F Jm£RS›_`zIJRd L, S L:cŽ‚+j`‚ŒJM|XktiJRdj`JM“JRUaIqk€U j`“‚

= (Rθ (s))




 ψ+ R−1~r
ψ− R−1~r

F ~r ‚+yhU q2k•L‡S›U JR“k•‚+j`‚ŒJM|Šej`k

R−1

zIe%OHk€U j`“‚ DJ
G IHG DJ
J €ML

^[„

-N),O+.-,;6Ž

ä ËÓ!˖å;Ñ ŒØbß _pj`zHd5F V 

ψ (|x| , t = 0) =

(  c 1− 0

|x| a



|x| ≤ a

|x| > a

Iz e§‚DcXyÛW¡eXf _`q€OP‚ p SdPz}~Djp~q€LTzGj`U‚+zI|Xkk‘zHj`Q JMQ“ÞW¡‚2f t=0 L:cŽzIJRQ “z}¢‡U S W¡v]f p k€JRvU dPe%k•zHJRQ “z}¢‡U S›zIe˜j`‚DcXyÛWâ~f _pj`UzHQ W¡eXf a

1 =

Z

∞

2

|ψ| dx −∞  Z a x x2 2 = 2c 1 − 2 + 2 dx a a 0   1 2 2 = 2c a 1 − 1 + = c2 a 3 3 r 3 c = 2a

φ (p, t = 0) = = = = = = = |φ (p, t = 0)|

2

=

kJRJMU j`Q5zIU q€zIk‘S“‚Œd™W¡‚!f

Z ∞ 1 i ψ (x, 0) e− ~ px dx 2π~ −∞ Z a 1 i √ ψ (x, 0) e− ~ px dx 2π~ −a Z 0  Z a 1 − ~i px − ~i px √ ψ (x, 0) e dx + dx ψ (x, 0) e 2π~ −a 0 Z ∞  Z a i 1 px − ~i px ~ √ ψ (x, 0) e dx + dx ψ (x, 0) e 2π~ 0 0 r Z a  2 px ψ (x, 0) cos dx π~ ~ 0 r r Z a   px x 2 3 cos dx 1− π~ 2a a ~ 0 r    3~3 1 a 1 − cos p πa3 p2 ~
3 sin4 pa 12~ 2~ πa3 p4 √

F JRUaIq2JR|«‚+yhU q2‚uJMv–j`OlPJReKU jYaIU Qj`e p cŽU yhe%q 58

hpi = 0

W¡v]f

‰

OPe ∂ψ ∂x

∂2ψ ∂x2

H=

P2 2m

  0 x < −a   q   1 3 −a ≤ x < 0 a q 2a = 1 3   − a 2a 0 ≤ x < a    0 x≥a r 3 1 = (δ (x + a) − 2δ (x) + δ (x + a)) a 2a

2

2

~ d = − 2m dx2

Wâ~f

OPe

hHi = hψ|H|ψi Z ∞ ∂2 ~2 ψ ∗ 2 ψdx = − 2m −∞ ∂x r Z ∞ 2 ~ 1 3 = −ψ ∗ (δ (x + a) − 2δ (x) + δ (x + a)) dx 2m a 2a −∞ r ~2 1 3 = (−ψ ∗ (−a) + 2ψ ∗ (0) − ψ ∗ (a)) 2m a 2a r r ~2 1 3 3 = 2 2m a 2a 2a 3~2 = 2ma2

RJ zHJMjpj`OISdPz˜tiS›SdINiJRNiL(y»F n √ zIJbcXe%U kqƒNiJRNiL(yk.NiyhU q!j ~ zIe§zIJRdmaIL(q2JR|ˆ~Œjp~q€L›_YwDzIJtie%kW¡eXf L r = ~r F L:cˆJMq€“Sk‚Œ“q€‚™|XQ e%zIq L:cˆzHJRQ “zIkƒ¢;US5JbwxyJRwŒj`kW¡‚!f tGJML‡~v‚™zHj`e~Djpj%LJMeƒL:cŽzIJRL(q2JRdPJRq2kk€L(QwDq€k‘zILe§kOI‚Œ“q2‚™‚DcXy hzi = a _pj`zHd‹W¡v]f     0 −i 0 1 0 0 I z § e % e  “ q zHj`dj`zIdšWâ~f L L =  i 0 i ,L =  0 0 0  0 −i 0 0 0 −1 _pj`UzHQ Ÿj`L‡JRy.|XyJ%Ngjp~‚Œd™W¡eXf ~ L

2

x

∆y∆Lx

z

y

y

~ 2

z

~ 2

j`L:c kJR“Nidj`Q›L:w™tGe§JRQj`L(JMy.tivHkJMkJ ~r tGeƒJMQj`L(JRytGe§L(‚Œe

[Lx , r]

= [ypz − zpy , r]

2

  ypz − zpy , x2 + y 2 + z 2

= 2i~yz − 2i~zy = 2i~ [y, z] = 0

Sjp~JuW¡‚!f

Lx |Xi = m~ |Xi [Lz , Lx ] = i~Ly i~ hX| Ly |Xi = hX| Lz Lx − Lx Lz |Xi = m~ hX| Lz |Xi − hX| Lz |Xi = 0

59

zHj`S ~Dj`k2lPJMe}_pj`UNiJRSŽW¡v]f ∆y∆Lx

1 |h| [y, Lx ] |i| 2

≥

‚ŒcXyhdi_YwDL

[y, Lx ] = [y, ypz − zpy ] = −z [y, py ] = −i~z

Wâ~f NGJMNGL(yk•L:cˆSdPz P UcXe2w W = vP tiv–j V = mω x _pj`zHd5F E ‚DcXyÛW¡eXf E E  JP£bSx_pjYcge%UxU ~D|X‚ tiJR‚Œ“q2k.zIe§‚DcXyÛW¡‚!f E ψ JRdmcŠU~|Xq‘k€JRvU dPe%L#_pj`NiJRzHkzIe˜j`‚DcXyÛW¡v]f zINiJMjp~q‘ke%“Dj`z  vHJRwuyJbwŒj`k€LŠWâ~f _pj`UzHQ OPe W¡eXf Lx =

1 i~

[Ly , Lz ]

1 2

2 2

%Š%

(1) n

(1) n

0 n

0

∆En

√





E (1) E n cnk

 v Ek0 |p|En0

tiv–j

1 A − A† p = P = √ 2i m~ω r
m~ω A − A† ⇒ p = −i 2

En0 |p|En0

En0 − Ek0

= hϕn |W |ϕn i

 = v En0 |p|En0

 En0 |αA + βA† |En0



 0 0 = α ˜ En0 |En−1 + β˜ En0 |En+1 = 0

vo£be%kq!j



=

_pjYcXe!UuU~|XL#ky|j`dPkW¡‚!f

 X (1) 0  = En0 + cnk Ek k6=n

=

v

Ek0 |p|En0 En0 − Ek0





= ~ω (n − k) r  m~ω 0 Ek |A − A† |En0 = iv 2 r
m~ω √ 0 0  √ n Ek |En−1 − n + 1 hEk |En+1 i = iv 2 r √
m~ω √ nδk,n−1 − n + 1δk,n+1 = iv 2 60

OPe

DK
G IHG DJ
J €ML

-N),O+.-,;˜;

OPe En(1)

=

∆En(2)

En(0)

=

+ iv

r

0 
m √ 0  √ n En−1 + n + 1 En+1 2ω~

kJRvU dPe%LT_pj`NGJMzIkW¡v]f



 X v 2 Ek0 |p|En0 En0 − Ek0

k6=n

=

X v 2 m~ω nδk,n−1 − (n + 1) δk,n+1 2 En0 − Ek0

k6=n

= =

H

X v 2 (n + 1) X v 2 m~ω n 2 δ − δk,n+1 k,n−1 E 0 − Ek0 En0 − Ek0 k6=n k6=n n   v2 m v 2 m~ω n n+1 m =− − = − v2 2 ~ω ~ω 2 2

tGJMS~Dj`Jj`dPe Wâ~f

= H0 + W 1 p2 + vp + mω 2 x2 = 2m 2 2 (p + mv) 1 m = + mω 2 x2 − v 2 2m 2 2

JMdPj`q2U kUjYaIL(JM|j`e!qK‚DwDUj`qƒ_`e%dMaHj`L‡JRq€k€k•U q!j`L:w

[x, p] = i~ ⇒ [x, p + mv] = i~


tiJRe%zIdIk€L U q2j`L:wšF kOPOPkq‘‚ŒwDU j`q!j

En = ~ω n +

En

_pj`UaINGL‡e!kckL;j`NGL‡j`q2k«‚Œ“q

1 2

  1 m = ~ω n + − v2 2 2

Ucge!wÜF tikJMdm‚ƒÙ;Q NGq€cÜUjYaINiL(ej›F tGJMq2jYaIeˆE›tGeˆkL‡j`NiL‡j`qŠk€dPj`zHd5F s F tGJM‚Œ“q2k|XJM|X‚Œ‚ukL‡j`NiL‡j`q€kƒ_`e%JRdjYaIL(JMq€kF i = 1, 2, 3 tijYaIe%L#UjYcgN

|ii |ii

H

= ε

3 X

|ii hi| + δ

X

|ii hj|

it JRdj`zId ε, δ U;cXe!w k€L‡j`NGL;j`q€L›~Djp~q2LTUcgQ e§zHj`JRvU dPe§j`L‡JRe W¡eXf j`L(L‡k‘tGJM‚Œ“q€k•zIe˜~Djp~q€LŽW¡‚!f zHj`djYcXk‘zHj`JMvU dme%kzIe˜~Djp~q€LTtiJRq!jYaIe%k~Dyhe%LTUjYcXN«ej`kU;cXe!w™zHj`U ‚ŒzI|XkkÚW¡v]f F U yeKtgjYaIe%L UjYcXN«kJMkJ t SvU ‚DcˆzHj`U ‚ŒzI|XkkÚWâ~f _pj`UzHQ i=1

61

i6=j

k“JMU;aIq€w H ‚+j`zIwDLŠW¡eXf 

ε =  δ δ

H



det 

ε − Ek δ δ

δ ε δ

 δ δ  ε

OPe



δ ε − Ek δ

δ  = 0 δ ε − Ek ε − E1 = δ E1 = ε − δ

Ome

 δ δ  = x3 − 2δx + 2δ 3 x



x δ det  δ x δ δ

= (x − δ) x2 + δx − 2δ 2 2

= (x − δ) (x + 2δ) = 0

x=ε−X




_`q€|XdItie

OPe

= E2 = ε − δ = ε + 2δ

E1 E3

tGJMdPjpj`dPq2k•tiJRq2“S;k•tiJR‚Œ“q2kW¡‚!f E1,2

  r   1 1 1  1  −2  √ 0 , 6 2 1 −1

=

E3

=



1 1 √  1  3 1

zHj`JRvU dPe%kqƒ~yeKL(w™~Djp~Dq€L P zGj`U‚+zI|Xkk k

|hpk |1i| 

 |ψ1 i  |ψ2 i  = |ψ2 i

  

2

|ψ (t = 0)i = |1i

= |h1|pk i|

√1 2 √1 6 √1 3

0 − √26 √1 3

P1

=

P2

=

P3

=

1 2 1 6 1 3

62

Uj`‚ŒS



2

√1 2 √1 6 √1 3

L UjYcgNˆ_pj`U;aINiL(e%kW¡v]f UcXQ eKtiv–j



 |1i  |2i   |3i

JROPe

t=0 |1i =

|ψ (t)i = =

1 1 1 √ |ψ1 i + √ |ψ2 i + √ |ψ3 i 2 6 3

_YwDL

i

i

i

e − ~ E1 t , e − ~ E2 t , e − ~ E3 t

‚Œ“qL:wDLTOPe

i i i 1 1 1 √ e− ~ E1 t |ψ1 i + √ e− ~ E2 t |ψ2 i + √ e− ~ E3 t |ψ3 i 2 6 3 1 − i E1,2 t 1 i e ~ (2 |1i − |2i + |3i) + e− ~ E3 t (|1i + |2i + |3i) 3 3

2

P2 (t) = |h2|ψ (t)i| 2 1 −iE t 1 − i E3 t 1,2 ~ ~ (2 |1i − |2i + |3i) + e (|1i + |2i + |3i) = h2| e 3 3 2 1 i 1 i = − e− ~ E1,2 t + e− ~ E3 t 3 3     E3 − E 1 4 3δ 1 2 − 2 cos t = sin2 t = 9 ~ 9 2~

†E

‚™‚Œ“q2kWâ~f

_YwDL