„h = 1; me

(1)

9

:#/!#;'))/:*<+,#;&=> :?=%@=AB&=# C'))/D&

EFHG<IKJ$L:M<NOLPNOQ<RSLDT M$UVMXWZY\[]T ^:Y`_KQbacYdTe^:YKI`Qgf&YKI`M<hWZY`Q`ijI`^!LPYeijkcMml

noqprts`uogv\ogwtvxy1oqz{y vrts |Ox}Xvtz{sdy~`x}~`}gx|owvrx1}gvxrtomw\gom}gvrtsdyz x1omw|sdS~`z{y1o xz~|o

x{vz{pomw~`p1p{z{}~`vzsdy1wogy}z{&zoogywt}gzogy1}momw|omwVS~`vtgrz~`xOogv|1~`y1womwVy~`y1s`wt}gzogy1}momw

,P1 Z ¡O¢:£¤

¥y~d|sdpvto,z}gz¦o,wt§ewvt¨go,|D©xyz{vtmw~`vtsd&zHx1omwsdªvtogy xogyz{&p!sw~`y v

¯

h = 1, m e = 1, e = 1, 1 4π² ₀ = 1,

sd«

¯ h = 1.054 × 10 ⁻³⁴

^¬ w|mwtz{®y1o,~}msdy1wv~`y vto$|o°¯±~`y}²brtm|Oxz{vto

m _e = 9.11 × 10 ⁻³¹

²e®&~

S~dwtwto,|o,c©gom}gvrtsdy¦

e = 1.602 × 10 ⁻¹⁹

n³~&}~`r®´o°ggogy v~`z{rto ogv

² 0 = 8.854 × 10 ⁻¹²

^µ±¶

~8p!ogr&z{vvz{·ez{vt$|Ozgom}gvrz x1o$|Oxq·ez|o

¥y¸}msdy1wz|¨grto&xy¹w§ewtvt¨go.|o&|ogxOgom}gvrtsdy1w°psdy®´mw$|1~`y1w,xyp!sdvtogy vzogoXevtgrzogxr

V

^Dogv°sdy

}1ogrt}1o.ºS|gvtogr&z{y1ogr,wtsdygv~`v]wz{y®xogvg»]|o$p{x1wª~dwtwo$gy1ogr®zoDnogv]gv~`vomwv|m}grz{vp~`rxy1o

¼½sdy1}gvzsdy|D©sdy1|o

ψ ₀ (x ₁ , x ₂ ) ∈ L ² (IR ³ × IR ³ )

º8·~`{ogxrwVrtmog{omw ·Hgrz¿¾*~`y v

ψ ₀ (x ₂ , x ₁ ) = ψ ₀ (x ₂ , x ₁ )

^ogv

Z

IR ³ ×IR ³ | ψ ₀ (x ₁ , x ₂ ) | ² dx ₁ dx ₂ = 1.

^ÀÂÁÄÃ

¥yÅsdªvzogy v

ψ 0

ogvwtsdyÅgy1ogr®zo

E 0

ogyÆ}1ogrt}~`y vos |obprtsdprto|op{x1w&ª~dwtwtobgy1ogr®zoq|o

c©mHx~`vzsdy|o Çe}rtÈ |Oz{y®´ogr

Hψ = Eψ,

^ÀÊÉ´Ã

sd«

H = − 1

2 ∆ _x 1 − 1

2 ∆ _x 2 + V (x ₁ ) + V (x ₂ ) + 1

| x ₁ − x ₂ |

|mwz{®y1o$c©~`&z{{vtsdyzomy|OxËw§ewtvt¨gosdxªzogy¦!|o°¼Ê~dÌmsdyËm xz{·d~`ogy vto!ogyrtmwtsd{·~`y vo$prtsdª¨go

|o]&z{yz{&zw<~`vzsdy

E ₀ = inf ⁿ h ψ, Hψ i , ψ ∈ H ¹ (IR ³ × IR ³ )

·Hgrz¿¾*~`y v ÀÂÁÄÃ

o

ÀÊÍ´Ã

sd«

h ψ, Hψ i = 1 2 Z

IR ³ ×IR ³ |∇ ψ(x ₁ , x ₂ ) | ² dx ₁ dx ₂ +

Z

IR ³ ×IR ³

(V (x ₁ ) + V (x ₂ ))ψ(x ₁ , x ₂ ) ² dx ₁ dx ₂ +

Z

IR ³ ×IR ³

ψ(x ₁ , x ₂ ) ²

| x 1 − x 2 | dx ₁ dx ₂ .

Î omw|ogxOqoXOog&pomw|1o,w§Owvt¨gomwº.|ogxOqgom}gvrtsdy1w x1o]y1sdx1w~`{sdy1w}msdy1wz|grtogrwtsdy v

ÏcÐÒÑÂÓ¦ÔÕÖÕcÓ¦ÓÊ×ÒØ`Ù<Ú`ÐÒÑÕcÓPÓÊÛ<ØmÕPÐjÑÓPÔÕÖtÕcÓÜÄÑ)ÓÊÝK×ÒØÕcÛÕÖ<Ð ÔÂÙXÖ<ÐeÞßÂÔàcÛ

(2)

Á

c©Ò~`vtsdo$|D©ág{z{x

À

xyy1sÄ§ ~`x|o°}~`r®´o

2

^Ã p!sdxromHx1og

V (x) = V He (x) = − 2

| x | ;

^{À£â Ã}

É

~.sdm}gxo$|D© §O|Ortsd®´¨gy1o

À

|ogxOby1sÄ§ ~`xOq|o,}~`r®´o

1

^Ã p!sdxr~dHx1og{o

V (x) = V _H ^a 2 (x) = − 1

| x − ^a ₂ e _z | − 1

| x + ^a ₂ e _z | ,

^ÀÊã´Ã

e z

|mwz{®y~`y vo]·Hom}gvtogxr|Oz{rtom}gvtogxr|o,c©Ò~KOo

Oz

^ogv

a

xyrtmog¦p!swz{vz¿¼

¥yp!ogxvsdy vrtogrHx1op!sdxr

V

|sdyy1p~`r À£â Ã

sdxp~`r

ÀÊã´Ã

eoprtsdª¨go

ÀÊÍ´Ã

~d|OogvVoX1~d}gvtogogy v

|ogxOwtsd{xvzsdy1w

ψ ₀

^ogv

− ψ ₀

^c©xy1o$|o$}momw|ogxOb¼Êsdy1}gvzsdy1w

À

p~`roXog&po

ψ ₀

^Ã gv~`y vwvrz}gvtogogy v p!swz{vz{·Ho,wxr

IR ³ × IR ³

ä 1y¦©omwvåp~dw±vrt¨mw±¼Ê~d}gz{o|oVrtmwtsdx|Orto

ÀÊÍ´Ã

|Oz{rtom}gvtogogy v}~`r}goprtsdª¨googvogy,uogxS|omw)¼Êsdy1}gvzsdy1w

|X¾1yzomw wtxr

IR ³ × IR ³

^Î ©Ò~`pprtsOz{S~`vzsdy|oSá]~`rvrtomoX

µ

s }²¸}msdy1wzwvto\º~`pprts }1ogr8~wtsd{xvzsdy

|o

ÀÊÍ´Ã

ogy\rtomwvroXz{®y~`yevc©ogy1wtog ªo]|omwå¼Êsdy1}gvzsdy1w

ψ

wxromwt x1ogwsdy\&z{yz{&zwto]º$c©ogy1wtog ªo|omw

¼½sdy1}gvzsdy1w xz¦w©m}grz{·Hogy vwtsdx1wV~8¼½sdro

ψ(x ₁ , x ₂ ) = φ(x ₁ )φ(x ₂ ), φ ∈ H ¹ (IR ³ ), Z

IR ³

φ(x) ² dx = 1.

^ÀÊæ´Ã

ç&èéHêëÄìZí!îðïñDò sdy vrtogrHx1o,o]prtsdª¨go$|o°á]~`rvrtomoX

µ

s }²qwÄ©m}grz{v

E ₀ ^HF = inf

½

E ^HF (φ), φ ∈ H ¹ (IR ³ ), Z

IR ³ φ(x) ² = 1

¾

ÀcóÃ

~·´om}

E ^HF (φ) = A ₁ Z

IR ³ |∇ φ | ² + A ₂ Z

IR ³ V φ ² + A ₃ Z

IR ³

Z

IR ³

φ(x) ² φ(y) ²

| x − y | dx dy.

¯)rtm}gzwtogr)omw·~`ogxrtw=|omw)}msdy1wv~`y vtomw

A 1

A 2

ogv

A 3

ogv=oXOp{zHx1ogr=p!sdxrt x1sdzsdy~vtsdxdusdxrtw

E ₀ ^HF ≥ E ₀

^ò sdy vrtogr°Hxo8p!sdxr

V

|sdyy1&p~`r À£â Ã

sdxp~`r

ÀÊã´Ã

:~\¼½sdy1}gvzsdyy1og{o

E ^HF

omwv]ªzogy¸|X¾1yzo wxr

H ¹ (IR ³ )

ô<õ*öd÷Êøtù´ú½÷½ûdõü=ûõýOú½÷£þÿ÷ùOûý ø Xþ{ù\þ÷£õ´ùdþ÷£ú8ö)ùdýOø!øåùgúåþ÷£õ´ùdþ÷£ú8ö°ùöbü

∀ u ∈ H ¹ (IR ³ ), Z

IR ³

u(x) ²

| x | ² dx ≤ 4 Z

IR ³ |∇ u(x) | ² dx.

¥yp!ogxvsdy vrtogrHx1op!sdxr

V

|sdyy1p~`r À£â Ã

sdxp~`r

ÀÊã´Ã

eoprtsdª¨go

ÀcóÃ

~d|OogvVoX1~d}gvtogogy v

|ogxOwsd{xvzsdy1w

φ ∗

ogv

− φ ∗

OogvV x¦©xy1o]|o]}momwV|ogxO\¼Êsdy1}gvzsdy1wVomwvVwvrtz{}gvtogogy vp!swz{vz{·Ho]wxr

IR ³

ç&èéHêëÄìZí!î! :ñ n~`}gxogr,~|Oz#"Dgrtogy vzog{o&|o8~S¼Êsdy1}gvzsdyy1og{o

E ^HF

oXv]sdy vrtogr$Hx1o8c©mHx~`vzsdy

|D©%$)xogrÂ Î ~d®r~`y1®´o8~dwtwts }gzmo$~`xqprtsdª¨go

ÀcóÃ

p!ogxvw©m}grz{rto

 

 

 

 



n1ogrt}1ogr

(², φ) ∈ IR × H ¹ (IR ³ )

vtogPHx1o

− 1

2 ∆φ + W φ = ²φ Z

IR ³

φ ² = 1 W (x) = V (x) +

Z

IR ³

φ(y) ²

| x − y | dy.

À'&´Ã

(3)

Î o]prtsdª¨go

À'&´Ã

omwvxyqprtsdª¨go$~`xO·~`ogxrtwprtsdp1rtomw õ*ûdõþÿ÷£õtùd÷(*)

φ

p!ogxv,+gvrto,}msdy1wz|grtmo }msd&o±xy1o)¼½sdy1}gvzsdy$prsdprtoå|o)c©sdp!gr~`vtogxr=~`xvts`~d|`usdz{y v

− ¹ ₂ ∆+W

mo±p!sdvtogy vzog

W

|gp!ogy1|1~`y v

{xz¿c+go|o

φ

^.-y s |¨go&|o&}mo.v§ p!o&w©Ò~`pp!og{o&ogyp §Owz x1o8xy s |¨go|o&}~`&psÄ§Hogy

~`xvts`}msd1grtogy v/-yrtmwx{v~`v&z{&p!sdrv~`y vomwtv. x1obwz

φ

omwv.xy&z{yz{&zwtogxrS|o ÀcóÃ

)~`sdrtw

²

^omwtv

ogy ¼c~`z{v]~ *þÿý*0mú½÷£ú ·~`ogxr,prtsdprto|o.c©sdp!gr~`vtogxr

− ¹ ₂ ∆ + W

Pnogvvto&rtogS~`rt x1o}msdy1|Oxz{v$|o

¼Ê~dÌmsdy¹y~`vxrtog{o\ºbc©Ò~`{®´sdrz{vo\|op!sdz¿yev,¾owxz{·~`y v ) sdy¹wto&|sdyyo

φ 0 ∈ H ¹ (IR ³ )

·Hgrz¿¾*~`y v

R

IR ³ φ ² ₀ = 1

ogvsdy}msdy1wtvrxz{v~&wxz{vto

(φ _n ) _n∈IN

|X¾1yzo°p~`r

 

 

 

 



n1ogrt}1ogr

(² _n+1 , φ _n+1 ) ∈ IR × H ¹ (IR ³ )

vtog¦ x1o

− 1

2 ∆φ _n+1 + W _n φ _n+1 = ² _n+1 φ _n+1

² _n+1

omwvV~.p{x1wVp!ogvz{vto°·d~`ogxrprtsdprto$|o

− ¹ ₂ ∆ + W _n φ n+1 > 0

Z

IR ³

φ ² _n+1 = 1 W _n (x) = V (x) +

Z

IR ³

φ _n (y) ²

| x − y | dy.

À'1´Ã

Î ~wxz{vto

(φ _n ) _n∈IN

omwtv)~`z{y1wzO|X¾1yzo|oS~`yz¨grtoxyz x1o$)y.o"DogvKov1msdrt¨go ó Í 1

|Ox&}msdxrtw32

Á4

w©gvtogy1| ~`xO¸sdp!gr~`vtogxrtw

− ¹ ₂ ∆ + W _n

⁾ ~bp{x1w$p!ogvz{vto·~`oXx1r°prtsdprto\omwv°y1sdyO|g®´gy1grtmoogv°o

·´om}gvtogxrp1rtsdprto$~dwtwts }gz,p!ogxv5+gvrto°}1sdzwzPwvrz}gvtogogy vp!swz{vz¿¼)wxr

IR ³

nsd&o8y1sdx1w]o ·Hogrrtsdy1w!}mogv°~`{®´sdrz{vo.y1o.}msdy ·´ogr®´o.p~dwvtsdxdusdxrtw

À

sdz{y|o8º36

Ã

S~`z{w,wx87v

p!sdxrvr~`z{vtogromw|ogxOq}~dw xzDy1sdx1wz{y vtgrtomwtwtogy v

g:9!';5<P¡O¢DZ¤Æ>=¸?ÿ¡e¢D@;A=B?#C 1 <D;FEÄ¡HGÅ1H;5=¤D¢D JIO£¤¦Z LKNM

¥y}1ogrt}1o$º&}msdy1wvrxz{rto°xy1o,wsd{xvzsdyy xgrz x1o°|Oxprtsdª¨go

ÀcóÃ

p!sdxrc©Ò~`vtsdo$|D©ág{z{x

ç&èéHêëÄìZí!îPODñ

¬

x1wtvz¿¾ogro¼Ê~`z{vHx1ortmwtsdx1|Orto

À'&´Ã

~·Hom}~$}msdy vr~`z{yevto,Q

²

omwv~°p{x1wp!ogvz{vto·~`ogxr prtsdprto°|o

− ¹ ₂ ∆ + W

^R rtog· zogy vº.rtmwtsdx1|Orto

 

 



 

 

n1ogrt}1ogr

(², u, f ) ∈ IR × H ₀ ¹ (]0, + ∞ [) × H _loc ¹ (]0, + ∞ [)

vtog¦ x1o

− 1

2 u ⁰⁰ (r) − 2 − f (r)

r u(r) = ²u(r) Z _+∞

0 u ² = 1

²

omwvV~.p{x1wp!ogvz{vto,·~`ogxrVprtsdprto$|o

− ¹ ₂ _dr ^d ² ² − ^2−f _r ^(r)

− f ⁰⁰ (r) = u(r) ²

r , f(0) = 0, lim

r→+∞ f (r) = 1.

ÀÂÁTS´Ã

¥y p!ogxv]sdy vtrtogr°Hx1o

ÀÂÁTS´Ã

p!swtw¨m|o8oX1~d}gvtogogy v°|ogxO wtsd{xvzsdy1w

(² _∗ , u _∗ , f _∗ )

^ogv

(² _∗ , − u _∗ , f _∗ )

Hx1oc©xy1o]|omw¼Êsdy1}gvzsdy1w

u ∗

ogv

− u ∗

omwvwvrz}gvtogogy vVp!swz{vz{·´oOogv x1oogw¼½sdy1}gvzsdy1w

u ∗

ogv

f ∗ − 1

|m}grtsdzwtwtogy v&oXep!sdy1ogy vzog{ogSogy v&· z{vto\·Hogrtw

0

^=¥y p!ogxv&|sdy1}omwp!grtogr.sdª1vtogyz{r8xy1ovrt¨mw ª!sdyy1o

~`pprtsez{S~`vzsdy|omwVwtsd{xvzsdy1wV|o

ÀÂÁTS´Ã

ogybvrtsdy1Hx~`y vVc©z{y vtogr·~`{o

]0, + ∞ [

^O¯sdxr

R > 0

^¾Osdy

}msdy1wz|¨grto,o,prtsdª¨go

 

 



 

 

n1ogrt}1ogr

(², u, f) ∈ IR × H ₀ ¹ (]0, R[) × H ¹ (]0, R[)

^vtogPHxo

− 1

2 u ⁰⁰ (r) − 2 − f (r)

r u(r) = ²u(r) Z _R

0

u ² = 1

²

omwvV~.p{x1wVp!ogvz{vto°·d~`ogxrprtsdprto$|o

− ¹ ₂ _dr ^d ² ² − ^2−f(r) _r

− f ⁰⁰ (r) = u(r) ²

r , f (0) = 0, f (R) = 1.

(4)

¥yð}msdy1wz|¨grto x1y S~`z{{~`®´oxyz¿¼Êsdro|o c©z{y vtogr·~`{o

[0, R]

p~`r|omwwtog®ogy vtw|o sdy®x1ogxr

∆r = R/(N + 1)

^~·Hom}

N ∈ IN ^∗

D¥yy1sdvto

r _j = j ∆r

^À

0 ≤ j ≤ N + 1

^Ã ^!ogv

V _N ⊂ H ₀ ¹ (]0, R[)

c©omwp~d}mo°|D©Ò~`pprtsOz{S~`vzsdyp~`rgXogyevtwV¾1yzw

IP 1

~dwtwts }gz$º.}mo,S~`z{{~`®´o )

V _N = ⁿ u ∈ H ₀ ¹ (]0, R[), u

~U7y1o°wxr}*~d}Xx1y|omwz{y vtogr·d~`{omw

[r _j , r _j+1 ] ^o .

¥yÆy1sdvtoogyO¾1y

(φ _i ) _1≤i≤N

^~ ª~dwto}~`y1sdyz x1o|o

V _N

^À

φ _i ∈ V _N

^ogv

φ _i (r _j ) = δ _ij

p!sdxrSvtsdxv

1 ≤ i, j ≤ N

^Ã ^ogv

φ _N+1

~\¼½sdy1}gvzsdy¹}msdy vz{y x1o~U7y1o&p~`r]sdrt}mo~`xOwxr°}~d}gxy¸|omw

[r _j , r _j+1 ]

vtog{o°Hx1o

φ _N ₊₁ (r _j ) = δ _N _+1,j

p!sdxrvtsdxv

0 ≤ j ≤ N + 1

*¥y|sdyy1o )

Z R 0

φ ² ₁ (r)

r dr = − 2+4 ln 2, Z R

0

φ ² _i (r)

r dr = − 2i+(i+1) ² ln µ

1 + 1 i

¶

− (i − 1) ² ln µ

1 − 1 i

¶

Z _R

0

φ ³ ₁ (r)

r dr = − 5+8 ln 2, Z _R

0

φ ³ _i (r)

r dr = − 5i+(i+1) ³ ln µ

1 + 1 i

¶

+(i − 1) ³ ln µ

1 − 1 i

¶

Z _R

0

φ i (r)φ i+1 (r)

r dr = i + 1

2 − i(i + 1) ln µ

1 + 1 i

¶

Z _R

0

φ _i (r) ² φ _i+1 (r)

r dr = 1 3 + 3i

2 + i ² − i(i + 1) ² ln µ

1 + 1 i

¶

Z _R

0

φ _i (r)φ _i+1 (r) ²

r dr = 1 6 − i

2 − i ² + i ² (i + 1) ln µ

1 + 1 i

¶

ç&èéHêëÄìZí!îWV¦ñ Çesdz{v

v ∈ H ₀ ¹ (]0, R[)

X]grz¿¾ogrHx1oV~¼Êsdy1}gvzsdy

v(r) ² r

omwvå}msdy vz{y x1owx1r

[0, R]

^´¯sdxr

v ∈ V _N

prtsdp!swtogrxy1o,gv1s |o$|o,rtmwtsd{xvzsdyp~`rggogy vtw¾1yzw|Oxqprtsdª¨go

 



n1ogrt}1ogr

g ∈ H ¹ (]0, R[)

vtogPHx1o

− g ⁰⁰ (r) = v(r) ²

r , g(0) = 0, g(R) = 1

xvz{{zw~`y vc©omwp~d}mo

V _N

0$±}grz{rto°o,prtsd®r~`&o UY:Z´¡HZ }msdrrtomwp!sdy1|1~`y v

ç&èéHêëÄìZí!î\[Dñ Çesdz{v

g ∈ φ _N+1 + V _N

^À^zco

g

|o]~8¼½sdo

φ _N+1 + g _e

^~·´om}

_e g ∈ V _N

^Ã ¯)rsdp!swtogrxy1o

gv1s |o$|o]rtmwsd{xvzsdyp~`rggogy vtw¾1yzw|Oxprtsdª¨go

 

 



 

 

n1ogrt}1ogr

(λ, v) ∈ IR × H ₀ ¹ (]0, R[)

vtogP x1o

− 1

2 v ⁰⁰ (r) − 2 − g(r)

r v(r) = λv(r) Z _R

0

v ² = 1

λ

omwvV~8p1¿xwVp!ogvz{vto,·~`ogxrprtsdprto°|1o

− ¹ ₂ _dr ^d ² ² − ^2−g(r) _r

xvz{{zw~`y vc©omwp~d}mo

V N

0$±}grz{rto°o,prtsd®r~`&o UY:Z´¡HZ }msdrrtomwp!sdy1|1~`y v

ç&èéHêëÄìZí!î^]¦ñ

$±}grz{rtoxySprtsd®r~`&o

_Y:Z´¡HZ

p!ogrogvv~`y vV|ortmwsdx1|Orto

ÀcóÃ

p!sdxrc©Ò~`vtsdo|D©áX

{z{x"ogvVprtmwtogy vtogrVomwVrtmwx{v~`vtwVy xgrz x1omw}msdrrtomwp!sdy1|1~`yevtw

À

·d~`ogxr|o

E ₀ ^HF

O·d~`ogxr|Oxq x¿

vz{p{z}~`vtogxr|o Î ~`®r~`y®´o

²

prts`¾1¦|Ox&z{yz{&zwtogxrogvV|Oxp!sdvtogy vzog

W

zwtvtsdrzHx1o]|o~ }msdy ·HogrÂ

®´ogy1}mo¦{{

Ã

(5)

2

¥yprtsdp!swto,S~`z{y vtogy~`y v|D©Ò~`pp1rts }1ogro,prtsdª¨go

ÀcóÃ p~`r

inf

½

E ^HF (φ), φ ∈ V _h , Z

IR ³

φ(x) ² = 1

¾

ÀÂÁÁÄÃ

sd«

V _h

omwvDxy$wtsdx1wÂomwp*~d}go±|o±|Oz{ogy1wzsdy°¾1yzo

N

^|o

H ¹ (IR ³ )

Ä¥y$wto)|sdyy1o±xy1o=ª~dwo

(χ ₁ , · · · , χ _N )

|o

V _h

ogvsdyqy1sdvto

S

^ogv

h

omwVS~`vrz}momw}~`rrtmomww§egvrz x1omw|o,v~`z{{o

N × N

|X¾1yzomwp~`r

S ij = Z

IR ³ χ i χ j

ogv

h ij = 1 2 Z

IR ³ ∇ χ i · ∇ χ j + Z

IR ³ V χ i χ j

ogv

A

o,vtogy1wtogxrº. x~`vrto,z{y1|Oz}momw|X¾1yzPp~`r

A _ijkl = Z

IR ³

Z

IR ³

χ _i (x) χ _j (x) χ _k (y) χ _l (y)

| x − y | dx dy.

ç&èéHêëÄìZí!îih*ñDò sdy vrtogrHx1o,o]prtsdª¨go

ÀÂÁÁÄÃ

inf ⁿ E(C), C ∈ IR ^N , C ^T SC = 1 ^o

^ÀÂÁÉ´Ã

sd«

C ^T

|mwz{®y1ooV·Hom}gvtogxrå¿z{®y1ovr~`y1wtp!swt|OxS·´om}gvtogxr}msdsdyy1o

C

^ogvsd«

E (C)

omwv±xy1oV¼½sdy1}gvzsdy

|o

C

h

^ogv

A

ÀÂÁÉ´Ã

 



 

n1ogrt}1ogr

(², C) ∈ IR × IR ^N

vtogPHx1o

F (C) C = ²SC C ^T SC = 1

sd«

F (C)

omwvxy1oS~`vrz}mo}~`rrtmow§egvrzHx1o|ov<~`z¿{o

N × N

|o

C

h

^ogv

A

l

~`y1w~SS~Kusdrz{vt.|omw]}~`}gxw:sdy xvz{{zwto8p!sdxr,}msdy1wvrxz{rto8~\ª~dwto

(χ 1 , · · · , χ N )

^y1sdy ^p~dwxy1o

wxz{·~`y v )

Á

nm }~d x1o±Xgogy v}z{&z x1o/m |Ox,v~`ªo~`x$p!grzs |OzHx1oÄsdy ~dwtwts }gzo)xy1o±}msd{og}gvzsdy

n ξ _µ ^A ^o

1≤µ≤n A

|o

n _A

¼Êsdy1}gvzsdy1w8|o

H ¹ (IR ³ )

{z{y1~`z{rtogogy v.z{y1|gp!ogy1|1~`y vtow ) }mo\wtsdy v omw8sdrªz{v~`omw&~`vts`

&z x1omw

À

¥,m

Ã

É

V¯=sdxro"Dom}gvx1ogrxy}~`}gxDwxrxybw§ewtvt¨gosdm}gx~`z{rto,|sdyy1esdyb}msdy1wvrxz{v~$ª~dwto

{ χ _i }

ogy$rtog®rtsdxp~`y vvtsdxvtomw¦omw¥,m roX~`vz{·Homwºvsdx1w¦omw~`vtsdomwP|Ox$wt§ewv¨XoTmz{y1wz´p~`rPoXOog&po

}gz¿|omwwx1wsdyprtogy1|

{ χ _i } =

½

ξ ₁ ^H (x − a

2 e _z ), · · · , ξ _n ^H _H (x − a

2 e _z ); ξ ₁ ^H (x + a

2 e _z ), · · · , ξ _n ^H _H (x + a 2 e _z )

¾ .

¥y~&|sdy1}$~`z{y1wz¦p!sdxr}mogvoXog&po

N = 2n _H

¼Êsdy1}gvzsdy1w|o,ª~dwto

n ξ ^A _µ ^o

1≤µ≤n A

p!sdxr\vtsdx1womw

(6)

xy1o]~`pprtsez{S~`vzsdyqvrt¨mwprtm}gzwto|Ox\¼Êsdy1|1~`ogy v~`Dá~`rvrtomoX

µ

s }²Sp!sdxrxywt§ewvt¨gosdm}gx~`z{rto

Hx1og}msdy1 x1o

z{y1|Oz}momw

A _ijkl

À

sdx|omw

prts |Oxz{vtwå|op!sd{§eybqdomwåogv±|oV®H~`x1wtwzogyy1omw

Ã

omw)vtogromwå|Oxvtogy1wtogxr

A _ijkl

p!ogx·Hogy ve+gvrto}~`}Xx1{mw

S

^ogv

h

Pnsdy1wz|grtsdy1w ogy¹o":ogv

Hx~`vrtomw¥,m>®H~`x1wtwzogyy1omw

χ 1 (x) = e ^−α ¹ ^|x−¯ ^x ¹ ^| ² , χ 2 (x) = e ^−α ² ^|x−¯ ^x ² ^| ² , χ 3 (x) = e ^−α ³ ^|x−¯ ^x ³ ^| ² , χ 4 (x) = e ^−α ⁴ ^|x−¯ ^x ⁴ ^| ² .

¯swtsdy1w

R _ij = | x ¯ _i − x ¯ _j |

β _ij = _α ^α ⁱ ^α ^j

i +α j

ogv

γ _ij = α _i + α _j

^ogv

y ¯ _ij = ^α ⁱ ^x _α ^¯ ⁱ ^+α ^j ^x ^¯ ^j

i +α j

¦¥yËp!ogxv]·Hgrz¿¾ogr

À

S~`zwsdyqy1o°|ogS~`y1|o°p~dw|o,o]¼Ê~`z{rto

Ã

Hxo

S ij = Z

IR ³ χ i χ j = Ã π

γ _ij

! 3/2

e ^−β ^ij ^R ² ^ij , 1 2

Z

IR ³ ∇ χ i ∇ χ j = β ij (3 − 2β ij R ² _ij ) S ij , Z

IR ³

χ _i (x)χ _j (x)

| x − x ¯ _k | = S _ij

| y ¯ _ij − x ¯ _k |

ogrÂ¼

¡ √ γ _ij | y ¯ _ij − x ¯ _k | ^¢

sdrtwt x1o

¯

y _ij − x ¯ _k 6 = 0

^ogv

A _ijkl = Z

IR ³

Z

IR ³

χ _i (x)χ _j (x)χ _k (x ⁰ )χ _l (x ⁰ )

| x − x ⁰ | dx dx ⁰ = S _ij S _kl

| y ¯ ij − y ¯ _kl |

ogrÂ¼

Ãs γ _ij γ _kl

γ ij + γ _kl | y ¯ _ij − y ¯ _kl |

!

sdrtwt x1o

¯

y _ij − y ¯ _kl 6 = 0

1sd«q~8¼½sdy1}gvzsdyogrÂ¼=omwv|X¾1yzo,p~`r

ogrÂ¼

(x) = 2

√ π Z _x

0

e ^−t ² dt.

2

omwvxy1o ¼Êsdy1}gvzsdy|o8~\|Ozwtv~`y1}mo8z{y vtogr~`vts`

&zHx1o

a

|sdyy1mo1p!sdxro]s |¨go$|o°á~`rvrtomoX

µ

s }²:p~`r

E (a) = E ^HF (a) + 1 a

sd«

E ^HF (a)

omwtv$o\&z{yz{ x |o ÀcóÃ

p!sdxr

V = V _H ^a ₂

H$±}grz{rtoxy prtsd®r~`&o 9tY:£¡DZ · zw<~`y v º\vr~d}mogr,~}msdxrª!o

a 7−→ E (a)

n _H = 5

ξ ^H ₁ (x) = e ^−33.8650 ^|x| ² , ξ ^H ₂ (x) = e −5.09479|x| ² , ξ ₃ ^H (x) = e −1.15879|x| ² , ξ ₄ ^H (x) = e ^−0.32584 ^|x| ²

^ogv

ξ ^H ₅ (x) = e ^−0.102741 ^|x| ² .

À'1´Ã

}msdy ·´ogr®´oVp!sdxr

a

~dwwtouVp!ogvz{vS~`zw)y1oV}msdy ·Hogr®´oVp~dw)p!sdxr

a

^~dwtwou

À'1´Ã

ogyËz{&p!sw~`yev

Hx1o,omw

φ _n

Oz

ç&èéHêëÄìZí!î³ïïyz|0~HèDZëT|OëÄì!é0Äñ om}1ogrt}1ogry xgrz x1ogogy v~°·d~`ogxr|o~$|Ozwv~`y1}moz{y vtogr~`vts`

E (a)

~`vvtogz{y vwtsdyb&z{yz{ x*nsd&p~`rtogrº ~$·~`ogxrVoXOp!grz{ogyO

v~`o

a exp = 1.4015

0.529 × 10 ⁻¹⁰

|1~`y1w,o&w§Owvt¨go&|omw,xyz{vtmw

~`vtsd&zHx1omw

Ã

(7)

ç&èéHêëÄìZí!îÆï8 dyz|0~ èHZëT|OëÄì*ébñ $±wvz{ogrå~]{z{&z{vto|o

E (a)

sdrtwt x1o

a

vtogy1|&·Hogrtw

+ ∞

^ÀsdySp!sdxrr~

æ´Ã

n~`}gxogr

2

sdrtwt x1o

a

^vtogy1| ^·´ogrtw

+ ∞

À

ÉU4

p~`roXOog&po

Ã w

µ

s }²b·´sdx1wp~`r~UvÂog{o ~d|1~`pvtmo ~`x}~`}gx|omwgy1ogr®zomw|o°|Ozwwts`

}gz~`vzsdyAw

!@fc@H@

2

Á4

bm{~`z{rtot

õ*ùdþ°õ1ýJ<÷'gýbmúû !ú½÷(.÷mùú½÷½ûõ

8$±}msdo°¯=sd{§ vtom}yz x1o$±|Oz{vzsdy

ÉNSNSÍ

2

ÉU4

¬

o~`yO Î sdx1zwÂV~dwt|og·~`y vogv

¬

o~`y l ~`{z{ª~`rt|DB tøùdõ1÷'gýbgýùdõú½÷'gý0 f$±}msdob¯sd{§evtom}yz x1o

$±|Oz{vzsdy

ÉNSNSÉ

„h = 1; me

¯

h = 1, m e = 1, e = 1, 1 4π² 0 = 1,

¯ h = 1.054 × 10 −34

m e = 9.11 × 10 −31

e = 1.602 × 10 −19

² 0 = 8.854 × 10 −12

V

ψ 0 (x 1 , x 2 ) ∈ L 2 (IR 3 × IR 3 )

ψ 0 (x 2 , x 1 ) = ψ 0 (x 2 , x 1 )

Z

IR 3 ×IR 3 | ψ 0 (x 1 , x 2 ) | 2 dx 1 dx 2 = 1.

ψ 0

E 0

Hψ = Eψ,

H = − 1

2 ∆ x 1 − 1

2 ∆ x 2 + V (x 1 ) + V (x 2 ) + 1

| x 1 − x 2 |

E 0 = inf n h ψ, Hψ i , ψ ∈ H 1 (IR 3 × IR 3 )

o

h ψ, Hψ i = 1 2 Z

IR 3 ×IR 3 |∇ ψ(x 1 , x 2 ) | 2 dx 1 dx 2 +

Z

IR 3 ×IR 3

(V (x 1 ) + V (x 2 ))ψ(x 1 , x 2 ) 2 dx 1 dx 2 +

Z

IR 3 ×IR 3

ψ(x 1 , x 2 ) 2

| x 1 − x 2 | dx 1 dx 2 .

2

V (x) = V He (x) = − 2

| x | ;

1

V (x) = V H a 2 (x) = − 1

| x − a 2 e z | − 1

| x + a 2 e z | ,

e z

Oz

a

V

ψ 0

− ψ 0

ψ 0

IR 3 × IR 3

IR 3 × IR 3

ψ

ψ(x 1 , x 2 ) = φ(x 1 )φ(x 2 ), φ ∈ H 1 (IR 3 ), Z

IR 3

φ(x) 2 dx = 1.

E 0 HF = inf

½

E HF (φ), φ ∈ H 1 (IR 3 ), Z

IR 3 φ(x) 2 = 1

¾

E HF (φ) = A 1 Z

IR 3 |∇ φ | 2 + A 2 Z

IR 3 V φ 2 + A 3 Z

IR 3

Z

IR 3

φ(x) 2 φ(y) 2

| x − y | dx dy.

A 1

A 2

A 3

E 0 HF ≥ E 0

V

E HF

H 1 (IR 3 )

∀ u ∈ H 1 (IR 3 ), Z

IR 3

u(x) 2

| x | 2 dx ≤ 4 Z

IR 3 |∇ u(x) | 2 dx.

V

φ ∗

− φ ∗

IR 3

E HF

h = 1, m e = 1, e = 1, 1 4π² ₀ = 1,

¯ h = 1.054 × 10 ⁻³⁴

m _e = 9.11 × 10 ⁻³¹

e = 1.602 × 10 ⁻¹⁹

² 0 = 8.854 × 10 ⁻¹²

ψ ₀ (x ₁ , x ₂ ) ∈ L ² (IR ³ × IR ³ )

ψ ₀ (x ₂ , x ₁ ) = ψ ₀ (x ₂ , x ₁ )

IR ³ ×IR ³ | ψ ₀ (x ₁ , x ₂ ) | ² dx ₁ dx ₂ = 1.

2 ∆ _x 1 − 1

2 ∆ _x 2 + V (x ₁ ) + V (x ₂ ) + 1

| x ₁ − x ₂ |

E ₀ = inf ⁿ h ψ, Hψ i , ψ ∈ H ¹ (IR ³ × IR ³ )

IR ³ ×IR ³ |∇ ψ(x ₁ , x ₂ ) | ² dx ₁ dx ₂ +

IR ³ ×IR ³

(V (x ₁ ) + V (x ₂ ))ψ(x ₁ , x ₂ ) ² dx ₁ dx ₂ +

IR ³ ×IR ³

ψ(x ₁ , x ₂ ) ²

| x 1 − x 2 | dx ₁ dx ₂ .

V (x) = V _H ^a 2 (x) = − 1

| x − ^a ₂ e _z | − 1

| x + ^a ₂ e _z | ,

ψ ₀

− ψ ₀

ψ ₀

IR ³ × IR ³

IR ³ × IR ³

ψ(x ₁ , x ₂ ) = φ(x ₁ )φ(x ₂ ), φ ∈ H ¹ (IR ³ ), Z

IR ³

φ(x) ² dx = 1.

E ₀ ^HF = inf

E ^HF (φ), φ ∈ H ¹ (IR ³ ), Z

IR ³ φ(x) ² = 1

E ^HF (φ) = A ₁ Z

IR ³ |∇ φ | ² + A ₂ Z

IR ³ V φ ² + A ₃ Z

IR ³

IR ³

φ(x) ² φ(y) ²

E ₀ ^HF ≥ E ₀

E ^HF

H ¹ (IR ³ )

∀ u ∈ H ¹ (IR ³ ), Z

IR ³

u(x) ²

| x | ² dx ≤ 4 Z

IR ³ |∇ u(x) | ² dx.

IR ³

E ^HF

(², φ) ∈ IR × H ¹ (IR ³ )

IR ³

φ ² = 1 W (x) = V (x) +

IR ³

φ(y) ²

− ¹ ₂ ∆+W

− ¹ ₂ ∆ + W

φ 0 ∈ H ¹ (IR ³ )

IR ³ φ ² ₀ = 1

(φ _n ) _n∈IN

(² _n+1 , φ _n+1 ) ∈ IR × H ¹ (IR ³ )

2 ∆φ _n+1 + W _n φ _n+1 = ² _n+1 φ _n+1

² _n+1

− ¹ ₂ ∆ + W _n φ n+1 > 0

IR ³

φ ² _n+1 = 1 W _n (x) = V (x) +

IR ³

φ _n (y) ²

(φ _n ) _n∈IN

− ¹ ₂ ∆ + W _n

IR ³

− ¹ ₂ ∆ + W