[PDF] Chapitre 4 Particule de spin 1/2

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jsi=aj+zi+bjzi 2 Hspin; a;b2C??????? ???? ?????? ??? ???? ?? ????jsi=aj+zi+bjzi? ?? ??????? ?? ???? ????? ?????z? P js>= cos2 j+z>+sin2 j z>??????? js>=^Ry()j+zi= cos2 j+zi+ sin2 j zi

Ry()jzi=sin2

j+zi+ cos2 j zi h

Ry()(????jz>)cos2

sin2 sin 2 cos2 exp i~ ^Sy dd ^Ry() =0=i~ ^Sy? ???? ?????

Sy=i~dd

^Ry() =0=~2 y; y:=0i i0|+ >z z|s >

θ|s >θ

|- >z|+ > zθθ/2 xz 3 2

Ry(4)j+z>=j+z>

Ry(2) =^??;^Ry(4) =^??:

z|+ >-

Spin Flipper

^Ry()????? ? ???? ?

8;^Ry() = expi~

^Sy

8;^Rx() = expi~

^Sx ^Rz() = expi~ djs>d i^Sy~ js>;????js>=^Ry()j+z> ?? ?? ???? ???? ??? ????x;z?

Sx;y;z(????jz>)~2

x;y;z; x=0 1 1 0 y=0i i0 z=1 0 01

Sxj+xi= +~2

j+xi;^Sxjxi= ~2 j xi;??? ????y;z ^Sx?????? ??? ? j+xi=1p2 (j+zi+jzi);jxi=1p2 (j+zi jzi) ^Sy? j+yi=1p2 (j+zi+ijzi);jyi=1p2 ~S:=^Sx;^Sy;^Sz Sy=i~ d^Ry()d =0=i~12 sin2 12 cos2 12 cos2 12 sin2 =0=~2 0i i0 =~2 y ?????z? ????xR ( ) xy|s >R ( ) yz z )^Ry()j+z>? ????? ^Rx()6=^Rz(2 )^Ry()? ? ??????? ??? ??????j+y>??

Sx=^Rz(2

)^Sy^Rz(+2 Rz(2 ) = exp i^Sz(=2)~ =ei=40 0ei=4 Sx=~2 ei=40

0ei=40i

i0 ei=40

0e+i=4

=~2 0 1 js;'>=^Rz(')^Ry()j+z> js;'>=ei'=2cos(=2)j+z>+ei'=2sin(=2)jz>

Ry()j+z>= cos2

j+zi+ sin2 j zi

Rz(')jzi= exp

i'^Sz~ j zi= exp (i)'2 j zi ~S2?

S2=~234

^I 24
(3)??? ~S2j >=j >??? ??????? ?????? ??j 0>=^Rj >? ?????~S2j 0>=~S2^Rj >= js >=aj+z>+bjz>2 Hspin; a;b2C js >? ~s=D^~SE ~s=8 >:s x= s y= s z= k~sk=~2 z=ab =????2 e i'2C:??b6= 0??????? sys z s z

Re(z)?

θSphère de la direction du spin

Plan complexe z=a/b(rayon 1/2)

2R3?? ??? ??????

θ/2 θ/2s

1/2 z/2 z

xz z z ????^Rz(')? j i=aj+zi+bjzi ???k~sk=~2 ?????? ?? ???????~s= (sx;sy;sz)2R3? [ ] =fj 0>;???? ???92C;j 0>=j >g: ^Rx()

Rx()^Ry()6=^Ry()^Rx()

?????? ??? ???h^Rx();Ry()i =^Rx()^Ry()^Ry()^Rx()6= 0? (Ry())1(Rx())1Ry()Rx() =I~

2h^Sy;^Sxi

+o(2;2;) yz R 1 R 2x Ry R R

1=Rx(2

)Ry(2 )6=R2=Ry(2 )Rx(2 )RxR yR x R y-1 -1 ^Rx() = exp i^Sx=~ = exp(A)'1 +A+12

A2+o(2)

^Rx()

1= exp

i^Sx()=~ = exp(A)'1A+12

A2+o(2)

^Ry()? ????? R yRx=

1 +B+12

B2

1 +A+12

A2 = 1 +A+B+BA+12 A2+12

B2+:::

R

1yR1x= 1AB+BA+12

A2+12

B2+:::

R

1yR1xRyRx= 1A+AB+BA2ABBAB2+BA+BA+A2+B2+o(2;2;)

= 1 +BAAB+:::= 1 + [B;A] +::: =I~

2h^Sy;^Sxi

h^Sx;^Syi =^Sx^Sy^Sy^Sx=i~^Sz??????? h ^Sy;^Szi =i~^Sx; h^Sz;^Sxi =i~^Sy

Sx^Sy^Sy^Sx~2

20 1 1 0 0i i0 ~2 20i i0 0 1 1 0 ~2 2 2i1 0 01 i~^Sz R

1y()R1x()Ry()Rx() =I+i~

^Sz+o(2;2;) =Rz() +o(2;2;) R ?????? ?? ?????z? R

S~u=~u:^~S=ux^Sx+uy^Sy+uz^Sz~2

(~u:~)??????? ???? ?? ?????2R??

R~u() = exp

i^S~u~ = 0???? Tr^Sx =Tr^Sy =Tr^Sz e

SU(2)?

S~U=Ux^Sx+Uy^Sy+Uz^Sz=~U^:~S2 R;~U= (Ux;Uy;Uz)2R3

1i h^S~U;^S~Vi

2 R???? ????~U;~V2R3?

1i~h^S~U;^S~Vi

=^S~U^~V

R~U= exp

i~ ^S~U = exp i~ ^S~u ^R~u() ^R~u(0) =Id:??????? ??????^R~u() M ~u() =h+j^R+i h+j^Ri hj ^R+i hj^Ri

SU(2) =M2Mat2(C)=det(M) = 1; M+=M1

R x;y;z??????? ??? ???? ?? ????? ???????su(2)? R H espace=L2R3 H spinC2 tot=Hespace j >=ajx1>j+z>+bjx2>jz>??????? j z>? ????? ???? H tot=Hespace Hspin ??? ?????? ???j~x;+>=j~x > j+z>??j~x;>=j~x > +(~x) =< ~x;+j > (~x) =< ~x;j > ~x

Hspin?

H (jn >+jp >)

00 = 0;10 = 1;01 = 1;11 = 0:

???????=??????? ??y??????? x zy y y|+ >? B x:j+xi 0;jxi 1 B y:j+yi 0;jyi 1 B xB xB yB xB yB yB B xB yB xB xB yB xB ???? ???j+xi?? ??Pjyi= 1=2? ?? ???? ?????? ??????? ??? ?B xB xB yB j i jRegistrei jEve0i ! j i j i jEve i j+xi jRegistrei jEve0i ! j+xi j+xi jEve+i j xi jRegistrei jEve0i ! jxi jxi jEvei (j+xi+jxi) j+yi jRegistrei jEve0i !1p2 (j+xi j+xi jEve+i+jxi jxi jEvei) j+yi jEve+i?? H tot=Hespace Hspin H=12m ~:^~pq~A(^~x;t)

2+qU(^~x;t)???????

H=12m^~pq~A(^~x;t)

2~Me:~B+qU(^~x;t)???????

Me=qm I=q =qv2r ~S=r2:~u?~u??? ?? ??????? ??????

L=~p^~r=m~v^~r=mvr~u:

Mclass=I:~S=qv2rr2:~u=qv2

r:~u=q2m~L

H=12m^~pq~A(^~x;t)

2~M:~B+qU(^~x;t)

M=g~S~

g n?? ^Hespace=12m^~pq~A(^~x;t)

2+qU(^~x;t)?

??????j (t)i=j 0;espacei H spin??????? ??? ? ^Hspin=g~ ~S:~B ??? ? ???? ??????? ???????E=12 ~!?!=g~ i~djs(t)>dt =^Hjs(t)> ~s(t) =D^~SE =< s(t)j^~Sjs(t)> ?????k~s(t)k=~2 d~s(t)dt =dHspind~s ^~s= g~ ~B ^~s??????? ??Hspin=g~ !=g~ ~B ~B? p! e=gpNg eB'5000 gB~ js(0)>=a(0)j+z>+b(0)jz> js(t)>= exp i^Ht=~ js(0)>=a(t)j+z>+b(t)jz> ????a(t) =a(0)eiEt=~?b(t) =b(0)e+iEt=~? ??E=gB2 z(t) =a(t)b(t)=z(0)e2iEt=~=z(0)ei!t ????'(t) =!t+'(0)??(t) =(0) =cste?!=2jEj~ =g~ ~B? ^S~u??????? ?????? ?? ?????~u??? ??? ??? ?????~B? ?? q= cos p=s:' dqdt =@Hspin@p dpdt =@Hspin@q = 1105?

B=B0~ez+B1cos(!1t)~ex

s(t)B

B?????? ??g?

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