Le couplage de deux spins ½

!"#$%&'()*"#+"#+"&,#-'./-#0#

!)#1)."#2#34#$5#+"#(6)7%5"#+689+1%*:/"#

!"-#$%11;()7.%/-#<./-7"./=>%+%(-?9=@%-"/#

A8)'.71"#4BC#')1)*1)'8"-#4#"7#B#

A8)'.71"#4DC#')1)*1)'8"#4# E/#F)#-6./7;1"--"1#.$.#2#############################################G#"-')$"#+"#+.5"/-.%/#D##

4H#

!6)++.7.%/#+"#+"&,#-'./-#0##

;("$71%/#I#'1%7%/#################################)7%5"#+689+1%*:/"#

/"&71%/#I#'1%7%/#################################/%9)&#+"#+"&7;1.&5#

J9-7:5"#$%5'%-;#G#

/%7)7.%/#-.5'(.K.;"#G#{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

!6%';1)7"&1#-'./#7%7)(#

AL"-7#&/"#%M-"1F)M("#+"#5%5"/7#$./;7.N&"#G#

O%/71%/-#N&"#("-#1;-&(7)7-#'%--.M("-#(%1-#+6&/"#5"-&1"#+"#!!####"7#!!!!###-%/7#

)F"$#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! (2)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

!L)$7.%/#+"#######-&1#("-###

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz (2)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2){|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

!"-#;7)7-##############-%/7#')1#$%/-71&$7.%/#;7)7-#'1%'1"-#+"##########"7#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

P(-#-%/7#+%/$#;7)7-#'1%'1"-#+"#()#-%55"#############################G#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz (3)

Q1%.-#F)("&1-#'1%'1"-#'%--.M("-#G##

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h (3)

)F"$##

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h (3)

Sz|+; +! = h |+; +! Sz|#;#! = h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

E/#F%.7#M."/#)'')1)R71"#&/#5%5"/7#$./;7.N&"#"/7."1C#+"#F)("&1#S#%&#4T#

!L)$7.%/#+"#######-&1#("-###

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)E/#+;F"(%''"#("#$)11;#G#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

4H#U$7.%/#+"########"7########-&1#("-##############G#K)$.("#V#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)!"-##############-%/7#;7)7-#'1%'1"-#+"########"7########'%&1#()#F)("&1#'1%'1"##

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 (7)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)

3H#U$7.%/#+"##############-&1#("-##############G#&/#'"&#'(&-#+.KK.$.("T#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)

"Sa · "Sb = SaxSbx + SaySby + SazSbz (9)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)

!L)$7.%/#+"#######-&1#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)

#SaxSbx + SaySby + SazSbz

$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

E/#&7.(.-"#G#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

=h2

4|+; +! (16)

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

=h2

4|+; +! (16)

+L%W#K./)("5"/7#G#

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

=h2

4|+; +! (16)

S2|+; +! =3h2

2|+; +! + 2

h2

4|+; +! = 2h2 |+; +! (17)

AL"-7#M."/#+"#()#K%15"#G##

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

=h2

4|+; +! (16)

S2|+; +! =3h2

2|+; +! + 2

h2

4|+; +! = 2h2 |+; +! (17)

S2|+; +! = h2 s(s + 1) |+; +! (18))F"$##

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

=h2

4|+; +! (16)

S2|+; +! =3h2

2|+; +! + 2

h2

4|+; +! = 2h2 |+; +! (17)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

<5"1*"/$"#+L&/#-'./#4#S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! (20)

X.()/#+"#$"#N&L%/#F."/7#+"#71%&F"1#'%&1#############G##

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! (20)

>%&1#############C#%/#5%/71"#+"#5Y5"#N&"#G#

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! (20)

|";"! (21)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

A%55"/7#71%&F"1#("#71%.-.:5"#5"5M1"#+"#()#K)5.(("############################Z#

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! (23)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

!L;7)7############################"-7#'1%'%17.%//"(#2##

"7##################################C#+%/$##

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

<5"1*"/$"#+L&/#-'./#S#

[)/-#(L"-')$"#+"#+.5"/-.%/#D#"/*"/+1;#')1#("-#F"$7"&1-##################C##{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

=h2

4|+; +! (16)

S2|+; +! =3h2

2|+; +! + 2

h2

4|+; +! = 2h2 |+; +! (17)

%/#)#.+"/7.K.;#&/#-%&-="-')$"#+"#+.5"/-.%/#B#$%11"-'%/+)/7#)&,#"#$#%!#&'()*#%!

{|a : ±! " |b : ±!} = {|!a;!b!} !a = ± , !b = ± (1)

Sz|!a;!b! = (Saz + Sbz)|!a;!b! =!

h!a

2+

h!b

2

"|!a;!b! (2)

Sz = Saz + Sbz h, 0, #h S2 (3)

Sz|+; +! = +h |+; +! Sz|#;#! = #h |#;#! Sz|+;#! = 0 Sz|#; +! = 0 (4)

S2 =#

"Sa + "Sb

$2

(5)

= S2a + S2

b + 2 "Sa · "Sb (6)

34

h2 |+; +! (7)

#S2

a + S2b

$|!a;!b! =

32

h2 |!a;!b! (8)


Sax|a : ±! =h

2|a : $! (10)

Say|a : ±! = ±ih

2|a : $! (11)

Saz|a : ±! = ± h

2|a : ±! (12)


$|+; +! =

!h

2

" !h

2

"|#;#! +

!ih

2

" !ih

2

"|#;#! +

!h

2

" !h

2

"|+; +! (13)

S2|+; +! =#S2

a + S2b + 2 "Sa · "Sb

$|+; +! (14)

=3h2

2|+; +! + 2 "Sa · "Sb|+; +! (15)

=h2

4|+; +! (16)

S2|+; +! =3h2

2|+; +! + 2

h2

4|+; +! = 2h2 |+; +! (17)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

\&"#'"/-"1#+&#F"$7"&1#%178%*%/)(#2#$"#-%&-="-')$"#G#

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

1$2

(|+;"! " |"; +!) (26)Z#

E/#F;1.K."1)#N&"##

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

1$2

(|+;"! " |"; +!) (26)

S2 (|+;"! " |"; +!) = 0 (27)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

1$2

(|+;"! " |"; +!) (26)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (27)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

"#$#!%'+,-)*#!

X.()/#+"#$"77"#;7&+"#

<-')$"#+"#+.5"/-.%/#D#G#

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

2 % 2 = 3 + 1 (29)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

2 % 2 = 3 & 1 (29)

-'./#4]3#-'./#4]3# 71.'("7#

^-'./#7%7)(#4_#

-./*&("7#

^-'./#7%7)(#S_#

J.#%/#)F).7#)`%&7;#+"&,#-'./-#4#^8%1-#'1%*1)55"_#G#

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

2 % 2 = 3 & 1 (29)

3 % 3 = 5 & 3 & 1 (30)

-'./#4#-'./#4# N&./7&'("7#

^-'./#7%7)(#3_#

71.'("7#

^-'./#7%7)(4_#

-./*&("7#

^-'./#7%7)(#S_#

3H#

!)#-71&$7&1"#89'"1K./"#+"#(6)7%5"#+689+1%*:/"#

^/.F")&#K%/+)5"/7)(_#

!6./7"1)$7.%/#5)*/;7.N&"#;("$71%/#=#'1%7%/#

@)''"(#+"#5)*/;7%=-7)7.N&"#G#

E1+1"-#+"#*1)/+"&1#G#

./7"1)$7.%/#5)*/;7.N&"###aa###./7"1)$7.%/#$%&(%5M."//"#

;$)17#79'.N&"##

"/71"#/.F")&,##

+6;/"1*."#4-#b#3-#

!"#/.F")&#K%/+)5"/7)(#+"#(6)7%5"#+689+1%*:/"#

["#(6"-')$"#+"-#;7)7-#7%7)(C#%/#/"#*)1+"#N&"#("#/.F")&#K%/+)5"/7)(#G#

;7)7#(.;#K%/+)5"/7)(#^4-_#

[;*;/;1"-$"/$"#D#'%&1#("#/.F")&#K%/+)5"/7)(#

<-')$"#+"-#;7)7-#'%&1#(6;("$71%/#G##

'%&1#("#'1%7%/#G#

U$7.%/#+"#######-&1#("#/.F")&#K%/+)5"/7)(#

E/#+%.7#71%&F"1#("-#;7)7-#'1%'1"-#+"#()#5)71.$"##

!"-#F)("&1-#'1%'1"-#$%11"-'%/+)/7"-#+%//"1%/7#("-#+;'()$"5"/7-#

+"-#-%&-=/.F")&,#+6;/"1*."#.--&-#+&#/.F")&#K%/+)5"/7)(#

U'1:-#$)($&(#+"#()#')17."#%1M.7)("C#%/#-"#1)5:/"#2#()#+.)*%/)(.-)7.%/##

+"#(6%';1)7"&1#+"#-'./#G#

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

2 % 2 = 3 & 1 (29)

3 % 3 = 5 & 3 & 1 (30)

H1 = "23

µ0

!a31

"µe · "µp (31)

A =163' 2.79

me

mp#2 EI (32)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

2 % 2 = 3 & 1 (29)

3 % 3 = 5 & 3 & 1 (30)

A =163' 2.79

me

mp!2 EI (31)

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

2 % 2 = 3 & 1 (29)

3 % 3 = 5 & 3 & 1 (30)

H1 = "23

µ0

!a31

"µe · "µp =A

h2"Se · "Sp (31)

A =163' 2.79

me

mp#2 EI (32)

)F"$#G#

S2|+; +! = h2 s(s + 1) |+; +! s = 1 (18)

S2|+; +! = 2h2 |+; +! Sz|+; +! = h |+; +! (19)

|s = 1, m = 1! = |+; +! |s = 1, m = "1! = |";"! |s = 1, m = 0! (20)

|";"! (21)

S2|";"! = 2h2 |";"! Sz|";"! = "h |";"! (22)

S!|s = 1, m = 1! S! = Sa! + Sb! (23)

S!|+; +! # |+;"! + |"; +! (24)

|s = 1, m = 0! =1$2

(|+;"! + |"; +!) (25)

|s = 0, m = 0! =1$2

(|+;"! " |"; +!) (26)

1$2

(|+;"! " |"; +!) (27)

S2 (|+;"! " |"; +!) = 0 Sz (|+;"! " |"; +!) = 0 (28)

2 % 2 = 3 & 1 (29)

3 % 3 = 5 & 3 & 1 (30)

H1 = "23

µ0

!a31

"µe · "µp =A

h2"Se · "Sp (31)

A =163' 2.79

me

mp#2 EI ( 5.87 10!6 eV (32)

!68)5.(7%/."/#+"#-71&$7&1"#89'"1K./"#

[.)*%/)(.-)7.%/#+"##

EM-"1F)7.%/#+"#()#1)."#89'"1K./"#

71.'("7#

-./*&("7#

/.F")&#4-#

[&1;"#+"#F."#+&##

/.F")&#71.'("7#G#4Sc#)/-#

<,';1."/$"-#+"#1;-%/)/$"#"/71"#/.F")&,#71.'("7#"7#-./*&("7#

.$%*&!/!012&3,4+*#G#

U#')17.1#+"-#-%&1$"-#)-71%'89-.N&"-#G#

#######%M-"1F)7.%/#+"-#'8%7%/-#;5.-#')1#;5.--.%/#-'%/7)/;"#2#

BH#

!"-#$%11;()7.%/-#<./-7"./#b#>%+%(-?9#b#@%-"/#

"7#(L./;*)(.7;#+"#X"((#

!"-#'1%'1.;7;-#;7%//)/7"-#+"-#;7)7-#./71.N&;-#$%55"#(6;7)7#-./*&("7#G#

!6./+;7"15./.-5"#+"#()#5;$)/.N&"#N&)/7.N&"#

P/+;7"15./.-5"#;N&.F)("/7#2#$"(&.#

+6&/#7.1)*"#2#'.("#%&#K)$"#Z#

@;'%/-"#N&)/7.N&"#G#d%/H##

A"7#./+;7"15./.-5"#"-7#./71./-:N&"##

"7#/"#1;-&(7"#')-#+6&/"#5)&F).-"###

$%//).--)/$"#+"-#$%/+.7.%/-#./.7.)("-##

%&#+&#5%&F"5"/7#&(7;1."&1#

<./-7"./#G#e[."&#/"#`%&"#')-#)&,#+;-e#

f/"#78;%1."#+;7"15./.-7"#+%//)/7#("-#5Y5"-#1;-&(7)7-##

N&"##()#5;$)/.N&"#N&)/7.N&"#'"&7="(("#",.-7"1#Z#

O).-#"-7=%/#-g1#N&"#()#5;$)/.N&"#N&)/7.N&"#-%.7#()#78;%1."#&(7.5"#Z#

\&"#-"1).7#&/"#78;%1."#+;7"15./.-7"##

1"'1%+&.-)/7#("-#1;-&(7)7-#N&)/7.N&"-#Z#

!L;7)7#+"#$8)N&"#)7%5"#./$.+"/7#

-"1).7#$)1)$7;1.-;#')1#&/#')1)5:71"#

##^F)1.)M("#$)$8;"_#

./)$$"--.M("#)&#hi'89-.$."/#N&)/7.N&"ij#

!)#$%//).--)/$"#+"######'"15"771).7#+"#+;7"15./"1#)F"$#$"17.7&+"#("#1;-&(7)7#+"##

()#5"-&1"#+"#!!!!!!!')1#(L./7"15;+.).1"#+L&/"#K%/$7.%/!

h#;(;5"/7#+"#1;)(.7;#j#

Az(!) = +1 (i.e. + h/2) (1)

Az(!) = !1 (i.e. ! h/2) (2)

{|a : ±" # |b : ±"} = {|"a;"b"} "a = ± , "b = ± (3)

Sz|"a;"b" = (Saz + Sbz)|"a;"b" =!

h"a

2+

h"b

2

"|"a;"b" (4)

Sz = Saz + Sbz h, 0, !h S2 (5)

Sz|+; +" = +h |+; +" Sz|!;!" = !h |!;!" Sz|+;!" = 0 Sz|!; +" = 0 (6)

S2 =#

#Sa + #Sb

$2

(7)

= S2a + S2

b + 2 #Sa · #Sb (8)

34

h2 |+; +" (9)

#S2

a + S2b

$|"a;"b" =

32

h2 |"a;"b" (10)

#Sa · #Sb = SaxSbx + SaySby + SazSbz (11)

Sax|a : ±" =h

2|a : $" (12)

Say|a : ±" = ±ih

2|a : $" (13)

Saz|a : ±" = ± h

2|a : ±" (14)


$|+; +" =

!h

2

" !h

2

"|!;!" +

!ih

2

" !ih

2

"|!;!" +

!h

2

" !h

2

"|+; +" (15)

S2|+; +" =#S2

a + S2b + 2 #Sa · #Sb

$|+; +" (16)

=3h2

2|+; +" + 2 #Sa · #Sb|+; +" (17)

Az(!) = +1 (i.e. + h/2) (1)

Az(!) = !1 (i.e. ! h/2) (2)

{|a : ±" # |b : ±"} = {|"a;"b"} "a = ± , "b = ± (3)

Sz|"a;"b" = (Saz + Sbz)|"a;"b" =!

h"a

2+

h"b

2

"|"a;"b" (4)

Sz = Saz + Sbz h, 0, !h S2 (5)

Sz|+; +" = +h |+; +" Sz|!;!" = !h |!;!" Sz|+;!" = 0 Sz|!; +" = 0 (6)

S2 =#

#Sa + #Sb

$2

(7)

= S2a + S2

b + 2 #Sa · #Sb (8)

34

h2 |+; +" (9)

#S2

a + S2b

$|"a;"b" =

32

h2 |"a;"b" (10)

#Sa · #Sb = SaxSbx + SaySby + SazSbz (11)

Sax|a : ±" =h

2|a : $" (12)

Say|a : ±" = ±ih

2|a : $" (13)

Saz|a : ±" = ± h

2|a : ±" (14)


$|+; +" =

!h

2

" !h

2

"|!;!" +

!ih

2

" !ih

2

"|!;!" +

!h

2

" !h

2

"|+; +" (15)

S2|+; +" =#S2

a + S2b + 2 #Sa · #Sb

$|+; +" (16)

=3h2

2|+; +" + 2 #Sa · #Sb|+; +" (17)

f/"#78;%1."#+;7"15./.-7"#;N&.F)("/7"##

2#()#OH\H#'"&7#"(("#",.-7"1#Z#

k!#>%&1#+"-#",';1."/$"-#$%5'%17)/7#&/"#-"&("#5"-&1"#^Sz_#-&1####&/"#')17.$&("#&/.N&"#^-'./#0_C#%/#/"#'"&7#')-#$%/$(&1"H#

k!#>%&1#+"-#",';1."/$"-#$%/-.-7)/7#2#5"-&1"1#+"-#$%11;()7.%/-##

##"/71"#%M-"1F)M("-C#()#1;'%/-"#"-7#dEd#G#

78;%1:5"#+"#X"((H##

O"-&1"#+"-#$%11;()7.%/-#"/71"#U(.$"#"7#X"1/)1+#

U(.$"# X"1/)1+#

"=# 'I#

'1%+&.7#+"-#+"&,#1;-&(7)7-#'%&1#$8)N&"#').1"#G#

K%/$7.%/#+"#$%11;()7.%/#G#

O"-&1"#()##

$%5'%-)/7"##

+"#######-"(%/###

O"-&1"#()##

$%5'%-)/7"##

+"#######-"(%/###

p = ±1 (i.e. ± h2/4) (1)

E(!a,!b) =!

A(",!a) B(",!b) P (") d" (2)

Az(") = +1 (i.e. + h/2) (3)

Az(") = !1 (i.e. ! h/2) (4)

{|a : ±" # |b : ±"} = {|#a;#b"} #a = ± , #b = ± (5)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (6)

Sz = Saz + Sbz h, 0, !h S2 (7)

Sz|+; +" = +h |+; +" Sz|!;!" = !h |!;!" Sz|+;!" = 0 Sz|!; +" = 0 (8)

S2 =$

!Sa + !Sb

%2

(9)

= S2a + S2

b + 2 !Sa · !Sb (10)

34

h2 |+; +" (11)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (12)

!Sa · !Sb = SaxSbx + SaySby + SazSbz (13)

Sax|a : ±" =h

2|a : $" (14)

Say|a : ±" = ±ih

2|a : $" (15)

Saz|a : ±" = ± h

2|a : ±" (16)

$SaxSbx + SaySby + SazSbz

%|+; +" =

"h

2

# "h

2

#|!;!" +

"ih

2

# "ih

2

#|!;!" +

"h

2

# "h

2

#|+; +" (17)

E(!a,!b) = !p" (1)

p = ±1 (i.e. ± h2/4) (2)

E(!a,!b) =!

A(",!a) B(",!b) P (") d" (3)

Az(") = +1 (i.e. + h/2) (4)

Az(") = #1 (i.e. # h/2) (5)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (6)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (7)

Sz = Saz + Sbz h, 0, #h S2 (8)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (9)

S2 =$

!Sa + !Sb

%2

(10)

= S2a + S2

b + 2 !Sa · !Sb (11)

34

h2 |+; +" (12)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (13)


Sax|a : ±" =h

2|a : %" (15)

Say|a : ±" = ±ih

2|a : %" (16)

Saz|a : ±" = ± h

2|a : ±" (17)

A(!,"a) , B(!,"b) = ±1 ± 2 (1)

E("a,"b) = !p" (2)

±1 (i.e. ± h/2) (3)

p = ±1 (i.e. ± h2/4) (4)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (5)

Az(!) = +1 (i.e. + h/2) (6)

Az(!) = #1 (i.e. # h/2) (7)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (8)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (9)

Sz = Saz + Sbz h, 0, #h S2 (10)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (11)

S2 =$

"Sa + "Sb

%2

(12)

= S2a + S2

b + 2 "Sa · "Sb (13)

34

h2 |+; +" (14)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (15)


Sax|a : ±" =h

2|a : %" (17)

A(!,"a) , B(!,"b) = ±1 ± 2 (1)

E("a,"b) = !p" (2)

±1 (i.e. ± h/2) (3)

p = ±1 (i.e. ± h2/4) (4)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (5)

Az(!) = +1 (i.e. + h/2) (6)

Az(!) = #1 (i.e. # h/2) (7)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (8)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (9)

Sz = Saz + Sbz h, 0, #h S2 (10)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (11)

S2 =$

"Sa + "Sb

%2

(12)

= S2a + S2

b + 2 "Sa · "Sb (13)

34

h2 |+; +" (14)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (15)


Sax|a : ±" =h

2|a : %" (17)

A%11;()7.%/-#+)/-#&/"#78;%1."#hi2#F)1.)M("#$)$8;"ij#

)F"$#

U(.$"#X"1/)1+#

e- p+

!"

["#5Y5"#'%&1#X"1/)1+#G#

#############./$%//&"-##

"/#OH\H#VVV#

!%$)(.7;#

I4#

=4#

E(!a,!b) =!

A(",!a) B(",!b) P (") d" (1)

Az(") = +1 (i.e. + h/2) (2)

Az(") = !1 (i.e. ! h/2) (3)

{|a : ±" # |b : ±"} = {|#a;#b"} #a = ± , #b = ± (4)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (5)

Sz = Saz + Sbz h, 0, !h S2 (6)

Sz|+; +" = +h |+; +" Sz|!;!" = !h |!;!" Sz|+;!" = 0 Sz|!; +" = 0 (7)

S2 =$

!Sa + !Sb

%2

(8)

= S2a + S2

b + 2 !Sa · !Sb (9)

34

h2 |+; +" (10)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (11)


Sax|a : ±" =h

2|a : $" (13)

Say|a : ±" = ±ih

2|a : $" (14)

Saz|a : ±" = ± h

2|a : ±" (15)

$SaxSbx + SaySby + SazSbz

%|+; +" =

"h

2

# "h

2

#|!;!" +

"ih

2

# "ih

2

#|!;!" +

"h

2

# "h

2

#|+; +" (16)

S2|+; +" =$S2

a + S2b + 2 !Sa · !Sb

%|+; +" (17)

!"#78;%1:5"#+"#X"((#

A%55"###############C#%/#)#M."/#-g1##############H####

\&)/7.7;#K%15;"#2#')17.1#+-&(7)7#+"#N&)71"#-;1."-#+"#5"-&1"-#G##

>%&1#7%&7"#78;%1."#2#F)1.)M("-#$)$8;"-#(%$)("-C##

lHJH#X"((#)#5%/71;#(L./;*)(.7;#M")&$%&'#'(&-#K%17"#G###

[;5%/-71)7.%/#+N%1:5"#+"#X"((#

Z#

S#%&# %&#

S#

E(!a,!b) = !p" (1)

p = ±1 (i.e. ± h2/4) (2)

E(!a,!b) =!

A(",!a) B(",!b) P (") d" (3)

Az(") = +1 (i.e. + h/2) (4)

Az(") = #1 (i.e. # h/2) (5)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (6)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (7)

Sz = Saz + Sbz h, 0, #h S2 (8)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (9)

S2 =$

!Sa + !Sb

%2

(10)

= S2a + S2

b + 2 !Sa · !Sb (11)

34

h2 |+; +" (12)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (13)


Sax|a : ±" =h

2|a : %" (15)

Say|a : ±" = ±ih

2|a : %" (16)

Saz|a : ±" = ± h

2|a : ±" (17)

A(!,"a) , B(!,"b) = ±1 (1)

E("a,"b) = !p" (2)

p = ±1 (i.e. ± h2/4) (3)

E("a,"b) =!

A(!,"a)B(!,"b) P (!) d! (4)

Az(!) = +1 (i.e. + h/2) (5)

Az(!) = #1 (i.e. # h/2) (6)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (7)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (8)

Sz = Saz + Sbz h, 0, #h S2 (9)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (10)

S2 =$

"Sa + "Sb

%2

(11)

= S2a + S2

b + 2 "Sa · "Sb (12)

34

h2 |+; +" (13)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (14)


Sax|a : ±" =h

2|a : %" (16)

Say|a : ±" = ±ih

2|a : %" (17)

A(!,"a) , B(!,"b) = ±1 ± 2 (1)

E("a,"b) = !p" (2)

p = ±1 (i.e. ± h2/4) (3)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (4)

Az(!) = +1 (i.e. + h/2) (5)

Az(!) = #1 (i.e. # h/2) (6)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (7)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (8)

Sz = Saz + Sbz h, 0, #h S2 (9)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (10)

S2 =$

"Sa + "Sb

%2

(11)

= S2a + S2

b + 2 "Sa · "Sb (12)

34

h2 |+; +" (13)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (14)


Sax|a : ±" =h

2|a : %" (16)

Say|a : ±" = ±ih

2|a : %" (17)

A(!,"a) , B(!,"b) = ±1 ± 2 (1)

E("a,"b) = !p" (2)

p = ±1 (i.e. ± h2/4) (3)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (4)

Az(!) = +1 (i.e. + h/2) (5)

Az(!) = #1 (i.e. # h/2) (6)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (7)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (8)

Sz = Saz + Sbz h, 0, #h S2 (9)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (10)

S2 =$

"Sa + "Sb

%2

(11)

= S2a + S2

b + 2 "Sa · "Sb (12)

34

h2 |+; +" (13)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (14)


Sax|a : ±" =h

2|a : %" (16)

Say|a : ±" = ±ih

2|a : %" (17)

A(!,"a) , B(!,"b) = ±1 ± 2 (1)

E("a,"b) = !p" (2)

p = ±1 (i.e. ± h2/4) (3)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (4)

Az(!) = +1 (i.e. + h/2) (5)

Az(!) = #1 (i.e. # h/2) (6)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (7)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (8)

Sz = Saz + Sbz h, 0, #h S2 (9)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (10)

S2 =$

"Sa + "Sb

%2

(11)

= S2a + S2

b + 2 "Sa · "Sb (12)

34

h2 |+; +" (13)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (14)


Sax|a : ±" =h

2|a : %" (16)

Say|a : ±" = ±ih

2|a : %" (17)

A%11;()7.%/-#N&)/7.N&"-#+)/-#(L;7)7#-./*&("7###

U(.$"# X"1/)1+#"=# 'I#

X"1/)1+#"-7#)(%1-#$"17)./#+"#71%&F"1#########+)/-#-)#5"-&1"#+"##

J.#U(.$"#71%&F"########C#(6;7)7#+&#-9-7:5"#)'1:-#-)#5"-&1"#"-7##

+1 (i.e. + h/2) (1)

A(!,"a) , B(!,"b) = ±1 ± 2 (2)

E("a,"b) = !p" (3)

±1 (i.e. ± h/2) (4)

p = ±1 (i.e. ± h2/4) (5)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (6)

Az(!) = +1 (i.e. + h/2) (7)

Az(!) = #1 (i.e. # h/2) (8)

{|a : ±" $ |b : ±"} = {|#a;#b"} #a = ± , #b = ± (9)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (10)

Sz = Saz + Sbz h, 0, #h S2 (11)

Sz|+; +" = +h |+; +" Sz|#;#" = #h |#;#" Sz|+;#" = 0 Sz|#; +" = 0 (12)

S2 =$

"Sa + "Sb

%2

(13)

= S2a + S2

b + 2 "Sa · "Sb (14)

34

h2 |+; +" (15)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (16)


[)/-#-)#5"-&1"#+"########C#U(.$"##

'"&7#71%&F"1#)F"$#'1%M)M.(.7;#0#G#

+1 (i.e. + h/2) (1)

!1 (i.e. ! h/2) (2)

A(!,"a) , B(!,"b) = ±1 ± 2 (3)

E("a,"b) = "p# (4)

±1 (i.e. ± h/2) (5)

p = ±1 (i.e. ± h2/4) (6)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (7)

Az(!) = +1 (i.e. + h/2) (8)

Az(!) = !1 (i.e. ! h/2) (9)

{|a : ±# $ |b : ±#} = {|#a;#b#} #a = ± , #b = ± (10)

Sz|#a;#b# = (Saz + Sbz)|#a;#b# ="

h#a

2+

h#b

2

#|#a;#b# (11)

Sz = Saz + Sbz h, 0, !h S2 (12)

Sz|+; +# = +h |+; +# Sz|!;!# = !h |!;!# Sz|+;!# = 0 Sz|!; +# = 0 (13)

S2 =$

"Sa + "Sb

%2

(14)

= S2a + S2

b + 2 "Sa · "Sb (15)

34

h2 |+; +# (16)

$S2

a + S2b

%|#a;#b# =

32

h2 |#a;#b# (17)


+1 (i.e. + h/2) Sez (1)

!1 (i.e. ! h/2) (2)

A(!,"a) , B(!,"b) = ±1 ± 2 (3)

E("a,"b) = "p# (4)

±1 (i.e. ± h/2) (5)

p = ±1 (i.e. ± h2/4) (6)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (7)

Az(!) = +1 (i.e. + h/2) (8)

Az(!) = !1 (i.e. ! h/2) (9)

{|a : ±# $ |b : ±#} = {|#a;#b#} #a = ± , #b = ± (10)

Sz|#a;#b# = (Saz + Sbz)|#a;#b# ="

h#a

2+

h#b

2

#|#a;#b# (11)

Sz = Saz + Sbz h, 0, !h S2 (12)

Sz|+; +# = +h |+; +# Sz|!;!# = !h |!;!# Sz|+;!# = 0 Sz|!; +# = 0 (13)

S2 =$

"Sa + "Sb

%2

(14)

= S2a + S2

b + 2 "Sa · "Sb (15)

34

h2 |+; +# (16)

$S2

a + S2b

%|#a;#b# =

32

h2 |#a;#b# (17)


+1 (i.e. + h/2) Sez (1)

!1 (i.e. ! h/2) (2)

A(!,"a) , B(!,"b) = ±1 ± 2 (3)

E("a,"b) = "p# (4)

±1 (i.e. ± h/2) (5)

p = ±1 (i.e. ± h2/4) (6)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (7)

Az(!) = +1 (i.e. + h/2) (8)

Az(!) = !1 (i.e. ! h/2) (9)

{|a : ±# $ |b : ±#} = {|#a;#b#} #a = ± , #b = ± (10)

Sz|#a;#b# = (Saz + Sbz)|#a;#b# ="

h#a

2+

h#b

2

#|#a;#b# (11)

Sz = Saz + Sbz h, 0, !h S2 (12)

Sz|+; +# = +h |+; +# Sz|!;!# = !h |!;!# Sz|+;!# = 0 Sz|!; +# = 0 (13)

S2 =$

"Sa + "Sb

%2

(14)

= S2a + S2

b + 2 "Sa · "Sb (15)

34

h2 |+; +# (16)

$S2

a + S2b

%|#a;#b# =

32

h2 |#a;#b# (17)


|e : !" # |p : +" (1)

+1 (i.e. + h/2) Sez (2)

!1 (i.e. ! h/2) (3)

A(!,"a) , B(!,"b) = ±1 ± 2 (4)

E("a,"b) = $p" (5)

±1 (i.e. ± h/2) (6)

p = ±1 (i.e. ± h2/4) (7)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (8)

Az(!) = +1 (i.e. + h/2) (9)

Az(!) = !1 (i.e. ! h/2) (10)

{|a : ±" # |b : ±"} = {|#a;#b"} #a = ± , #b = ± (11)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (12)

Sz = Saz + Sbz h, 0, !h S2 (13)

Sz|+; +" = +h |+; +" Sz|!;!" = !h |!;!" Sz|+;!" = 0 Sz|!; +" = 0 (14)

S2 =$

"Sa + "Sb

%2

(15)

= S2a + S2

b + 2 "Sa · "Sb (16)

34

h2 |+; +" (17)

$S2

a + S2b

%|#a;#b" =

32

h2 |#a;#b" (18)

A%11;()7.%/-#<>@###^<./-7"./=>%+%(-?9=@%-"/_#

J.#U(.$"#71%&F"########C#(6;7)7#+&#-9-7:5"#)'1:-#-)#5"-&1"#"-7##

X"1/)1+#"-7#)(%1-#$"17)./#+"#71%&F"1###########+)/-#-)#5"-&1"#+"##

+1 (i.e. + h/2) Sez (1)

!1 (i.e. ! h/2) (2)

A(!,"a) , B(!,"b) = ±1 ± 2 (3)

E("a,"b) = "p# (4)

±1 (i.e. ± h/2) (5)

p = ±1 (i.e. ± h2/4) (6)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (7)

Az(!) = +1 (i.e. + h/2) (8)

Az(!) = !1 (i.e. ! h/2) (9)

{|a : ±# $ |b : ±#} = {|#a;#b#} #a = ± , #b = ± (10)

Sz|#a;#b# = (Saz + Sbz)|#a;#b# ="

h#a

2+

h#b

2

#|#a;#b# (11)

Sz = Saz + Sbz h, 0, !h S2 (12)

Sz|+; +# = +h |+; +# Sz|!;!# = !h |!;!# Sz|+;!# = 0 Sz|!; +# = 0 (13)

S2 =$

"Sa + "Sb

%2

(14)

= S2a + S2

b + 2 "Sa · "Sb (15)

34

h2 |+; +# (16)

$S2

a + S2b

%|#a;#b# =

32

h2 |#a;#b# (17)


+1 (i.e. + h/2) Sez (1)

!1 (i.e. ! h/2) (2)

A(!,"a) , B(!,"b) = ±1 ± 2 (3)

E("a,"b) = "p# (4)

±1 (i.e. ± h/2) (5)

p = ±1 (i.e. ± h2/4) (6)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (7)

Az(!) = +1 (i.e. + h/2) (8)

Az(!) = !1 (i.e. ! h/2) (9)

{|a : ±# $ |b : ±#} = {|#a;#b#} #a = ± , #b = ± (10)

Sz|#a;#b# = (Saz + Sbz)|#a;#b# ="

h#a

2+

h#b

2

#|#a;#b# (11)

Sz = Saz + Sbz h, 0, !h S2 (12)

Sz|+; +# = +h |+; +# Sz|!;!# = !h |!;!# Sz|+;!# = 0 Sz|!; +# = 0 (13)

S2 =$

"Sa + "Sb

%2

(14)

= S2a + S2

b + 2 "Sa · "Sb (15)

34

h2 |+; +# (16)

$S2

a + S2b

%|#a;#b# =

32

h2 |#a;#b# (17)


|e : !" # |p : +" (1)

|e : +" # |p : !" (2)

+1 (i.e. + h/2) Sez (3)

!1 (i.e. ! h/2) (4)

A(!,"a) , B(!,"b) = ±1 ± 2 (5)

E("a,"b) = $p" (6)

±1 (i.e. ± h/2) (7)

p = ±1 (i.e. ± h2/4) (8)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (9)

Az(!) = +1 (i.e. + h/2) (10)

Az(!) = !1 (i.e. ! h/2) (11)

{|a : ±" # |b : ±"} = {|#a;#b"} #a = ± , #b = ± (12)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (13)

Sz = Saz + Sbz h, 0, !h S2 (14)

Sz|+; +" = +h |+; +" Sz|!;!" = !h |!;!" Sz|+;!" = 0 Sz|!; +" = 0 (15)

S2 =$

"Sa + "Sb

%2

(16)

= S2a + S2

b + 2 "Sa · "Sb (17)

34

h2 |+; +" (18)

|e : !" # |p : +" (1)

|e : +" # |p : !" (2)

+1 (i.e. + h/2) Sez (3)

!1 (i.e. ! h/2) (4)

A(!,"a) , B(!,"b) = ±1 ± 2 (5)

E("a,"b) = $p" (6)

±1 (i.e. ± h/2) (7)

p = ±1 (i.e. ± h2/4) (8)

E("a,"b) =!

A(!,"a) B(!,"b)P (!) d! (9)

Az(!) = +1 (i.e. + h/2) (10)

Az(!) = !1 (i.e. ! h/2) (11)

{|a : ±" # |b : ±"} = {|#a;#b"} #a = ± , #b = ± (12)

Sz|#a;#b" = (Saz + Sbz)|#a;#b" ="

h#a

2+

h#b

2

#|#a;#b" (13)

Sz = Saz + Sbz h, 0, !h S2 (14)

Sz|+; +" = +h |+; +" Sz|!;!" = !h |!;!" Sz|+;!" = 0 Sz|!; +" = 0 (15)

S2 =$

"Sa + "Sb

%2

(16)

= S2a + S2

b + 2 "Sa · "Sb (17)

34

h2 |+; +" (18)

A%11;()7.%/-#N&)/7.N&"-#+)/-#(L;7)7#-./*&("7#^-&.7"_#

A8%.,#+6)/*("-##

./7;1"--)/7#G#

A%11;()7.%/#7%7)("#"/71"#("-#1;-&(7)7-#+LU(.$"#"7#X"1/)1+#G#

<,"1$.$"#4G#5%/71"1#N&"#$"77"#$%11;()7.%/#7%7)("#1"-7"#F)()M("#7)/7#N&LU(.$"#"7##

###################X"1/)1+#$8%.-.--"/7#("#5Y5"#),"#+L)/)(9-"#######G##

<,"1$.$"#3G#5%/71"1#N&L%/#)#'(&-#*;/;1)("5"/7#G#

m.%()7.%/#+"#(6./;*)(.7;#+"#X"((#################VVV#

!"-#",';1."/$"-#+6E1-)9#^4_#

A)-$)+"#)7%5.N&"##

^)7%5"#+"#$)($.&5_#

<7)7#+"#'%()1.-)7.%/#+"#()#').1"#+"#'8%7%/-#G#

f (J = 0)

e1 (J = 0, # = 15 ns)

Laser

e2 (J = 1, # = 5 ns)

!a = 551 nm

!b = 422 nm

UH#U-'"$7C#>H#n1)/*."1C#nH#@%*"1#o#lH#[)(.M)1+#

!"-#",';1."/$"-#+6E1-)9#^3_#

Compteurs

Sexp=2,697 (15)

Stheo M.Q.= 2,70 ! Sexp : 71.%5'8"#+"#()#5;$)/.N&"#N&)/7.N&"#

Source

|Sexp| > 2 : +;K).7"#+"-#78;%1."-#2#F)1.)M("-#$)$8;"-#(%$)("-#

DH#

!)#1)."#2#34#$5#+"#(L89+1%*:/"#"/#)-71%'89-.N&"#

71.'("7#

-./*&("7#

/.F")&#4-#

!#p#34#$5#

5)--"-#+L89+1%*:/"#"/#`"&#$%/-.+;1)M("-#G#%/#1"q%.7#&/#-.*/)(#-&KK.-)/7#

+"#'8%7%/-#;5.-#(%1-#+"#()#+;-",$.7)7.%/#+&#/.F")&#71.'("7#

(# = 107 ans)

d%71"#*)(),."#

3SS#5.((.)1+-#+6;7%.("-#

r%15"#-'.1)("#

[.)5:71"#G##

4SS#SSS#)//;"-=(&5.:1"#

<').--"&1#G##

4SSS#)//;"-=(&5.:1"#

F&"#+L)17.-7"C#sssH/)-)H*%F#

!689+1%*:/"#+)/-#/%71"#*)(),."#

</71"#("-#;7%.("-C#5)7.:1"#+.KK&-"#G#4S#t#+"#()#5)--"#7%7)("#

[)/-#$"#*)u#./7"1-7"(().1"C#vSt#+"-#)7%5"-#-%/7#+"#(689+1%*:/"#

SCB#)7%5"]$5B#"/#5%9"//"#

d&)*"-#+"#5)--"#$%5'1.-"#"/71"#SC4#"7#4SSS#5)--"-#-%().1"-##

5$#'4&*!*%%*+#'*))*!(3-&!)$!63&.$#'3+!2*!+3-7*))*%!,"+"&$#'3+%!28"#3')*%9!*#!:-'!*%#!#&3(!;6&3'2*;!(3-&!<#&*!7-*!3(#':-*.*+#!

EM-"1F)7.%/#+"#()#1)."#2#34#$5#

E/#%M-"1F"#)F"$#+"-#1)+.%=7;("-$%'"-#()#1)."#;5.-"#2#34#$5#')1#$"-#/&)*"-#

<KK"7#[%''("1##

7"5';1)7&1"#G#3S#2#4SS#w#

F.7"--"#5%9"//"G##

`&-N&62#3xS#?5]-#

<KK"7#y""5)/##

$8)5'#5)*/;7.N&"#(%$)(#

!"#5"--)*"##

+"#>.%/""1#

Le couplage de deux spins ½

Documents

Transcript of Le couplage de deux spins ½