Postulats mathématiques de la mécanique quantique

H
|ψi H
H
+ : H×H→H . :C×H→H
(,) : H×H→C
|ψ1i |ψ2i
H |ψ1+2i=|ψ1i+|ψ2i
|ψi H |ψi
H H → C
|ψi hψ| ∀ |φi∈H,hψ|φi= (ψ, φ) (ψ, φ)
|ψi |φi
H=L2R3
R3C
ψ:R3→C
−→
r→ψ(−→ r)
ψ1+ψ2:R3→C
−→
r→ψ1(−→ r) + ψ2(−→ r)
λψ :R3→C
−→
r→λψ(−→ r)
(ψ1, ψ2) = hψ1|ψ2i=´R3ψ∗
1(−→
r)ψ2(−→ r)d3−→ r
H=R2| i =1
0|−i =
0
1X|ψi=ψ+|+i+ψ−|−i =ψ+ ψ−
ψ+∈Rψ−∈R
|ψ1i+|ψ2i=ψ+
1+ψ+ 2|+i+
ψ−
1+ψ− 2|−i =ψ+
1+ψ+ 2
ψ−
1+ψ− 2
λ|ψi=λψ+|+i+
λψ−|−i =λψ+
λψ−
(ψ1.ψ2) =
hψ1|ψ2i=ψ+
1∗ψ+ 2+ψ− 1∗ψ− 2

•
H=L2R3⊗R2L2R3
R2
• H =R2⊗R2
O
ˆ
O:H → H
ψ→ˆ O(ψ)
Oˆ O
Hˆ O O
ˆ
O
ˆ
O†=ˆ Oˆ Aˆ B ψ1, ψ2
ψ1,ˆ
A(ψ2)=ˆ B(ψ1), ψ2hψ1|ˆ A|ψ2i=hψ1|ˆ B|ψ2i
ˆ
B=¯
tˆ
A
{|ψii} H ˆ Ohψi|ψji=δij
ˆ
O|ψi=Pµi|φii
|φiiˆ O aiˆ O|φii=ai|φii
µi=hψ|φii
H=L2R3ˆxˆx:H → H
|ψi → x|ψi
x|ψi:R3→C
−→
ρ→ρxψ(−→ ρ)y z
H=L2R3ˆpxˆpx:H → H
|ψi → ˆpx|ψi
ˆpx|ψi:R3→C
−→
ρ→ −i~∂xψ(−→ ρ)y z
H=R2Szzˆ Sz|+i=
+~
2|+iˆ
Sz|−i =−~
2|−i
• H =R2⊗R2S1,z =Sz⊗Id
S2,z =Id⊗Sz
Stot,z =S1,z +S2,z
• H =L2R3⊗R2
ˆpx⊗Id
ˆ
O
ai
ˆ
O|ψi hψ|ˆ O|ψi

• |φiiˆ O O
ai
•ˆ O O
aj∈Sp(ˆ O)
ˆ
O
aihψ|ˆ O|ψi
xˆx
{|x0i} x0(x) = δ(x−x0)
{x0} hxi=
´x|ψ(x)|2dx
ˆpx|px,iipx,i(x) = 1
√Leipx,ix/~
{x0}
hpxi=−i~´ψ∗(x) (∂xψ(x)) dx
|ψi=1
√2|+i+1 √2|−i
1
2(h+|+h−|)Sz(|+i+|−i) = 1 21 2+0+0+−1 2= 0
|ψiai
|hψ|φii|2
|ψiα
dα |hψ|φ(α)i|2dα
x0x0+dx0|hψ|x0i|2dx0=
´ψ∗(x)δ(x−x0)
2dx =|ψ(x0)|2dx0
px,i dpx,i |ψi |hψ|px,ii|2dpx,i =
1
L
´eipx,ix/~ψ∗(x)dx
2dpx,i
1
2|ψi=1
√5|+i+2 √5|−i +~
2
|hψ|+i|2=1
5−~ 2|hψ|−i|2=4 5
ai
Eaiai
aiai
x0x0+dx0x0x0+dx0
+~
2|+i

i~∂|ψi
∂t =ˆ
H|ψi
ˆ
H