Mécanique Quantique : Résumé de Cours

HC
H × H → C
l(2)
n
|φi=
a1
.
.
an
|ψi=
b1
.
.
bn
ai, bi∈C∀i
|φi |ψi
hφ|ψi=
n
X
i=1
a∗
ibi
a∗
iai
hφ|= (a∗
1, . . . , a∗ n)
hφ|ψi= (hψ|φi)∗
µ∈Chµφ|ψi=µ∗hφ|ψi hφ|µψi=µhφ|ψi

L(2)
R3→C
φ(~r, t)∈ L(2)
ZR3|φ(~r, t)|2dxdydz < ∞
φ(~r, t)ψ(~r, t)
hφ|ψi=ZR3 φ∗(~r, t)·ψ(~r, t)dxdydz
l(2)
l(2) L(2)
L(2) R C
|x0i ∈ l(2)
ˆx x0∈R
ˆx|x0i=x0|x0i
|φi ∈ l(2) L(2)
φ(x, t)
φ(x0) = hx0|φi
ˆ
A φ(x0) = hx0|ˆ A|φi
=

~x(t)~p(t)E=H
~
L
Hˆx
ˆpˆ Hˆ L
~
L
ˆ
~
L
ˆ
~r ˆ ~p
ˆ
~
L=
ˆ
Lx
ˆ
Ly
ˆ
Lz
ˆ
~r =
ˆx
ˆy
ˆz
ˆ
~p =
ˆpx
ˆpy
ˆpz
ˆ
~
L=ˆ ~r ׈ ~p
ˆ
Aˆ B
[ˆ A, ˆ B] = ˆ Aˆ B−ˆ Bˆ A
ˆ
AHˆ A†|φi,|ψi ∈ H
hφ|ˆ A|ψi= (hψ|ˆ A†|φi)∗
H
A A†A
aij A a†
ij
A†
aij = (a†
ji)∗

{|φni} l(2)
ˆ
Bˆ A
A
Aij =hφi|ˆ A|φji
ˆ
B
ˆ
AH
Hˆ A
|φiˆ A λ
hφ|ˆ A|φi= (hφ|ˆ A†|φi)∗= (hφ|ˆ A|φi)∗
hφ|ˆ A|φi=λhφ|φi
(hφ|ˆ A|φi)∗=λ∗(hφ|φi)∗=λ∗hφ|φi
λ=λ∗⇐⇒ λ∈R

{|φni} n
l(2) nˆ H
L(2) ˆx
ˆx L(2)
ˆx x0
L(2) φ(x) = δ(x−x0) 0 x0
|φiP|φi
P|φi|Ψi=hφ|Ψi
|{z}
scalaire
|φi= (|φihφ|)|Ψi
|Ψi |φi |φihφ|
|Ψi |φi
{|φni} l(2)
X
n
(|φnihφn|) = Id
Id
{|xi} ˆx
Zdx|xihx|= Id