Enoncé

o1
m x
x= 0 x=L V (x) = 0
0< x < L V (x) = +∞ψ(0) = ψ(L) = 0
ψnEn
x t = 0
ψ(x) = Cexp ip0x
~exp −(x−x0)2 2σ2!
x0, p0∈Rσ > 0C > 0
ψ(x) = hx|ψi |ψi ∈ L2(R)
C|ψi kψk2=hψ|ψi= 1
P(x) = |ψ(x)|2
Z+∞
−∞
exp −Ax2+Bx +Cdx =rπ
Aexp −C+B2 4A, A, B, C ∈C,
<(A)>0
|pi
hx|pi=1
√2π~exp ipx
~
˜
ψ(p) = hp|ψi˜ P(p) = ˜ ψ(p)
2
|ψiσ→ ∞ σ→0

hxi=RxP (x)dx =hψ|ˆxψi
P(x) = |ψ(x)|2∆x
(∆x)2:= D(x− hxi)2E
(∆x)2=x2− hxi2
hxi,∆x, hpi,∆p∆x∆p
p=mv ∆x∆v
∆x
h= 2π~= 6.6 10−34J.s
m= 9 10−31 '
1cm m ≥10−15
∆x(t)'∆x(0) et/τ ∆v(t)'∆v(0) et/τ
τ τ '1s. ∆x(t)
TE
TEτ'1m= 10−2
t= 0 ψ(x)˜
ψ(p)
p'p0x
ˆ
H=ˆp2
2me
ψ(p, t)t
e
ψ(p, 0)
p H p0
H(p)'H(p0)+(p−p0)∂H
∂p p=p0
ψ(x, t)t ψ (x, t = 0)
|ψ(x, t)|
p