mirror of
https://asciireactor.com/otho/phy-4600.git
synced 2024-12-04 19:05:07 +00:00
70 lines
1.6 KiB
Plaintext
70 lines
1.6 KiB
Plaintext
❙Ψ❭ ≐ Ψ(x)
|
||
Ψ(x) = ❬x❙Ψ❭
|
||
𝓟(x) = │Ψ(x)│²
|
||
𝓟(x) = ⎮Ψ(x)⎮²
|
||
|
||
⌠ ∞
|
||
1 = ❬Ψ❙Ψ❭ = ⎮ │Ψ(x)│² dx = 1
|
||
⌡-∞
|
||
|
||
❙Ψ❭ → Ψ(x)
|
||
❬Ψ❙ → Ψ⃰(x)
|
||
|
||
 → A(x)
|
||
⌠b
|
||
𝓟(a<x<b) = ⎮ │Ψ(x)│² dx
|
||
⌡a
|
||
|
||
│⌠∞ │²
|
||
𝓟(Eₙ) = │❬Eₙ❙Ψ❭│² = │⎮ Eₙ⃰(x) Ψ(x) dx │
|
||
│⌡-∞ │
|
||
|
||
x̂ = x
|
||
|
||
p̂ = ι͟ ∂͟
|
||
ħ ∂x
|
||
|
||
|
||
⎛- ħ͟²͟ d͟²͟ + V(x)⎞ φₙ(x) = E φₙ(x)
|
||
⎝ 2m dx² ⎠
|
||
|
||
Boundary conditions:
|
||
|
||
1) φₙ(x) is continuous.
|
||
2) d φₙ(x) is continuous unless V = ∞.
|
||
dx
|
||
|
||
Infinite square potential energy well:
|
||
|
||
Eₙ = n͟²͟π͟²͟ħ͟², n = 1, 2, 3, ...
|
||
2mL²
|
||
|
||
φₙ(x) = √⎛2͟⎞ sin⎛n͟π͟x͟⎞, n = 1, 2, 3, ...
|
||
⎝L⎠ ⎝ L ⎠
|
||
|
||
|
||
Energy eigenstates obey the following properties:
|
||
|
||
Bra-ket Notation Wavefunction Notation
|
||
|
||
Normalization
|
||
|
||
⌠∞
|
||
❬Eₙ❙Eₙ❭ = 1 ⎮ │φₙ(x)│² dx = 1
|
||
⌡-∞
|
||
|
||
Orthogonality
|
||
|
||
⌠∞
|
||
❬Eₙ❙Eₘ❭ = δₙₘ ⎮ φₙ⃰(x) φₘ(x) dx = δₙₘ
|
||
⌡-∞
|
||
|
||
Completeness
|
||
|
||
❙Ψ❭ = ∑ cₙ ❙Eₙ❭ Ψ(x) = ∑ cₙ φₙ(x)
|
||
ⁿ ⁿ
|
||
|
||
|
||
|
||
|