phy-4600/notes/particles in a box
2016-02-28 17:15:29 -05:00

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❙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