phy-4600/notes/particles in a box

❙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