Quantum Mechanics

An introduction to and derivation of the fundamental equations of Schrödinger’s quantum mechanics.
Tyson Jones

Initialization

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    Note: we’ll work in natural and nondimensional units where
    ℏ=m=ω=1
    .
    ClearAll["Global`*"]
    AutoCollapse[]:=(​​If[$FrontEnd=!=$Failed,​​SelectionMove[EvaluationNotebook[],All,GeneratedCell];​​FrontEndTokenExecute["SelectionCloseUnselectedCells"]])
    colorBar[title_:"arg[ψ]"]:=​​ BarLegend[​​ {"Rainbow",{-π,π}},​​ LegendLabeltitle,​​ "Ticks"{-3.14,-3.14/2,0,3.14/2,3.14},​​ "TickLabels"{"-π","-π/2","0","π/2","π"}​​ ]​​​​colorBarNorm[title_]:=​​ BarLegend[​​ "Rainbow",​​ LegendLabeltitle​​ ]​​​​plotWavefunction[psi_,{r_,range__},showbar_:True,title_:
    2
    Abs[ψ]
    ,legendpsi_:ψ,plotrange_:All,ticks_:Automatic]:=​​ ReplaceAll[​​ Plot[​​
    2
    Abs[psi]
    ,{r,range},​​ AxesLabel{r,title},​​ PlotRangeplotrange,​​ ColorFunction(ColorData["Rainbow"][Rescale[Arg[psi/.r#],{-π,π}]]&),​​ ColorFunctionScalingFalse,​​ Ticksticks,​​ FillingAxis,​​ PlotLegendsIf[showbar,colorBarNorm["arg["<>ToString[legendpsi]<>"] / 2π"],None]​​ ],​​ Line[pts_,_]{Black,Line[pts]}​​ ]
    simulateWavefunction[psi_,potential_,{r_,domain__},{_,times__}]:=​​ NDSolveValue​​ ​​ 
    (0,1)
    ψ
    [r, τ]-
    (2,0)
    ψ
    [r, τ]  2+ψ[r, τ]potential,​​ ψ[r, 0]  psi​​ ,​​ ψ,​​ {r, domain},​​ {τ, times},​​ Method{"FiniteElement"}​​ ​​ ​​​​Off[NDSolveValue::bcart]​​Off[NDSolveValue::femcscd]​​
    hamiltonian[psi_,potential_,r_]:=​​ -
    1
    2
    D[psi,{r,2}]+potentialpsi
    expectedEnergy[psi_, potential_, r_] :=​​ Integrate[​​ (Conjugate[psi] /. Conjugate[r]  r)​​ (hamiltonian[psi, potential, r]),​​ {r, -∞, ∞}​​ ]
    quickNormalise[psi_,r_:x]:=​​ psiSqrt[NIntegrate[​​
    2
    Abs[psi]
    ,{r,-∞,∞},​​ Method{Automatic,"SymbolicProcessing"0}]]​​

    The Wavefunction

    The Schrödinger Equation

    The Quantum Harmonic Oscillator

    Quantum Tunneling

    AUTHORSHIP INFORMATION
    Tyson Jones