Remove["Global`*"] hbar:=\[HBar] H[V_] @psi_:=-hbar^2/(2m) D[psi,{x,2}] + V psi SchD[V_] @psi_:=H[V] @ psi - En psi SchD[V[x]] @ psi[x]==0 // TeXForm

phi[n_,x_]:=Cn Sin[n Pi x/L] SchD[0] @ phi[n,x]==0 // ExpandAll SchD[0] @ phi[n,x] / phi[n,x]==0 // Simplify // TeXForm

Starting with a linear superposition

A E^(Ikx) + B E^(-Ikx)of independent plane waves, whereAandBare constants, verify the box eigen-functions and eigen-energies given above.Thus, show that this superposition is a solution of the Schrodinger equation and, by invoking the boundary condition, that

k -> n Pi/L.— Quantum Methods with Mathematica

phi[k_,x_]:=A E^(I k x) + B E^(- I k x) eq1:=SchD[0] @ phi[k,x]/phi[k,x]==0 // Simplify Ek[k_]:=En/.Solve[eq1, En] [[1]] Ek[k] // TeXForm

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The boundary conditions are .

So

Solve[E^(I k L) - E^(-I k L)==0, k] // Simplify

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phi[n_,x_]:=Cn Sin[n Pi x/L] norm[n_]=Cn/.Solve[ Integrate[ phi[n,x]^2, {x,0,L}]==1/.Sin[m_Integer n Pi]->0, Cn ] [[1]] // TeXForm

It’s interesting to note in passing that the 1D box eigenfunctions are also classically the eigenfunctions of a taut string. However, whereas the quantum mechanical energies scale as

n^2, the classical eigenfrequencies of the string’s normal modes are linear inn. This is a consequence of the classical wave equation beingsecond orderin time in contrast to the quantum wave equation beingfirst order.— Quantum Methods with Mathematica

— Me@2022-12-16 10:23:45 AM

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2022.12.17 Saturday (c) All rights reserved by ACHK