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Find the Energy Spectrum for a Finite Potential Well
Find the energy spectrum for particles of mass m in a potential well of width L and height Γ . Show the solutions normalized to be 1 at L/2.
Find the Energy Spectrum for a Finite Potential Well
Find the energy spectrum for particles of mass m in a potential well of width L and height Γ . Show the solutions normalized to be 1 at L/2.
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Remove["Global`*"]α=Sqrt[2m(Γ-Ε)]/ℏ;k=Sqrt[2mΕ]/ℏ;Subscript[ψ,1]=GExp[αx];Subscript[ψ,2]=ASin[kx]+BCos[kx];Subscript[ψ,3]=HExp[-αx];bconds={(Subscript[ψ,1]/.x->-L/2)==(Subscript[ψ,2]/.x->-L/2),(Subscript[ψ,2]/.x->L/2)==(Subscript[ψ,3]/.x->L/2),(/.x->-L/2)==(/.x->-L/2),(/.x->L/2)==(/.x->L/2)};d=Det[CoefficientArrays[bconds,{A,B,G,H}][[2]]];ℏ=1;L=1;Γ=100;m=1;spec=Reduce[d==0&&Ε>1/1000,Ε,Reals]
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Ε||Ε||Ε||Ε||Ε
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sols=(Piecewise[{{Subscript[ψ,1],x<-L/2},{Subscript[ψ,2],-L/2<x<L/2}},Subscript[ψ,3]]/.#/.Solve[bconds/.#,{A,B,G}]/.H->(Exp[Sqrt[(Γ-Ε)/2]]/.#))&/@N[{ToRules[spec]},20];Plot[sols,{x,-L,L},PlotRange->All]
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ψ[x_]:=Piecewise[{{Subscript[ψ,1],x<-L/2},{Subscript[ψ,2],-L/2<x<L/2}},Subscript[ψ,3]]CostNorm:=1/Sqrt[NIntegrate[Abs[ψ[x]]^2,{x,-L,L}]];ψ[x_]:=CostNormψ[x];sols=(ψ[x]/.#/.Solve[bconds/.#,{A,B,G}]/.H->(Exp[Sqrt[(Γ-Ε)/2]]/.#))&/@N[{ToRules[spec]},20]Plot[sols,{x,-L,L},PlotRange->All]
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{TerminatedEvaluation[RecursionLimit],TerminatedEvaluation[RecursionLimit],TerminatedEvaluation[RecursionLimit],TerminatedEvaluation[RecursionLimit],TerminatedEvaluation[RecursionLimit]}
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