Tunneleffekt

Transmission & Reflexion

solpar=Simplify[Solve[E^(-I k L/2) +R E^(I k L/2)==A E^(-κ L/2)+B E^(κ L/2)&& I k (E^(-I k L/2) -R E^(I k L/2))==κ( A E^(-κ L/2)-B E^(κ L/2))&&T E^(I k L/2)==A E^(κ L/2)+B E^(-κ L/2)&& I k T E^(I k L/2)==κ(A E^(κ L/2)-B E^(-κ L/2)),{A,B,R,T}]]

Wellenfunktion

ψI[x_,L_,k_,κ_]=E^(I k x) +(R/.solpar[[1,3]])  E^(-I k x);

ψII[x_,L_,k_,κ_]=(A/.solpar[[1,1]] )E^(κ x)+(B/.solpar[[1,2]])E^(-κ x);

ψIII[x_,L_,k_,κ_]=(T/.solpar[[1,4]])  E^(I k x);

ψ[x_,L_,k_,κ_]=If[x<-L/2,ψI[x,L,k,κ],If[x<L/2,ψII[x,L,k,κ],ψIII[x,L,k,κ]]];

Plots

Plot[Re[ψ[x,1,40,1]],{x,-2,2}]

Plot[Re[ψ[x,1,2,1]],{x,-20,20}]

Plot[Im[ψ[x,1,2,1]],{x,-20,20}]

Plot[Abs[ψ[x,1,2,1]]^2,{x,-10,10}]

Checks

Simplify[(T/.solpar[[1,4]]) *((T/.solpar[[1,4]]) /.k->-k)+(R/.solpar[[1,3]]) *((R/.solpar[[1,3]]) /.k->-k)]

Simplify[(T/.solpar[[1,4]]) *((T/.solpar[[1,4]]) /.k->-k)]

κ=Sqrt[2 m/hbar^2 E0]; k=Sqrt[2 m/hbar^2 (V0-E0)];

Simplify[(κ^2+k^2)^2/(k^2 κ^2)]