Potential Barrier

Explanation

Hamilton-Function for scattering at potential barrier

H  =  - ((p^2)/(2 m))   +   V(
              x) ;      (V(x) = (Subscript[V, 0]) for  |x| ≤  a;  0 otherwise)

Schrödinger-Equation 

\(TraditionalForm\`\(-\(h\&~\^2\/\(2\ m\)\)\)\ d\^2\/dx\^2\) Ψ(x)  +  \(TraditionalForm\`\(\(V\_0\)\(\ \ \ \)\(Θ(\(|\)\(x\)\(|\)\(\(<\)\(a\)\))\)\(\ \)\)\) Ψ(x)  =  E Ψ(x) 

dimensionsless quantities

x       =    ξ   (Subscript[x, 0])
a         =  α  (Subscript[x, 0])
\(TraditionalForm\`x\_0\ \ \ \  = \ \((h\&~\^2\/\(2\ m\^2 | V\_0 | \))\)\^\(1/2\)\)
E       =  ϵ  Abs[(Subscript[V, 0]) ]
σ        =  
        sign((Subscript[V, 0]))

\(TraditionalForm\`Φ(ξ)\  = \ Ψ(ξ\ x\_0)\)

Schrödinger-Equation

-\(TraditionalForm\`Φ'' \((ξ)\) + \ σ\ \ \(Θ(\(|\)\(ξ\)\(|\)\(-α\))\)\ \ \(Φ(ξ)\)\ \ \  = \ \ \ ϵ\ \ \ \(Φ(ξ)\)\) ;

Discretization

ξ → (Subscript[ξ, j]) := j Δ + (Subscript[ξ, 0]);    j=1,2,..,n;  Δ = ((Subscript[ξ, max])-(Subscript[ξ, 0]))/n

Φ(ξ) → (Subscript[Φ, j]) := Φ((Subscript[ξ, j]))

- ((Subscript[Φ, j - 1])  -  
                2  (Subscript[Φ, j])  +  (Subscript[Φ, j + 1]))/(Δ^2)+ σ Θ(|(Subscript[ξ, j])|-α) \(TraditionalForm\`Φ\_j\ \  = \ ϵ\ \ Φ\_j\)

Hamilton-Matrix

Definition

Clear["Global\.b4*"];
V[ξ_, α_, σ_]   =  σ UnitStep[α - 
              Abs[ξ]];
\(\(M[n_, ξ0_, ξ1_, α_, σ_] := 
        Block[{H, 
            a, Δ}, Δ = \ \((ξ1 - ξ0)\)/
              n; 
							a = 
            1/Δ\^2; 
				H = 
            Table[If[i == j, 
                2  a\  + \ 
                  V[ξ0\  + \ i\ Δ\ , α, σ], 
                If[Abs[i - j] == 1, \(-a\), 0]], {i, 1, n}, {j, 1, 
                n}];
			];\)\)

Meaning of Parameters:
n:		dimension of Hamilon-matrix
Δ:		step size of finte differences in spacial derivatives

Display Matrix

p[α_, σ_] :=  
        Plot[V[x, α, σ], {x, -2 α, 
            2  α}];
M[20, -2, 2, 1, 1] // MatrixForm
p[1, -1]

Dynamics

Definition:

OneStep[DM_, x_, Δ_, Φ_, n_, ξ0_] := 
        Block[{}, 
          ListPlot[Transpose[{x, Abs[Φ]^2}], 
            PlotRange -> {{ξ0, n Δ  +  ξ0}, {0, 
                  1}}]; DM . Φ];

\(\(dyn[n_, ξ0_, ξ1_, α_, σ_, tmax_, dt_, x0_, 
          k_, σ0_] := 
        Block[{nt = tmax/dt, Δ = \((ξ1 - ξ0)\)/n}, 
          H = M[n, ξ0, ξ1, α, σ]; 
									x = 
            Table[ξ0\  + \ i\ Δ, {i, 1, 
                n}]; 
									Φ = \(\(E\^\(-\(\((x - x0)\)\^2\/\(2\ σ0\^2\)\)\)\) E\^\(\(-I\)\ k\ x\)\)\/\((π\ σ0)\)\^\(1/4\)\ ; 
Plot[
            V[x, α, σ], {x, ξ0, ξ1}]; 
									DM = 
            MatrixExp[I\ dt\ H]; 
									\ Table[\(Φ = 
                OneStep[DM, x, Δ, Φ, 
                  n, ξ0]\ ;\)\ , {i, 1, 
              nt}]
								];\)
\)
\(\(DynMean[n_, Δ_, tmax_, dt_, x0_, σ_] := 
        Block[{x, Φ, DM, tmean, tarr, tp, p1, p2nt = tmax/dt, 
            tp}, 
									H\ \ \ \ \ \ \ \ \ \ \ \ \  = M[n, Δ]; 
									x\ \ \ \ \ \ \ \ \ \ \ \ \  = 
            Table[i\ Δ, {i, 1, 
                n}]; 
									Φ\ \ \ \ \ \ \ \ \ \ \ \ \  = \ E\^\(-\(\((x - x0)\)\^2\/\(2\ σ\^2\)\)\)\/\((π\ σ)\)\^\(1/4\)\ ; 
									DM\ \ \ \ \ \ \ \ \ \  = 
            MatrixExp[I\ dt\ H]; 
									tmean\ \ \  = 
            Table[Φ = DM . Φ; 
              x\  . Abs[Φ]^2/\((Abs[Φ]\  . 
                    
                    Abs[Φ])\), {nt}]; 
									tarr\ \ \ \ \  = \ 
            Table[\((nt - 1)\) dt, {nt, 1, 
                tmax/dt}]; 
									tp\ \ \ \ \ \ \ \ \ \  = \ \@x0; 
									p1\ \ \ \ \ \ \ \ \ \  = Plot[x0 - \((Mod[t - tp, 2\ tp] - tp)\)\^2, {t, 0, 
                tmax}]; 
									p2\ \ \ \ \ \ \ \ \ \  = ListPlot[Transpose[{tarr, tmean}], 
              PlotRange -> {{0, tmax}, {0, 
                    x0*1.1}}]; 
									Show[{p1, 
              p2}];
							];\)

\)

Meaning of parameter:
n:		dimension of Hamilon-matrix
		in DynExact only needed for plot-resolution
Δ:		step size of finte differences in spacial derivatives
tmax:		time-evolution up to time tmax
dt:		time-steps for plot
x0:		center of initial gaussian wave-package
σ:		width of initial gaussian wave-package
Mmax:		max # of states included in spectral theorem

Time-evolution of gaussian wave-package

dyn[100, -10, 20, 4, 
      1, 10,  .01, -5, 4,  .8]