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I'm trying to solve the Schrödinger equation using NDSolve in a case where there is a potential box barrier. The initial condition is a cosine wave, and for boundary conditions I require that the solution on the ends of domain is 0. Essentially I am encapsulating a potential barrier within infinite walls. (I am also hoping this requirement allows for reflections off the walls.) I'm getting a solution that oscillates but it doesn't behave like what I would expect. Increasing the barrier size I expect the wavefunction to go to 0 within the barrier, but instead the oscillations that happen in the barrier are more frequent the larger the barrier. This doesn't make sense, and I don't know of a different way of solving the equation in Mathematica.

My code is this:

ClearAll["Global`*"]
e = 1; (* energy *)
k1[e_] = Sqrt[2 e];

s = NSolve[Cos[k1[e] x] == 0, x, WorkingPrecision -> 16];
L = N[s[[1, 1, 2, 1]] /. C[1] -> -5, 16] (* left boundary *)
R = N[s[[1, 1, 2, 1]] /. C[1] -> 5, 16] (* right boundary *)

width = 2; (* potential height and width *)
height = 2;
a = 1/Sqrt[Integrate[(Cos[k1[e] x])^2, {x, L, R}]]; (* normalization of input *)

eqn = D[ψ[x, t], t] == 
I/2 D[ψ[x, t], {x, 2}] - 
I*ψ[x, 
  t]*(Piecewise[{{0, x <= 0}, {height, 0 < x < width}, {0, 
     x >= width}}]);

bc = {ψ[L, t] == 0, ψ[R, t] == 0};

ic = {ψ[x, 0.] == a Cos[k1[e] x]};

sol = NDSolve[{eqn, bc, ic}, ψ, {x, L, R}, {t, 0, 20}, 
Method -> {"MethodOfLines", 
"SpatialDiscretization" -> {"TensorProductGrid", 
"DifferenceOrder" -> "Pseudospectral"}}, AccuracyGoal -> 10, 
PrecisionGoal -> 10]

Manipulate[
Plot[{Re[ψ[x, t] /. sol], 
Piecewise[{{0, x <= 0}, {height, 0 < x < width}, {0, 
x >= width}}]}, {x, L, R}, PlotRange -> {-2, 3}], {t, 0, 20, 
0.1}]

I'm using Mathematica 11 on Linux. Any help or direction is appreciated.

Edit: Complex valued 2+1D PDE Schrödinger equation, numerical method for `NDSolve`? This is a similar post, and one of the answers implements Dirichlet boundary conditions. I'm having trouble understanding how the boundary conditions were constructed but they are probably important to solving my problem.

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  • 1
    $\begingroup$ What do you mean by "Increasing the barrier size"? Make height larger, or width larger? $\endgroup$ – xzczd Nov 26 '16 at 13:09
  • $\begingroup$ Make height larger $\endgroup$ – Buddhapus Nov 26 '16 at 18:25
  • $\begingroup$ According to your description, I guess we can use C[1] -> -1 for L, C[1] -> 1 for R, too? $\endgroup$ – xzczd Nov 27 '16 at 3:25
  • $\begingroup$ You could. I was just trying to force the boundary conditions on the initial condition. I'm not sure how else I would do that. $\endgroup$ – Buddhapus Nov 27 '16 at 5:03
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This problem can be solved quite quickly and accurately by computing its eigenfunctions, projecting the initial condition onto the eigenfunctions, and then summing them. With parameters,

e = 1; k1[e_] = Sqrt[2 e];L = -(2 Pi + Pi/2)/ k1[e]; R = (2 Pi - Pi/2)/ k1[e]; 
    w = 2; a = 1/Sqrt[(R - L)/ k1[e]]; ic = a Cos[k1[e] x]

the solution is given by

ss[hh_, t_] := 
  Module[{ms, mc, op = - D[ψ[x], {x, 2}]/2 + ψ[x]*Piecewise[{{hh, 0 < x < w}}, 0]}, 
    ms = NDEigensystem[{op, DirichletCondition[ψ[x] == 0, True]}, ψ[x], {x, L, R}, 30, 
      Method -> {"SpatialDiscretization" -> {"FiniteElement", 
      {"MeshOptions" -> {"MaxCellMeasure" -> (R - L)/1000}}}}]; 
    mc = Table[Quiet@NIntegrate[ms[[2, i]] ic, {x, L, R}, 
      Method -> "ClenshawCurtisRule", MaxRecursion -> 18], {i, 30}]; 
    mc.(Exp[-I First@# t] Last@# & /@ Transpose@ms)]

sol[2] = ss[2, 20];
sol[5] = ss[5, 20];
sol[8] = ss[8, 20];
Plot[Evaluate[Re@sol[#] & /@ {2, 5, 8}], {x, L, R}, PlotRange -> 1]

enter image description here

Increasing the number of eigenfunctions (here, 30) or decreasing the MaxCellMeasure (equivalent to about 1000 grid points) has no visible impact on the solution.

To illustrate the process in more detail, consider h == 2. The eigenvalues (the temporal frequencies of eigenfunctions) and the projection of the initial condition on the eigenfunctions are

ListLogPlot[First@ms]

enter image description here

ListPlot[mc, PlotRange -> All]

enter image description here

Eigenfunctions 3 and 4 are the primary contributors to the solution. This is so, because their wavelengths are comparable to the wavelength of the initial condition. The first eight eigenfunctions are plotted next.

Plot[Evaluate[ms[[2, 1 ;; 8]]], {x, L, R}, PlotLegends -> Range[8]]

enter image description here

The first three eigenfunctions are restricted mostly to the left of the barrier, because they do not fit in the region to the right of the barrier, although the third shows signs of leaking through. The fourth eigenfunction is quite large in the region to the right of the barrier, probably because a half-wavelength of the eigenfunction matches fairly well the width of the region on the right. (This behavior of eigenfunctions four becomes even more pronounced for larger h.) Higher modes, with frequencies exceeding h penetrate the barrier.

Incidentally, DEigensystem is unable to obtain a symbolic expression for the eigenfunctions, although a symbolic solution does exist and can be obtained using DSolve in each region separately and matching the results across the edges of the barrier. The resulting dispersion function still must be solved numerically.

Addendum

Sample temporal results can be obtained from

sol1[2] = ss[2, t];
sol1[5] = ss[5, t];
sol1[8] = ss[8, t];

Plot[Evaluate[Re@(sol1[#] /. x -> -2.5) & /@ {2, 5, 8} ], {t, 0, 20}]

enter image description here

Plot[Evaluate[Re@(sol1[#] /. x -> 2.5) & /@ {2, 5, 8} ], {t, 0, 20}]

enter image description here

Plot[Evaluate[Re@(sol1[#] /. x -> 1) & /@ {2, 5, 8} ], {t, 0, 20}]

enter image description here

As expected, behavior to the left of the barrier is dominate by the third eigenfunction, to the right by the fourth eigenfunction, and in the barrier by high frequency eigenfunctions that can penetrate the barrier.

Second Addendum - Large Domain

The computation above used a fairly small domain in order to facilitate comparison with the answer by xzczd. The domain presented in the question is substantially larger,

L = -(10 Pi + Pi/2)/ k1[e]; R = (10 Pi - Pi/2)/ k1[e]; 

Results for this case too can be obtained, although doing so requires n == 120 eigenfunctions and Method adjustments to NDEigensystem.

ss[hh_, t_] := 
  Module[{ms, mc, op = - D[ψ[x], {x, 2}]/2 + ψ[x]*Piecewise[{{hh, 0 < x < w}}, 0]}, 
    ms = NDEigensystem[{op, DirichletCondition[ψ[x] == 0, True]}, ψ[x], {x, L, R}, 120, 
    Method -> {"Eigensystem" -> "Arnoldi", "MaxIterations" -> 2000}, 
      "SpatialDiscretization" -> {"FiniteElement", 
        {"MeshOptions" -> {"MaxCellMeasure" -> (R - L)/1000}}}}]; 
    mc = Table[Quiet@NIntegrate[ms[[2, i]] ic, {x, L, R}, 
      Method -> "ClenshawCurtisRule", MaxRecursion -> 30], {i, 120}]; 
    mc.(Exp[-I First@# t] Last@# & /@ Transpose@ms)]

The resulting curves are somewhat different from those above,

Plot[Evaluate[Re@sol[#] & /@ {2, 5, 8}], {x, L, R}]

enter image description here

Because the barrier occupies only about 5% of the computational region, a blowup is helpful.

Plot[Evaluate[Re@sol[#] & /@ {2, 5, 8}], {x, -w, 2 w}]

enter image description here

Eigenfunctions in the range 16 - 23 dominate these plots, except inside the barrier, where higher frequency eigenfunctions dominate. Only about 15 minutes was required to compute these two plots.

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  • $\begingroup$ Wow beautiful job! Just to clarify you solve the eigenvalue/eigenvector problem for x, and then you manually add the time dependence. The last line of the module outputs a sum of: coefficients multiplied by the time dependence (where eigenvalues are the frequencies), multiplied by the eigenfunctions. So the solution is a superposition of eigenfunctions. $\endgroup$ – Buddhapus Nov 29 '16 at 17:53
  • $\begingroup$ @Buddhapus Exactly, although I would not describe adding the time dependence as "manual". I hope you find it helpful. $\endgroup$ – bbgodfrey Nov 29 '16 at 17:57
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You can use a appropriate 4th order spatial grid:

e = 1;(*energy*)
k1[e_] = Sqrt[2 e];
s = NSolve[Cos[k1[e] x] == 0, x, WorkingPrecision -> 16];
L = N[s[[1, 1, 2, 1]] /. C[1] -> -1, 16] (*left boundary*)
R = N[s[[1, 1, 2, 1]] /. C[1] -> 1, 16] (*right boundary*)    
a = 1/Sqrt[Integrate[(Cos[k1[e] x])^2, {x, L, R}]];    
bc = {ψ[L, t] == 0, ψ[R, t] == 0};
ic = {ψ[x, 0.] == a Cos[k1[e] x]};    
Clear@eqn;
width = 2;
eqn[height_] := 
  D[ψ[x, t], t] == 
   I/2 D[ψ[x, t], {x, 2}] - 
    I*ψ[x, t]*(Piecewise[{{0, x <= 0}, {height, 0 < x < width}, {0, x >= width}}]);

mol[n_Integer, o_: "Pseudospectral"] := {"MethodOfLines", 
    "SpatialDiscretization" -> {"TensorProductGrid", "MaxPoints" -> n, 
        "MinPoints" -> n, "DifferenceOrder" -> o}}    
showStatus[status_] := 
  LinkWrite[$ParentLink, 
   SetNotebookStatusLine[FrontEnd`EvaluationNotebook[], ToString[status]]];
clearStatus[] := showStatus[""];
opt = EvaluationMonitor :> showStatus["t = " <> ToString[CForm[t]]];    
solfunc[height_, points_, order_: "Pseudospectral", tbegin_: 0] := (clearStatus[]; 
  NDSolveValue[{eqn[height], bc, ic}, ψ, {x, L, R}, {t, tbegin, 20}, 
   Method -> mol[points, order], MaxSteps -> Infinity, opt])

sol[2] = solfunc[2, 550, 4]; // AbsoluteTiming
(* {17.080840, Null} *)
sol[5] = solfunc[5, 550, 4]; // AbsoluteTiming
(* {29.171855, Null} *)   
sol[8] = solfunc[8, 550, 4]; // AbsoluteTiming
(* {54.542178, Null} *)

Plot[sol[#][x, 20] & /@ {2, 5, 8} // Re // Evaluate, {x, L, R}, PlotRange -> 1]

Mathematica graphics

Remark

To obtain a reasonable result, the grid should be dense enough i.e. points should not be too small. However, it seems that "dense enough grid" isn't the only requirement, for example solfunc[8, 552, 4, 19] will lead to a catastrophically wrong result after a very slow solving process. I'm not sure why this happens.

I think it's relatively clear from the picture that the function value is going to 0 when height gets larger. If you still feel worried, you can check their integration:

NIntegrate[sol[#][x, 20] & /@ {2, 5, 8} // Re // Abs, {x, 0, 2}]
(* {0.245181, 0.176277, 0.0766969} *)
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  • $\begingroup$ Can you please explain why you chose an infinite number of steps but you specify the number of points? Is one for time and the other for space discretization? $\endgroup$ – Buddhapus Nov 27 '16 at 9:05
  • $\begingroup$ @Buddhapus Yes, MaxSteps->Infinity allows NDSolve to use infinite steps on the direction of t (The solution of your equation is so complicated that without this option NDSolve isn't able to evaluate to $t=20$ in 10000 steps), "MaxPoints" and "MinPoints" specify the number of points used for spatial discretization. Notice "method of lines" is a method solving PDE by discretizing it to a set of ODEs with finite difference formula. For more information you can search MethodOfLines in the document. $\endgroup$ – xzczd Nov 27 '16 at 9:21
  • $\begingroup$ Okay, thanks! I keep running and re-running it for a larger domain, and with more MinPoints and MaxPoints, but either of two things happen - either I run out of memory because there are too many points, or there is error that accumulates because there are too few points. In general the trends definitely look better though. Before I accept your answer would you have any recommendations to help larger domain calculations? The only thing I can think of to try is increase the memory that Mathematica uses on my computer. Not sure of the default. $\endgroup$ – Buddhapus Nov 27 '16 at 23:47
  • $\begingroup$ @Buddhapus Do you need the solution for the whole time range? If you just need the result at the end time, you can use something like solfunc[8, 550, 4, 20], this will save a lot of memory. Also, I'd like to point out (of course I think you've already noticed) even if the eerr warning comes out, the result is still not too bad when the error isn't too large. $\endgroup$ – xzczd Nov 28 '16 at 2:31
  • $\begingroup$ Nice computation. However, when I increased the grid size to 1000, the height == 2 calculation became very slow at around t == 4.8, and produced a seriously incorrect answer at t == 5. Perhaps, this is related to the issue you had with a grid size of 552. $\endgroup$ – bbgodfrey Nov 28 '16 at 2:39

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