Mathematica unable to evaluate function which it can write explicitly in simple form

Question

I have a function $\psi(x,n)$ in mathematica which is quite complicated. However, plugging in $n=0$ gives a simple form. Mathematica knows this (see below), however it is still unable to even evaluate the function at a point. Doing so gives no errors but the output never returns, even though it is a calculation I can do in my head.

At this point I am stumped... here is the output which has me confounded:

Here is the function definition. $\psi$ is defined recursively as a function that remembers values it has found.

ψ0[x_] := 1/(2 π)^(1/4) Exp[-(x^2/4) - I k x];
ψr[x_?NumericQ, n_?NumericQ] := ψr[x, n] = 
   NIntegrate[
    SK[y, x, Δt] (
      Projector[x]/
      Sqrt[NIntegrate[Abs[Projector[x] ψr[x, n - 1]]^2, {x, -∞, ∞}]]
      ) ψr[x, n - 1], {y, -∞, ∞}];
ψr[x_, 0] := ψ0[x];

Here are the definitions which indirectly enter into the definition of the function:

ε = 0.2; 
Δt = 1.0;
L = 8.0;
tTypical = 10.0;
tMax = 20.0;
k = L/tTypical;
nMax = Floor[tMax/ Δt];
Projector[x_] := 1/(1 + Exp[-((L - x)/ε)]);
SK[x1_, x2_, t_] := 1/Sqrt[2 π I t] Exp[-((x1 - x2)^2/(2 I t))];

Edit: Here is a stack trace of the evaluation using TracePrint[ψr[0,0]]. Mathematica seems not smart enough to just plug in x, rather it is trying to do an integration. An elegant solution doesn't come to mind.

---

Answer 1

Order of definition matters, oddly enough. If you switch the order of definitions, things will work OK. I always define the specific cases first, then the general.

ψ0[x_] := 1/(2 π)^(1/4) Exp[-(x^2/4) - I k x];

ψr[x_, 0] := ψ0[x];

ψr[x_?NumericQ, n_?NumericQ] := ψr[x, n] = NIntegrate[SK[y, x, Δt] (Projector[x]/
   Sqrt[NIntegrate[Abs[Projector[x] ψr[x, n - 1]]^2, {x, -∞, ∞}]]) ψr[x,  n - 1], {y, -∞, ∞}];

ψr[10, 0]

$$ \frac{e^{-25-10 i k}}{\sqrt[4]{2 \pi }} $$

Set the constants and functions

ε = 0.2;
Δt = 1.0;
L = 8.0;
tTypical = 10.0;
tMax = 20.0;
k = L/tTypical;
nMax = Floor[tMax/Δt];
Projector[x_] := 1/(1 + Exp[-((L - x)/ε)]);
SK[x1_, x2_, t_] := 1/Sqrt[2 π I t] Exp[-((x1 - x2)^2/(2 I t))];

Try again

 ψr[10, 0]
(* -1.27631*10^-12 - 8.67854*10^-12 I *)

Note that this does not work

ψr[1, 2]
(* aborted *)

EDIT

For your function ψr[x, n] you have x as an input and also x as the iterator for the NIntegrate. Not sure what exactly you mean, but this runs...note the xx as an iterator, replacing x.

ψr[x_?NumericQ, n_?NumericQ] := ψr[x, n] = NIntegrate[SK[y, x, Δt] (Projector[x]/
   Sqrt[NIntegrate[Abs[Projector[xx] ψr[xx, n - 1]]^2, {xx, -∞, ∞}]]) ψr[x,  n - 1], {y, -∞, ∞}];

Running...

ψr[2, 2]
(* -0.00678479 - 0.23226 I *)