

Eigenfunctions f_{k} and probability
densities f_{k}^{2}.
f0[x_]:=Exp[x^2/2];
A[g_]:=Function[y,D[g[x],x]+x g[x] /. x>y];
l[f_]:=Integrate[f[x]^2,{x,Infinity,Infinity}];
S[f_]:=Module[{},ff[x_]:=Evaluate[
Simplify[A[f][x]/l[f]]];ff];
f[0_]:=f0; f[k_]:=S[f[k1]]; f[7][x]


The partial differential equation
d/dt f(t) = i T(f) = i (d^{2}/dx^{2} f(x) + x^{2} f(x) )

is called the Schrödinger equation for the quantum harmonic oscillator.
For each eigenvalue L_{n} with eigenfunction f_{n}
of T, the time evolution is f_{n}(t) = exp(i L_{n} t) f_{n}(0).
If f(0) = a_{1} f_{1} + a_{2} f_{2} + ...
then
f(t) = exp(i L_{1} t) f_{1}(0) + exp(i L_{2} t) f_{2}(0) + ...

We can write T(f) = P^{2} + Q^{2},
where P(f)(x) = i f'(x) and Q(f) = x f(x)(x). The Hamiltonian T has
the same structure as the Hamiltonian of the harmonic oscillator giving the
quantum system the name "quantum harmonic oscillator".
T has the eigenvalues L_{n} = 1+2n, with eigenvectors f_{n}, which
are recursively be defined by f_{n+1} = (xD) f_{n} starting with f_{0}(x)=exp(x^{2}/2).
Simlilar as in Fourier theory, it is possible to write any function f for which f^{2} has a
finite integral over the real line as a sum of such functions and so solve the Schrödinger evolution
explicitely.
