Are normalization constants of wave equation time dependent?

[singularity93]

The wave function solution psi is a function of time and position. Hence the integral of its square over all x will, in general, give a function of time. To normalize this, we must multiply with the inverse of the function. Therefore it seems that the normalization constant does not remain constant over all time. Please tell if my understanding is correct or whether I have made a mistake somewhere.

 

[dextercioby]

The trick is that probability conservation in time forces stationary states into an exp (i Et) time factor which can be 'absorbed' into the normalization constant, if you please.

 

[singularity93]

That the time is an exp(i Et) is true only in stationery states. What will happen when the wave equation isn't separable? It may not simply get absorbed while normalizing anymore.

 

[atyy]

The Schroedinger equation is unitary, so it will preserve the length of the vector, so normalizing once is enough.

Actually, this is the same as dextercioby's reply. For a time-independent Hamiltonian, the Schrödinger equation is something like |ψ(t)> = exp(iHt)|ψ(0)>, which is unitary. Because the Hamiltonian H is generates time evolution, one can expand the initial state in energy eigenstates, which will each then evolve with exp(iEt).

There's a slightly different formula for time-dependent Hamiltonians, but the idea is the same:
http://ocw.mit.edu/courses/nuclear-...ng-2012/lecture-notes/MIT22_02S12_lec_ch6.pdf