Eigenfunction, Eigenvalue, Wave Function and collapse

[birulami]

Reading Sam Treiman's http://books.google.de/books?id=e7fmufgvE-kC" he nicely explains the dependencies between the Schrödinger wave equation, eigenvalues and eigenfunctions (page 86 onwards). In his notation, eigenfunctions are u:R^3\to R and the wavefunction is \Psi:R^4\to R, i.e. in contrast to the eigenfunctions it depends on time.

Then on page 94 he says:

Whatever the state of the system was just before the measurement, during the measurement process it "collapses" into the eigenstate u that corresponds to the eigenvalue \lambda obtained in the measurement.

With "state of the system" he refers of course to \Psi, so during the measurement, the jump or collapse is from \Psi to u.

The one thing I don't understand here is: u does not depend on time, so how is the development of the new \Psi over time governed? Is it that every solution of the Schrödinger equation is uniquely determined as soon as the value at just one point in time is known?

Harald.

 

[Nugatory]

Yes, the time-dependent Schrodinger equation is sufficient to determine the future evolution of the wave function if its value is known at a particular time.

The easiest way to do this is generally to write the initial wave function as a sum of the energy eigenfunctions you find by solving the time-independent Schrodinger equation, because the time evolution of these eigenfunctions is particularly simple: $\psi_N(x,t)=\psi_N(x)e^{-iE_Nt/\hbar}$ where $\psi_N(x)$ is an eigenfunctions of the time-independent equation with eigenvalue $E_N$: $H\psi_N(x)=E_N\psi_N(x)$.

It is also worth googling for "quantum time evolution operator"