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 Schrödinger 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 Schrödinger 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"