Basic questions about QM computations

[gulsen]

1. How can we calculate expectation values of an arbitrary Q, even if \psi is not an eigenfunction of Q?

2. (Fourier transform related) Suppose I have piecewise wavefunction. \psi_{I} at (-\infty,-L), \psi_{II} at (-L,+L) and \psi_{III} at (L,+\infty). I can compute entire \phi(k) by taking the Fourier transform's integral from -\infty to +\infty. But what if I try to calculate \phi(k) only between (-L,L)? Is it \phi_{II}(k) = \int_{-L}^L \psi(x) e^{ikx}dx?

3. After I solved the time-independent SE, I get a series of solutions. I plug the time dependent part after I find E_ns, and \Psi(x,t) = \sum c_n \psi(x)e^{-i E_n t/\hbar}, where \sum c_n = 1 But how on Earth do I get c_n. Is there a realistic example (i.e., I'm not talking about examples "let's say c_0 = 0.3 and c_2 = 0.7, calculate bla bla bla") where c_ns are computed by us?

Thanks in advance!

 

[dextercioby]

3. The coefficients are computed from the normalization condition for the wave-function. The typical examples are the H-atom amd the 1-dim harmonic oscillator.

2. Yes.

1. Expand \psi in terms of (possibly generalized) eigenfunctions of Q.

Daniel.