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 $$E_n$$s, 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 $$c_n$$s 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.