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.

 

[denian]

1. The expectation value of an operator Q can be calculated using the formula <Q>=<psi|Q|psi>, where |psi> is the wavefunction and Q is the operator. This formula holds even if the wavefunction is not an eigenfunction of Q. In this case, the wavefunction can be expressed as a linear combination of eigenfunctions of Q and the expectation value can be calculated by taking the sum of the expectation values for each eigenfunction with their respective coefficients.

2. Yes, you are correct in your calculation for \phi_{II}(k). The Fourier transform can be used to calculate the wavefunction in any region, including between (-L,L).

3. The coefficients c_n represent the probability amplitudes for each eigenfunction in the wavefunction. They can be calculated by solving the time-independent Schrodinger equation and applying normalization conditions, such as the sum of the probabilities equaling 1. A realistic example would be the calculation of the coefficients for a particle in a one-dimensional infinite square well potential. The coefficients can be found by solving the Schrodinger equation and applying normalization conditions, and the resulting wavefunction will have a finite number of non-zero coefficients.