Recent content by R P Stone

@BvU @Isaac0427 @vanhees71 @A. Neumaier Any new ideas? Please help! Thanks!

Really many thanks. But I don’t think that strategy works here. That average would not yield zero for the integral.

Thanks! But I don't have the book at hand. Would you sketch how to do it?

Unfortunately, it won't work. You'll run into the same kind of problems described in in the original question. In fact, it has to do the principal parts @ vanhees71.

The wave function has to satisfy $$\psi'(0_+) - \psi'(0_-) = - 2mV_0\psi(0)/\hbar^2.$$ Using your expression, $$\psi'(0_+) = N_k \sqrt{1-R_k} ik, \quad \psi'(0_-) = N_k(1-\sqrt{R_k}) ik, \quad \psi(0) = N_k \sqrt{1-R_k}.$$ Substituting this in the above condition leads to $$1 - \frac{1 -...

Many thanks! But you can’t assume that ##R_k## is real. It is in general complex.

@Isaac0427 @A. Neumaier : Taking Isaac's wave function, $$I = \int^\infty_{-\infty} dx ~ \psi^*_{k'}(x) \psi_k(x) = I_> + I_<,$$ where $$I_> = \int^\infty_{0} dx ~ \psi^*_{k'}(x) \psi_k(x), \quad I_< = \int^0_{-\infty} dx \psi^*_{k'}(x) \psi_k(x) .$$ Substituting the expressions, $$I_> = N^2_k...

@A. Neumaier I could not get @Isaac0427 ‘s result. I ran into the same problem as described in the question.

@A. Neumaier Thanks. That’s what I did, but it did not work out.

@Isaac0427 really appreciate your effort. While the wave function you gave is certainly proper, the one I gave is also legitimate. yours can be obtained by combining an odd and an even parity wave functions mentioned in my question. i would like to have wave functions with definite parity.

@Isaac0427 thanks for your help. As you might see in my question, I’ve already tried what you suggested, but went nowhere.

@vanhees71 thanks for your suggestion. Actually I tried it but it did not work: I could not get rid of the principal parts.

You mean they can not be normalised even to the delta function? Thanks.

Thanks for your suggestion. However, the article deals with bound states and the regularization issue seems present only for D>1. Moreover, such issue does not exist if one looks at a discrete version of the Hamiltonian. Whatever, the wave functions must be orthogonal, as physically required...

Hi, thanks for your responses. I know that they are not normalized to unity, but they can still be normalized to Dirac function, which is what's sought for.