If I found all the bound states for a certaing potential in 2 or 3 dimensions (numerically), can I immediately obtain some information about the propagating solutions for the same potential (such as transmission and reflection coefficients in certain directions, scattering angle distribution)? I will gladly accept the answer for 1 dimensional problem as well.

The thing is, the numerical approach I use is very convenient, but only for the bound states since it uses the wave function expansion in the basis of eigenfunctions of quantum box or harmonical oscillator. This means that while I can easily obtain all the bound states with arbitrary accuracy, I can't solve for the propagating states. Or can I?

If you know some good approach that is useful mainly for finding the propagating solutions numerically, I would be grateful too.

Edit based on the comment below.

For clarification, the method I use to find the energy level and wavefunctions of the bound states can be summarized as follows:

We expand the wavefunction using the complete orthonormal basis of the known solutions (quantum box for example, any other basis can be used as well).

$$ \Psi( \vec{r} )=\sum_{j,k,l}^\infty C_{jkl} \psi_{jkl}( \vec{r} ) $$

We substitute this expansion into Schrodinger equation, calculate all the matrix elements then numerically solve the resulting matrix equation for $\{C_{jkl}\}$ and the corresponding eigenenergies $E_{jkl}$ for some finite number of basis functions $N$.

This method was first proposed in 1988 for nanowires and is very useful because there is no need to explicitly define the boundary conditions (they are incorporated during the calculation of matrix elements $<\psi^*_{JKL}| U(\vec{r})|\psi_{jkl}>$), it gives all energy levels and normalized wavefunctions and can be used for almost any potential in any number of dimensions.

But it seems to me that I can't use it to find the propagating wavefunction for $E>0$ because the system is still bound because we use the finite expansion, and our basis potential well kind of 'surrounds' the electron even if its energy is positive.

So, do I need to use a completely different method to calculate (for example) the scattering of electrons by the layer of quantum dots and such?

  • $\begingroup$ This would get more attention if you showed some of your work in the question. $\endgroup$ – Gert Oct 2 '15 at 13:46
  • $\begingroup$ Thank you for the advice. I elaborated on the method and the question, but I do not see the point of showing any calculations here. The question is general enough. $\endgroup$ – Yuriy S Oct 2 '15 at 14:48
  • $\begingroup$ You can use a similar method, by imposing periodic boundary conditions you get a basis of propagating states and you can then calculate the transmission and reflection amplitudes. $\endgroup$ – Count Iblis Oct 2 '15 at 21:52
  • $\begingroup$ @CountIblis, thank you very much! It's just what I needed. And it will work in 2D and 3D too! $\endgroup$ – Yuriy S Oct 3 '15 at 11:45

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