Solving Schrödinger equation numerically we will have finite size matrix, which has finite number of eigenvalues, that is - discrete spectrum. But we know, that atoms also have continuous spectrum. Is it possible somehow modify algorithm for quantum mechanic calculations to account continuous spectrum?
Not that I know of.
A continuous spectrum is obtained through theory calculations, when you assume plane wave for free electrons say, this implies an infinite wave extent.
Alternatively , in the tight binding approximation, you
broaden the individual atomic levels due to interaction with
the neighbours, this gives rise to bands. Again you use an infinite crystal.
Boundary conditions will restore a discrete spectrum. This can also be done theoretically.
But if there is say disorder that does not make a perfect crystal then you resort to numerical calculations, using finite size samples.
You can use a perfect crystal of finite size, but you will always get a discrete spectrum.
This is not correct.
1. Atoms has two types of spectrum: discrete when E0. Wave functions in both cases goes to zero if r -> infinity.
2. Continuous spectrum is important in particles scattering.
Do you want to tell me, that if we will think, that our Universe has finite size, we will not have continuous spectrum (like quantize in the box)?
Well, correct, but then you talk about conduction electrons
free from the atoms, this does not usually need detailed calculation. You use the well known theory of nearly free electrons, as in metals. Im talking about solids, not single atoms.
I cannot imagine numerical calcultions for electrons free from individual atoms either. The spectrum would be continuous, size would not be enough and eigenvalues too many.
something like collision theory might hold.
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