Finding Spectrum of Hamiltonians

[curtdbz]

I wanted to know if there was a standard way of finding out the spectrum of a Hamiltionian given a specific $$H$$. For example, $$H = -\Delta - 10|x|^{3} + |x|^{4}$$ or $$H = -\Delta -(1+|x|)^{-2}$$.

I don't need answers to the above, I just want to know if one can describe the spectrums of Hamiltinian's such as those, in an easy fashion. I'm looking at examples from the textbook and something as simple as the operator $$x$$ or even momentum $$p$$ have a fairly complicated derivation to show their spectrum is $$[0,\infty)$$.

Does the variation principle come into play at all? Can anyone find examples on the web, like a ton of them, so that I can learn and get the hang of it? Thanks!

 

[cgk]

As long as you are dealing with one-dimensional one-body problems, the simplest way of getting an impression of what is going on is probably to just code up a scheme to calculate the eigenvalues numerically on a fixed grid (with a finite-difference approach). In python with scipy or MATLAB this can be done in less than 100 lines of code.

Say, you have an finite x space of -5.0 to 5.0 in steps of h=0.01, denoted as x_i, then you can setup the (approximate) Hamiltonian as
<i|H|i> = V(x_i) + 2*z
<i|H|j> = -z | i is next to j
<i|H|j> = 0 | otherwise
where i,j denote grid points and z = 0.5 * h^2 and V(x_i) is the potential function at grid point x_i.
Then you can just use a dense matrix diagonalization routine to get the eigenvalues.

This is not exactly a mathematical proof, but it might give you some impression of the behavior of the system.

 

[curtdbz]

As long as you are dealing with one-dimensional one-body problems, the simplest way of getting an impression of what is going on is probably to just code up a scheme to calculate the eigenvalues numerically on a fixed grid (with a finite-difference approach). In python with scipy or MATLAB this can be done in less than 100 lines of code.

Say, you have an finite x space of -5.0 to 5.0 in steps of h=0.01, denoted as x_i, then you can setup the (approximate) Hamiltonian as
<i|H|i> = V(x_i) + 2*z
<i|H|j> = -z | i is next to j
<i|H|j> = 0 | otherwise
where i,j denote grid points and z = 0.5 * h^2 and V(x_i) is the potential function at grid point x_i.
Then you can just use a dense matrix diagonalization routine to get the eigenvalues.

This is not exactly a mathematical proof, but it might give you some impression of the behavior of the system.

That's great, thanks so much! However, I can't use programming for this. Since they might be potential test questions or exam questions, and we're just supposed to use a pen and paper only. So I'm looking for a more mathematical way of solving these problems, fairly rigorously. In the future I will for sure use your python method though! Thanks

 

[martinbn]

My understanding of these questions is very superficial, so I may be wrong, but I think it is a safe bet that in general these problems are very hard. In your examples the potential depend only on |x| so it is invariant under rotations and it may be a good idea to use spherical coordinates. Also the whole space will decompose as a sum of irreducible representations of the rotation group and you can restrict to those subspaces and get ordinary differential equations, which i presume will still be hard. You can take a look at the case of the Coulomb potential, which should be in most books.

 

[genneth]

Finding the spectrum of a Hamiltonian amounts to completely solving the system, so in general it is not possible at all, analytically. For one dimensional problems this can often be reduced to considering the mathematical properties of some differential operator on a given interval, which is a fairly well studied (i.e. extremely hard) field.