Using the Variational Method to get higher sates

[ExplosivePete]

TL;DR

An intro quantum course will teach you the Variational Method for approximating the ground state by minimizing an ansatz. But what about higher states?

In a typical quantum course we learn how to approximate the ground state of a particular Hamiltonian by making an educated guess at an ansatz with a tunable parameter then calculating the expectation energy for the ansatz. The result will depend on the tunable parameter if done correctly. Then we can minimize the energy with respect to that parameter, and that can give nice results approximating the ground state and energy. Cool stuff.

I have been wondering about what kind of algorithm we could cook up to find higher energy states. Say you have some Hamiltonian, H. We can apply the Variational method to get the ground state ψ₀ and E₀. Since the true energy states span the hilbert space, then we would hope that the expansion of ψ₀ would mostly be composed of the true ground state. We could then apply the variational method again, but this time choose an ansatz, ψ₁, which is orthogonal to ψ₁. Assuming that ψ₀ was a good guess, then ψ₁ will be "mostly" orthogonal to the true ground state. Then the application of the variational method will result in an energy that will be closer to the first excited state energy. In fact, it would be easy to prove that if ψ₀ is the exact ground state, then the energy of ψ₁ will be bounded below by the first excited state energy.

This kind of has the feel of applying the Graham Schmidt process for finding orthogonal basis vectors, except the resulting basis is ordered by the energy. I assume there is some sort of algorithm out there to do this. The tricky part is how to handle the step of choosing the ansatz.

Let me know your thoughts and if you have seen something like this before.

 

[Orodruin]

The easier way of doing this is to start by picking a set of functions that you can guess are going to be close to spanning the ground state and the first excited state. You can then compute the matrix elements of the Hamiltonian in this subspace and you will get a 2x2 matrix that you can diagonalise to get approximations for the energies.

If you wish you can always let the functions in this procedure depend on some parameters and minimize the energies with respect to those parameters as well.

 

[hilbert2]

To avoid having the ansatz being "accidentally" orthogonal to the first excited state too, it's good to remember how the number of nodes depends on the level of excitation. At least this works in single-particle problems where there are no complications of making the wave function antisymmetric in the exchange of identical fermions.

 

[ExplosivePete]

Good point. There are a lot of nice tricks that would be nice to see summarized somewhere as far as techniques for choosing the ansatz. It is also useful to look at when the particle is in a "classically restricted region" (E < V(x)). In these regions, the wave function will be concave down (as easily shown from the Schrödinger equation).

Coming up with the trial functions can be a fun game, but unfortunately most assigned problems tell you what ansatz to use, which is all the fun. The rest is tedious calculation.