- #1
ExplosivePete
- 13
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- TL;DR Summary
- 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.
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.