Why energy expectation values (Energy) are functions of coefficients of corresponding trial wave functions in variational principle?

Question

The method consists of choosing a "trial wavefunction" depending on one or more parameters and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy.

[Source: Wikipedia - Variational method (quantum mechanics)]

What is the physical interpretation of it?

Answer 1

I don't think I have a physical interpretation. But I like to think about it as a two-step process:

i. Showing that the expectation value of the energy for the ground state wavefunction is the lowest energy possible. The proof is elegant and short. Computing the inner product of the Hamiltonian (finding the expectation value of the energy) with any other wavefunction gives a higher energy.

ii. Using this principle in practice. This is hard because you don't know a priori what the ground state wavefunction looks like -- it could be any kind of crazy function! So one thing people do is assume some ansatz, meaning you assume it has a certain form. Because we have algorithms to help us minimize functions, we find the minimum energy with that ansatz. This cannot be lower than the global minimum because the wavefunction that achieves the global minimum may not be of the same form.

I like to think about it like approximating functions. A function can have any form you want. Introducing an ansatz is like saying I want to approximate the function by a quadratic or cubic polynomial. There are some coefficients that you can optimize to make it resemble the function you want (in your case, you would want to use an algorithm like gradient descent so that the wavefunction you get has the lowest energy for that given ansatz).

Answer 2

The underlying principles are straightforward. The ground state is the solution of the Schrödinger equation with the lowest possible energy. One way to find a function that approximates the ground state is to create some form of expansion, such as a Fourier series, with variable expansion coefficients. Any given set of expansion coefficients equates to a particular wave function, for which you can readily calculate an associated energy expectation value. You can now vary the expansion coefficients to see what effect the variation has on the energy expectation value. If the change in the coefficients reduces the energy, then the expansion must have become closer to the ground state (ie to the function that has the lowest possible energy). In practice what you do is to program an algorithm which varies the coefficients in a systematic way to home in on a set that appears to minimise the energy expectation value. Once you've arrived at a set that seems to give the lowest energy value, then you can take the Fourier expansion with those coefficients as being your best approximation to the ground state.

I've mentioned a Fourier series because that's a familiar example of an expansion that most readers of this answer are likely to understand. In practice, however, you will pick an expansion that you think will be a good fit to the physical arrangement you are trying to model. For example, to model electrons in a spherical potential, plane waves are unlikely to be a good basis for an expansion, so you might use spherical Bessel functions, say, instead.