Ritz Variational Principle

Given the same Hamiltonian \(\mathcal{H}\), the energy of an arbitrary (normalized) state \(|\psi\rangle\) is guaranteed to be no lower than the ground-state energy, simply because it mostly likely contains some components of an excited-state.

Suppose the normalized eigenvectors and eigenvalues of the \(\mathcal{H}\) are \(\{|\psi_i\rangle\}\) and \(\{E_i\}\), respectively. I will assume \(i=0\) labels the ground state, and \(i\) can go up to \(\infty\) for all I care. An arbitrary normalized state can be written as a linear combination of the ground and excited states since the Hamiltonian has full rank

$$ |\psi\rangle=\sum_i \alpha_i |\psi_i\rangle, $$

with \(\sum_i |\alpha_i|^2=1\). The energy of this state can be calculated as

$$ E=\langle \psi |\mathcal{H}|\psi\rangle =\sum_i |\alpha_i|^2E_i. $$

By definition, the ground-state energy is the lowest eigenvalue \(E_0\le E_i~\forall i\). Therefore

$$ E = \sum_i |\alpha_i|^2E_i \ge \sum_i |\alpha_i|^2E_0 = E_0. $$

Q.E.D.

An important correct statement one can make from the above proof is that: IF we use enough parameters in a trial wave function such that it can span the ENTIRE space of wave functions, THEN after minimizing the energy of this wave function, (and thereby obtaining the best parameters) we WILL obtain the ground-state wave function and energy.

This is the motivation for the Variation Monte Carlo method. Of course, what we actually do is imperfect. In the most naive implementation of VMC, we use a small number of parameters in our trial wave function and ASSUME that after energy minimization we obtain the “closest” wave function in our parametrized subspace to the true ground-state wave function. We also try to minimize the variance of the wave function, which is defined in the context of Monte Carlo integration. From a mathematical perspective, it seems absurd to use two scalar quantities to define the closeness of wave functions, which often live in a vast Hilbert space of hundreds if not thousands of dimensions. However, in practice, given a well-parametrized trial wave function, energy and variance optimization almost always give a VERY good approximation to the true ground state.