Variational Theorem

The expectation value of the Hamiltonian, computed with any trial wave function, is always higher or equal than the energy of the ground state. Mathematically,

$<\psi\vert\hat{H}\vert\psi> \geq E_0$,

where $\hat{H}\phi_j =E_j\phi_j$.

Proof:

$\psi=\sum_j C_j \phi_j$, where $\{\phi_j\}$ is a basis set of orthonormal eigenfunctions of the Hamiltonian $\hat{H}$.

$$\begin{aligned}<\psi\vert\hat{H}\vert\psi> &= \sum_j \sum_k C_k... ... C_j E_j \qquad \geq \qquad E_0 & \sum_j C_j^* C_j, \end{aligned}$$

where, $\sum_j C_j^* C_j=1$.

Variational Approach

Starting with an initial trial wave function $\psi$ defined by the expansion coefficients $\{C_j^{(0)}\}$, the optimum solution of an arbitrary problem described by the Hamiltonian $\hat{H}$ can be obtained by minimizing the expectation value $<\psi\vert\hat{H}\vert\psi>$ with respect to the expansion coefficients.