The Variational Theorem in Quantum Mechanics

The Schrödinger equation is separable only for hydrogen-like atoms, with a single electron. For systems with multiple electrons, the interelectronic repulsion term present in the potential prevents the equation from being solved analytically. In these cases, trial functions, dependent on variational parameters, are used to calculate an approximation to the system’s ground-state energy.

But how should we vary the parameters to get close to the true energy? Is there some defined criterion? The answer is yes, and it’s called the variational theorem. In this article, we will demonstrate it.

Let E_0 denote the true value of the ground-state energy, the lowest eigenvalue of the Hamiltonian \hat{H}. Let us now define the following integral:

\begin{equation}I=\int_{-\infty}^\infty\varphi^*(\hat{H}-E_0)\varphi\,d\mathbf{x}\end{equation}

where \varphi is a variational test function and the integration extends over the entire space for all independent variables x_i, being d\mathbf{x}=\prod_i x_i. If \varphi is normalized, that is,

\int_{-\infty}^\infty\varphi^*\varphi\,d\mathbf{x}=1

then I is reduced to

I=\int_{-\infty}^\infty\varphi^*\hat{H}\varphi\,d\mathbf{x}-E_0\int_{-\infty}^\infty\varphi^*\varphi\,d\mathbf{x}=\int_{-\infty}^\infty\varphi^*\hat{H}\varphi\,d\mathbf{x}-E_0

Let \psi_i and E_i be the true eigen functions and true eigen values ​​of \hat{H}

\begin{equation}\hat{H}\psi_i=E_i\psi_i\end{equation}

If the \psi_i form a complete set and \varphi satisfies the same boundary conditions as the \psi_i, then we can develop \varphi as the following linear combination

\begin{equation}\varphi=\sum_k a_k\psi_k\end{equation}

Replacing (3) in (1)

\begin{equation}I=\int_{-\infty}^\infty\sum_k a^*_k\psi^*_k\sum_j (\hat{H}-E_0)a_j\psi_j\,d\mathbf{x}=\sum_k\sum_j a^*_k a_j\int_{-\infty}^\infty (\hat{H}-E_0)\psi^*_k\psi_j\,d\mathbf{x}\end{equation}

Using (2), considering \psi_i orthonormal, and taking advantage of the properties of the Kronecker delta, \delta_{kj}, we have

I=\sum_k\sum_ja^*_ka_j(E_j-E_0)\delta_{kj}=\sum_k|a_k|^2(E_k-E_0)

All terms of the final sum are positive, since E_k-E_0\ge0, with E_0 being by hypothesis the lowest eigen value. This gives us the final expression of the variational theorem.

\int_{-\infty}^\infty\varphi^*\hat{H}\varphi\,d\mathbf{x}\ge E_0

If \varphi is not normalized, the variational theorem takes the form

\begin{equation}\int_{-\infty}^\infty\frac{\varphi^*\hat{H}\varphi}{\varphi^*\varphi}\,d\mathbf{x}\ge E_0\end{equation}

The procedure is developed using a trial function \varphi that depends on a series of parameters. These parameters are then varied to obtain the lowest value of the integral (5).