Variational method

The variational method is a method of finding approximate solution to quantum problems, typically finding eigenvalues and eigenfunctions of a time-independent Hamiltonian.

Method#

Consider some time-independent Hamiltonian $H$ of a function $\phi$ which can vary freely, called a trial function. We define the Functional:

E[\phi]=\frac{\braket{ \phi | H|\phi }}{\braket{ \phi | \phi } }

This is the mean value of $E$ in the state $\ket{\phi}$ . As usual, if $\ket{\phi}$ is an eigenstate, then $E(\phi)$ is an energy eigenvalue. The heart of the method is this:

To prove this, let's consider a small variation $\delta \phi$ of $\phi$ ¹:

E[\phi+\delta \phi]\braket{ \phi+\delta \phi | \phi+\delta \phi } =\braket{ \phi+\delta \phi | H|\phi+\delta \phi }

We now look for the eigenvalues of $E$ for the varied function:

\begin{align} \delta E\int \phi^{*}\phi d\tau&=-E\int \delta\phi^{*}\phi d\tau-E\int \phi^{*}\delta\phi d\tau+\int \delta \phi^{*}H\phi d\tau +\int \phi^{*}H\delta \phi d\tau \\ &=\int \delta \phi^{*}(H-E)\phi d\tau+\int \phi^{*}(H-E)\delta \phi d\tau \end{align}

But we want $E$ to be stationary, in which case $\delta E=0$ always. Thus, the left hand side must be zero. This is our condition to find $\phi$ ; the rest is "just" a matter of solving the equation.

For example, consider the Schrödinger equation $H\psi_{n}=E_{n}\psi_{n}$ , indexed by a quantum number $n\geq0$ . The ground state is $n=0$ with $E_{0}$ . We now consider a generic function $\phi$ . This can be expressed in Fourier series in the $\{ \psi_{n} \}_{n}$ basis as

\phi=\sum_{n}a_{n}\psi_{n}

The Expected value of $E(\phi)$ for this function is

E[\phi]=\frac{\sum_{n}\lvert a_{n} \rvert ^{2}E_{n}}{\sum_{n}\lvert a_{n} \rvert ^{2}}=E_{0}+ \frac{\sum_{n}\lvert a_{n} \rvert ^{2}(E_{n}-E_{0})}{\sum_{n}\lvert a_{n} \rvert ^{2}}\geq E_{0}

We hit on a lower bound. In fact, while this doesn't give us a solution, it does gives a bound to restrict what $E[\phi]$ can be:

\boxed{E[\phi]=\frac{\braket{ \phi | H|\phi }}{\braket{ \phi | \phi } }\geq E_{0}}

The variational method consists in minimizing this functional. Thus, the approximations given by the variational method are always overestimates and the actual ground state energy is guaranteed to be lower (or technically equal is the best case scenario).

Since $\phi$ is a function, $\delta \phi$ is a directional derivative. ↩

Method#

Footnotes#