The Variational Principle (Rayleigh-Ritz Approximation)

Because the ground state has the lowest possible energy, we can vary a test wavefunction, minimizing the energy, to get a good estimate of the ground state energy.

\begin{displaymath}\bgroup\color{black}H\psi_E=E\psi_E\egroup\end{displaymath}

for the ground state \bgroup\color{black}$\psi_E$\egroup.

\begin{displaymath}\bgroup\color{black}E={\int{\psi_E^*H\psi_Edx} \over{\int{\psi_E^*\psi_Edx}}}\egroup\end{displaymath}

For any trial wavefunction \bgroup\color{black}$\psi$\egroup,

\begin{displaymath}\bgroup\color{black}E'={\int{\psi^*H\psi dx} \over{\int{\psi^...
...rt H\vert\psi\right> \over{ \left<\psi\vert\psi\right>}}\egroup\end{displaymath}

We wish to show that \bgroup\color{black}$E'$\egroup errors are second order in \bgroup\color{black}$\delta\psi$\egroup

\begin{displaymath}\bgroup\color{black}\Rightarrow\qquad {\partial E\over{\partial\psi}}=0\egroup\end{displaymath}

at eigenenergies.

To do this, we will add a variable amount of an arbitrary function \bgroup\color{black}$\phi$\egroup to the energy eigenstate.

\begin{displaymath}\bgroup\color{black} E'= {\left<\psi_E+\alpha\phi \vert H\ver...
...{ \left<\psi_E+\alpha\phi\vert\psi_E+\alpha\phi\right>}}\egroup\end{displaymath}

Assume \bgroup\color{black}$\alpha$\egroup is real since we do this for any arbitrary function \bgroup\color{black}$\phi$\egroup. Now we differentiate with respect to \bgroup\color{black}$\alpha$\egroup and evaluate at zero.

\begin{displaymath}\bgroup\color{black} \left. {dE'\over{d\alpha}}\right\vert _{...
...ght>\right)
\over {\left<\psi_E\vert\psi_E\right>^2 }} \egroup\end{displaymath}


\begin{displaymath}\bgroup\color{black} = E\left<\phi\vert\psi_E\right> + E\left...
...i\vert\psi_E\right> - E\left<\psi_E\vert\phi\right> = 0 \egroup\end{displaymath}

We find that the derivative is zero around any eigenfunction, proving that variations of the energy are second order in variations in the wavefunction.

That is, \bgroup\color{black}$E'$\egroup is stationary (2nd order changes only) with respect to variation in \bgroup\color{black}$\psi$\egroup. Conversely, it can be shown that \bgroup\color{black}$E'$\egroup is only stationary for eigenfunctions \bgroup\color{black}$\psi_E$\egroup. We can use the variational principle to approximately find \bgroup\color{black}$\psi_E$\egroup and to find an upper bound on \bgroup\color{black}$E_0$\egroup.

\begin{displaymath}\bgroup\color{black} \psi=\sum_E c_E\psi_E\egroup\end{displaymath}


\begin{displaymath}\bgroup\color{black} E'=\sum_E \vert c_E\vert^2 E\geq E_0 \egroup\end{displaymath}

For higher states this also works if trial \bgroup\color{black}$\psi$\egroup is automatically orthogonal to all lower states due to some symmetry (Parity, \bgroup\color{black}$\ell$\egroup ...)

* Example: Energy of 1D Harmonic Oscillator using a polynomial trail wave function.*
* Example: 1D H.O. using Gaussian.*

Jim Branson 2013-04-22