Variational Method#

  • Variational method provides a powerful tool tool to (a) Make systematic approximations and quantiatively measure convergence of predictions towards true values.

  • In variational method one first makes an “educated” guess by picking trial functions for the hamiltonian. One then minimizes parameters of the trial function to get solutions closer to the truth.

  • Variational method, when applied to linear combination of trial functions can turn hard QM problem into an easier linear algebra task: solution of systems of linear equations. Instead of solving differentiation equations for eignefunctions/eigenvalues we instead are solving for matrix eigevnalues and eigenvectors.

Variational theorem#

Any trial function \(\mid \phi \rangle\) we come up with the energy computed with it will always be greater or equal to exact or true energy.

\[ E_{\phi}=\frac{\langle \phi \mid \hat{H} \mid \phi\rangle}{\langle \phi \mid \phi\rangle} \geq E_0 \]

  1. Ground state energy is the lowest possible energy for the system.

  2. By minimizing the energy functions we can make most accurate prediction for a given trail function.

  3. More parameters give us more handles to vary and get more acurate solutions.

Example: H-atom trial function#

  • Exact solution for ground state: \(\psi(r)=\frac{1}{(\pi a^3_0)^{1/2}}e^{-r/a_0}\) and \(E_1 = -0.5 h\)

  • Let’s test a trail function \(\phi=e^{-\alpha r^2}\) and predict energy:

\[ E_{trial} = \frac{\langle \phi \mid \hat{H} \mid \phi\rangle}{\langle \phi \mid \phi\rangle} \]
\[ E_{trial}(\alpha) = \frac{4\pi \int^{\infty}_0 \hat{ e^{-\alpha r^2}[-\frac{\hbar^2}{2\mu}\nabla_r^2}-\frac{e^2}{4\pi \epsilon_0 r}] e^{-\alpha r^2} r^2 dr }{4\pi \int^{\infty}_0 e^{-2\alpha r^2} r^2 dr} \]
\[ E_{trial}(\alpha)=\frac{3\hbar^2 \alpha}{2m_e}-\frac{e^2 \alpha^{1/2}}{\sqrt{2}\epsilon_0 \pi^{3/2}} \]

Minimization gives the best values of paramaters in trial functions#

\[ E_{trial}=\frac{3\hbar^2 \alpha}{2m_e}-\frac{e^2 \alpha^{1/2}}{\sqrt{2}\epsilon_0 \pi^{3/2}} \]
\[ \frac{\partial E_{trial}(\alpha)}{\partial \alpha} = 0 \]
\[ \alpha_{min} = \frac{m_e e^4}{18 \pi^3 \epsilon^2_0 \hbar^4} \]

  • We need to plug this parameters back into energy function to get the energy minimized with respect to \(\alpha\), e.g \(E(\alpha_{min})\)

Comparison of optimized vs true values#

\[ E_0 = -0.5 h \]
\[ E_{trial}(\alpha_{min}) = -0.424 \]

  • About 15% error. But hey it is not too bad for a start!

  • Adding more parameters and functions will reduce the error.