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Basically we are assuming a wave function form and then seeing how it differs from the exact form. We cannot do this if we have nothing to compare it against.
But, we note that the variance of the number operator in the systems eigenstate is zero. So we can still calculate the variance and try to minimize it.
Approximate Methods in Quantum Mechanics
This is one way of coming up with an approximate wave function. This does not necessarily give the ground state wave function though.
For this we need to minimize the energy itself. Modern quantum mechanics. Pearson Higher Ed, Notify me of follow-up comments by email. Notify me of new posts by email.
Chapter WKB Approximation and Electron Tunnelling - Principles of Quantum Mechanics [Book]
The resulting trial wavefunction and its corresponding energy are variational method approximations to the exact wavefunction and energy. Why would it make sense that the best approximate trial wavefunction is the one with the lowest energy? This results from the Variational Theorem, which states that the energy of any trial wavefunction is always an upper bound to the exact ground state energy. This can be proven easily.
Let the trial wavefunction be denoted. Any trial function can formally be expanded as a linear combination of the exact eigenfunctions. Of course, in practice, we do not know the , since we are assuming that we are applying the variational method to a problem we can not solve analytically. This is a preview of subscription content, log in to check access.
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