Electronic Properties of Materials/Quantum Mechanics for Engineers/Variational Methods
This is the eighth chapter of the first section of the book Electronic Properties of Materials.
An extremely useful fact is that the time-independent Schrödinger Equation is equivalent to a variation principle. The energy is a functional, or a function of functions, of the wavefunction.
The is minimized when is the ground state wavefunction. This can be proven by calculus of variation methods or by methods of Lagrange multipliers. Here, we're going to show this by a practical example.
A Practical ExampleEdit
Say are a complete set of orthonormal eigenfunctions of .
is an arbitrary square-integrable function, in that you can take the integral of without singularity.
We can write as:
Further subtracting the lowest possible energy, called the ground state ( ), from both sides gets us:
Since is always greater than or equal to for all , the right side of this equation must always be greater than zero.
This equality has a very practical importance. It means that if is not the ground state wave function the energy will be larger than . As it also happens, if and , then is . (for many non-degenerate )
So... say you have a difficult and can't solve it, but you have a "good" guess for , say . If you can find some way to tweak to minimize then . This allows for Rayleigh-Ritz Variational Method
Rayleigh-Ritz Variational MethodEdit
- Guess: which has a good form, where are a set or variational parameters.
- Solve , for each
To find the set of that minimizes and returns the best given the chosen form of .
Take as an Example, the Hydrogen atom. What if we can't solve for it? We can try making a good guess. Let's see how close a reasonable guess is.
Excellent Guesses will get you close to the true ground state. Good guesses will still do "ok" but not great. The shortcoming of this method is that you can't know if your guess for is close unless you already know the general solution. The best approach is to make several educated guesses based on asybiotic behavior of in the extreme limits.