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.

**INCOMPLETE**

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.

${\displaystyle \langle E\rangle \ =\ \langle \psi \ H\ \psi \rangle \ =\ E[\psi ]}$

The ${\displaystyle E[\psi ]}$ is minimized when ${\displaystyle \psi }$ 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 Example

Say ${\displaystyle \psi _{n}}$ are a complete set of orthonormal eigenfunctions of ${\displaystyle H}$.

${\displaystyle H\psi _{n}=E_{n}\psi _{n}}$

${\displaystyle \phi }$ is an arbitrary square-integrable function, in that you can take the integral of ${\displaystyle \phi ^{*}\phi }$ without singularity.

We can write ${\displaystyle \phi =\sum _{n}a_{n}\psi _{n}}$ as:

{\displaystyle {\begin{aligned}E[\phi ]&={\int \phi ^{*}H\phi \over \int \phi ^{*}\phi }\\&={\sum _{n}\sum _{n'}\int a_{n}^{*}\ \phi _{n}^{*}\ H\ a_{n'}\ \psi _{n'} \over \sum _{n}\sum _{n'}\int a_{n}^{*}\ \psi _{n}^{*}\ \psi _{n'}\ a_{n'}}\\&={\sum |a_{n}|^{2}E_{n} \over \sum |a_{n}|^{2}}\end{aligned}}}

Further subtracting the lowest possible energy, called the ground state (${\displaystyle E_{o}}$), from both sides gets us:

${\displaystyle E[\phi ]-E_{o}={\sum _{n}|a_{n}|^{2}E_{n} \over \sum _{n}|a_{n}|^{2}}-E_{o}}$

Since ${\displaystyle E_{n}}$ is always greater than or equal to ${\displaystyle E_{o}}$ for all ${\displaystyle n}$, the right side of this equation must always be greater than zero.

This equality has a very practical importance. It means that if ${\displaystyle \phi }$ is not the ground state wave function the energy will be larger than ${\displaystyle E_{o}}$. As it also happens, if ${\displaystyle \phi H=E\phi }$ and ${\displaystyle E=E_{o}}$, then ${\displaystyle \phi }$ is ${\displaystyle \psi _{o}}$. (for many non-degenerate ${\displaystyle \psi }$)

So... say you have a difficult ${\displaystyle H}$ and can't solve it, but you have a "good" guess for ${\displaystyle \Psi }$, say ${\displaystyle \phi }$. If you can find some way to tweak ${\displaystyle \phi }$ to minimize ${\displaystyle E}$ then ${\displaystyle \phi \rightarrow \psi }$. This allows for Rayleigh-Ritz Variational Method

Rayleigh-Ritz Variational Method

1. Guess: ${\displaystyle ({\vec {\phi }},\ {\vec {\alpha }})}$ which has a good form, where ${\displaystyle {\vec {\alpha }}}$ are a set or variational parameters.
2. Calculate ${\displaystyle \langle E\rangle ={\int \phi ^{*}H\phi \over \int \phi ^{*}\phi }}$
3. Solve ${\displaystyle {\partial \langle E\rangle \over \partial \alpha _{i}}=0}$, for each ${\displaystyle \alpha _{i}}$

To find the set of ${\displaystyle {\vec {\alpha }}}$ that minimizes ${\displaystyle \langle E\rangle }$ and returns the best ${\displaystyle \phi }$ given the chosen form of ${\displaystyle \phi (x)}$.

Rayleigh-Ritz Example

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.

<INSERT MATH>

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 ${\displaystyle \phi }$ is close unless you already know the general solution. The best approach is to make several educated guesses based on asybiotic behavior of ${\displaystyle \psi }$ in the extreme limits.

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