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The Average Value Theorem and Variational Principle

Even with the separation of the nuclear and electronic wave functions provided by the BO approximation, the electronic Schrödinger equation does not have any known solution, and theorists are forced to resort to more approximations. The fundamental postulate of quantum mechanics that starts one down the road to obtaining actual answers is the Average Value Theorem. The Average Value Theorem states that the expectation value of any physical observable A is given by the Rayleigh quotient

\begin{displaymath}\langle\hat{A} \rangle = \frac{\int_{-\infty}^{\infty} \Psi^\...... \vert \Psi(r) \rangle}{\langle \Psi(r) \vert \Psi(r) \rangle}\end{displaymath} (5)

when the system is in a state described by the wave function $\Psi(r)$. In equation (1.5), $\hat{A}$ is the Hermitian operator which corresponds to the observable A whose expectation value is desired. In addition, the denominator of each quotient accounts for the fact that the wave functions might not be normalized and is equal to 1 if $\Psi$ is indeed normalized. The quotient on the far right is written in the shorthand notation introduced by Dirac. For a summary of the notational conventions used in this dissertation, please see section 1.11 toward the end of this chapter. If the operator used in the Average Value Theorem is the Hamiltonian, then one has a means of obtaining the expectation value of the energy even if the wave function, which must be known, is not an eigenfunction of the Hamiltonian and does not satisfy equation (1.1). However, if the wave function is indeed an eigenfunction of $\hat{H}$, then the energy obtained using either the wave equation or the Average Value Theorem is the same.
 

It may not seem that the Average Value Theorem gets one any closer to solving the electronic Schrödinger equation because to use it one must already know the wave function, and all that one obtains from it is an average energy. Moreover, the wave function used in equation (1.5) need not satisfy the very equation we wish to solve. So, how does this get us anywhere? The answer lies in a mathematical principle known as the Variational Principle. In simplest terms, the Variational Principle states that any arbitrary wave function,$\Phi$, which satisfies the same set of boundary conditions as imposed upon the exact wave function, $\Psi$, will result in an expectation value of the Hamiltonian,  $\langle \hat{H} \rangle$, that is always greater than or equal to the exact ground state energy. The proof of the Variational Principle relies on the fact that the Hamiltonian is a Hermitian operator with the two properties mentioned in section 1.2. The expectation value of the energy of the system in the arbitrary state $\phi$ as given by the average value theorem is

\begin{displaymath}E_{\Phi} = \langle \Phi \vert \hat{H} \vert \Phi \rangle= \int \Phi^{*} \hat{H} \Phi \; dr.\end{displaymath} (6)

If the arbitrary state, $\Phi$, is equal to the ground state of the system,$\Psi_0$, then the energy of the system described by $\Phi$ is simply equal to that of the ground state E0:

\begin{displaymath}E_{\phi} = E_0 = \int \Psi_{0}^{*} \hat{H} \Psi_{0} \; dr.\end{displaymath} (7)

However, if $\Phi$ is not equal to $\Psi_0$, one can expand $\Phi$ in terms of the eigenfunctions of the Hamiltonian,  ${\Psi_0, \Psi_1, \Psi_2, \cdots,\Psi_n}$, which are guaranteed to be orthogonal because the Hamiltonian is a Hermetian operator; thus, one obtains

\begin{displaymath}\phi = \sum_{n} a_{n} \Psi_{n} \;\; {\rm with}\;\;\sum_{n} a_{n}^{*} a_{n} = 1.\end{displaymath} (8)

Substituting (1.8) into (1.6), leads to the equation

\begin{displaymath}E_{\phi} = \int \sum_{n} a_{n}^{*} \Psi_n \hat{H} \sum_{m} a_{m}\Psi_m dr = \sum_{n} a_{n}^{*} a_{n} E_n.\end{displaymath} (9)

Subtracting the ground state energy from both sides of (1.9) yields

\begin{displaymath}E_{\phi}-E_0 = \sum_n a_{n}^{*} a_{n} (E_n - E_0).\end{displaymath} (10)

By definition, E0 is the ground state (i. e. lowest possible) energy of the Hamiltonian for the system; therefore, the right side of (1.10) must be equal to either a positive number or zero. This result leads directly to the Variational Principle

\begin{displaymath}E_{\phi} \geq E_0.\end{displaymath} (11)

The significance of the Variational Principle is that it provides a method of judging the quality of a trial state function relative to the exact wave function. The lower the value of the expectation energy resulting from a particular trial state function, the closer that function is to the actual state function. This allows theorists to propose a trial wave function in which a number of parameters are left variable and minimize the expectation value of the energy with respect to these parameters. The resulting values of the parameters when placed back into the trial wave function yield the best approximation to the exact wave function. The larger the number of the variable parameters, i. e. the more flexibility incorporated into the trial function, the more accurate the resulting trial function will be. Methods that use this kind of scheme to determine wave functions are known as variational methods, of which the methods discussed in this dissertation are only a few.

Table 2: General procedure for variational methods
1 Construct a trial state function containing one or more variable parameters.
2 Construct the equation for the expectation value of the energy  $\langle \hat{H} \rangle$ using the average value theorem.
3 Vary the parameters in the trial state function until a minimum value of  $\langle \hat{H} \rangle$ is found. The easiest method of doing this is to differentiate the equation for $\langle \hat{H} \rangle$ with respect to each of the parameters, set each derivative to 0, and solve the resulting equations for the parameters.
4 Plug the parameters back into the constructed trial state function. According to the Variational Principle, the result is one's best guess for the actual wave function.
5 Utilize the resulting trial state function to determine approximate values of the energy or other desired quantities.

This page maintained by Brian C. Hoffman
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