1.61: Energy Expectation Values and the Origin of the Variation Principle

A system is in the state \(|\Psi\rangle\), which is not an eigenfunction of the energy operator, \(\hat{H}\). A statistically meaningful number of such states are available for the purpose of measuring the energy. Quantum mechanical principles state that an energy measurement must yield one of the energy eigenvalues, \(\epsilon_{i}\), of the energy operator. Therefore, the average value of the energy measurements is calculated as,

\[
\langle E\rangle=\frac{\sum_{i} n_{i} \varepsilon_{i}}{N} \tag{1}
\nonumber \]

where n_i is the number of times \(\epsilon_{i}\) is measured, and N is the total number of measurements. Therefore, p_i = n_i/N, is the probability that \(\epsilon_{i}\) will be observed. Equation (1) becomes

\[
\langle E\rangle=\sum_{i} p_{i} \varepsilon_{i} \geq \varepsilon_{1}=\varepsilon_{g s} \tag{2}
\nonumber \]

where gs stands for ground state. As shown above, it is clear that the average energy has to be greater than (p₁ < 1) or equal to (p₁ = 1) the lowest energy. This is the origin of the quantum mechanical variational theorem.

According to quantum mechanics, for a system in the state \(|\Psi\rangle, p_{i}=\langle\Psi | i\rangle\langle i | \Psi\rangle\), where the \(|i\rangle\) are the eigenfunctions of the energy operator. Equation (2) can now be re-written as,

\[
\langle E\rangle=\sum_{i}\langle\Psi | i\rangle\langle i | \Psi\rangle \varepsilon_{i}=\sum_{i}\langle\Psi | i\rangle \varepsilon_{i}\langle i | \Psi\rangle \tag{3}
\nonumber \]

However, it is also true that

\[
\hat{H}|i\rangle=\varepsilon_{i}|i\rangle=|i\rangle \varepsilon_{i} \tag{4}
\nonumber \]

Substitution of equation (4) into (3) yields

\[
\langle E\rangle=\sum_{i}\langle\Psi|\hat{H}| i\rangle\langle i | \Psi\rangle \tag{5}
\nonumber \]

As eigenfunctions of the energy operator, the \(|i\rangle\) form a complete basis set, making available the discrete completeness relation, \(\sum_{i}| i \rangle\langle i|=1\), the use of which in equation (5) yields

\[
\langle E\rangle=\langle\Psi|\hat{H}| \Psi\rangle \geq \varepsilon_{g s} \tag{6}
\nonumber \]

Chemists generally evaluate expectation values in coordinate space, so we now insert the continuous completeness relationship of coordinate space, \(\int|x\rangle\langle x| d x=1\), in equation (6) which gives us,

\[
\langle E\rangle=\int\langle\Psi | x\rangle\langle x|\hat{H}| \Psi\rangle d x=\int\langle\Psi | x\rangle \hat{H}(x)\langle x | \Psi\rangle d x \tag{7}
\nonumber \]

where

\[
\hat{H}(x)=-\frac{\hbar^{2}}{2 m} \frac{d}{d x^{2}}+V(x) \tag{8}
\nonumber \]