# 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 ni is the number of times $$\epsilon_{i}$$ is measured, and N is the total number of measurements. Therefore, pi = ni/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

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

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