10.3: The Variation Theorem in Dirac Notation

The recipe for calculating the expectation value for energy using a trial wave function is,

\[ \langle E \rangle = \langle \psi | \hat{H} | \psi \rangle \label{1} \]

Now suppose the eigenvalues of \( \hat{H}\) are denoted by \( |i \rangle\). Then,

\[ \hat{H} |i \rangle = \varepsilon_{i} |i \rangle = |i \rangle \varepsilon _{i} \label{2} \]

Next we write \( | \psi \rangle\) as a superposition of the eigenfunctions \( |i \rangle\),

\[ | \psi \rangle = \sum_{i} |i \rangle \langle I| \psi \rangle \nonumber \]

and substitute it into Equation \ref{1}.

\[ \langle E \rangle = \sum_{i} \langle \psi | \hat{H} | i \rangle \langle i | \psi \rangle \nonumber \]

Making use of Equation \ref{2} yields,

\[ \langle E \rangle = \sum_{i} \langle \psi |i \rangle \varepsilon_{i} \langle i | \psi \rangle \nonumber \]

After rearrangement we have,

\[ \langle E \rangle = \sum_{i} \varepsilon_{i} | \langle i| \psi \rangle |^{2} \nonumber \]

However, \( | \langle i | \psi \rangle |^{2}\) is the probability that \( \varepsilon_{i}\) will be observed, \(p_i\).

\[ \langle E \rangle = \sum_{i} \varepsilon_{i} p_{i} \geq \varepsilon_{0} \nonumber \]

Thus, the expectation value obtained using the trial wave function is an upper bound to the true energy. In other words, in valid quantum mechanical calculations you can't get a lower energy than the true energy.