Consider the functional \(S [ \psi , \psi ^ { \dagger } ] = \int _ { - \infty } ^ { \infty } d x \psi ^ { \dagger } ( E - \hat { H } ) \psi\) , where
\[ { \hat { H } } = - { \frac { \hbar ^ { 2 } } { 2 m } } { \frac { \partial ^ { 2 } } { \partial x ^ { 2 } } } + { \frac { m \omega ^ { 2 } x ^ { 2 } } { 2 } } .\tag{19} \]
Find an extremum of \(S [ \psi , \psi ^ { \dagger } ]\) in a class of \(C ^ { 1 } ( \mathbb { R } )\) functions of the form \(\psi ( x ) = { \sqrt { \pi \sigma ^ { 2 } } } e ^ { - { \frac { x ^ { 2 } } { 2 \sigma ^ { 2 } } } }\)
Find an eigenvalue of \(\hat { H }\) corresponding to this extremum.