Operator Bonanza

From the relation $\hat{H} {\psi}_{n} = {E}_{n} {\psi}_{n}$ , we can clearly see that ${\psi}_{n}$ is an eigenfunction for the Hamiltonian operator $\hat{H}$ , i.e. an operator for energy, and so, ${\psi}_{n}$ is an energy eigenstate. Hence, it must be time independent and therefore, a stationary state.

Secondly, if we assume ${\hat{a}}_{+} {\psi}_{n}$ is directly proportional to ${\psi}_{n}$ , we can write ${\hat{a}}_{+} {\psi}_{n} = k{\psi}_{n}$ , for some constant $k$ . But, this must imply that:

$\hat{H} {\hat{a}}_{+} {\psi}_{n} = \hat{H} k {\psi}_{n} = k \hat{H} {\psi}_{n}$

Now, from the first relation, we can write, $\hat{H} {\psi}_{n} = {E}_{n} {\psi}_{n}$ , which gives us:

$\hat{H} {\hat{a}}_{+} {\psi}_{n} = {E}_{n} k{\psi}_{n} = {E}_{n} {\hat{a}}_{+} {\psi}_{n}$

But, from the question, this is not true, as $\hat{H} {\hat{a}}_{+} {\psi}_{n} = ({E}_{n} + \hbar \omega) {\hat{a}}_{+} {\psi}_{n}$

Hence, by proof with contradiction, we can clearly say that, our initial assumption is wrong and so, ${\hat{a}}_{+} {\psi}_{n}$ is not directly proportional to ${\psi}_{n}$ .