Show that the rotational wave function is the eigenfunction of the total energy operator of a schrodinger equation determine the energy eigenvalue?

Answer 1

In quantum mechanics, the rotational wave function for a rigid rotor is given by ( \psi(\theta) = e^{im\theta} ), where ( m ) is the magnetic quantum number. The total energy operator, for a rigid rotor, is expressed as ( \hat{H} = -\frac{\hbar^2}{2I} \frac{d^2}{d\theta^2} ), where ( I ) is the moment of inertia. Applying the energy operator to the wave function yields ( \hat{H} \psi(\theta) = \frac{\hbar^2 m^2}{2I} \psi(\theta) ), demonstrating that ( \psi(\theta) ) is indeed an eigenfunction of the total energy operator with energy eigenvalue ( E_m = \frac{\hbar^2 m^2}{2I} ).