Ladder Operators Eigenvalues

Overview


Solution


Starting from the Time independent Schrodinger Equation
{% H|E_n \rangle = E_n |E_n \rangle %}
multiply by {% a^\dagger %}
{% a^\dagger H|E_n \rangle = a^\dagger E_n |E_n \rangle %}
{% a^\dagger E_n |E_n \rangle = (Ha^\dagger + [a^\dagger, H]) |E_n \rangle = (H - \hbar \omega) a^\dagger |E_n \rangle %}
{% H(a^\dagger |E_n \rangle) = (E_n + \hbar \omega) a^\dagger | E_n \rangle %}