Quantum Mechanics – Ladder Operator

The operator,  when operating on a harmonic energy eigenstate ψ_n with energy E_n, produces another energy eigenstate whose energy is E_n − ħω₀. Which of the following is true? 

  I.  commutes with the Hamiltonian. 

 II.  is a Hermitian operator and therefore an observable. 

III. The adjoint operator  

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

(GR9677 #100)

Solution:

I. FALSE
Commutes if [H, a] = 0
But a is a ladder operator, a raises the energy level so that [H, a] = − ħωa

II. FALSE
Hermitian operator if  





III. TRUE
See II.

Answer: C