Question

a) The operator aˆ satisfies the equation [aˆ, aˆ†] = n. Prove that n is a real number. [Hint: Apply the Hermitian conjugation operation “†” to the above equation.]

b) Consider the Hamiltonian Hˆ = h ̄ωˆa†ˆa, where ω > 0 is a real parameter. Prove that Hˆ is Hermitian.

c) Let |ψ〉 be an eigenstate of the above Hamiltonian with the energy ε. Use the commutation relations from part a) to prove that aˆ†|ψ〉 is also an eigenstate of Hˆ. Find the energy of the state aˆ†|ψ〉. You do not need to prove that aˆ†|ψ〉 ≠ 0.

d) Let n > 0. Use the commutation relations to find the ground state energy of Hˆ .

Answer 1