a) The operator aˆ satisfies the equation [aˆ, aˆ†] = n. Provethat n is a real number. [Hint: Apply the Hermitian conjugationoperation “††to the above equation.]
b) Consider the Hamiltonian Hˆ = h ̄ωˆa†ˆa, where ω > 0 is areal parameter. Prove that Hˆ is Hermitian.
c) Let |ψ〉 be an eigenstate of the above Hamiltonian with theenergy ε. Use the commutation relations from part a) to prove thataˆ†|ψ〉 is also an eigenstate of Hˆ. Find the energy of the stateaˆ†|ψ〉. You do not need to prove that aˆ†|ψ〉 ≠ 0.
d) Let n > 0. Use the commutation relations to find theground state energy of Hˆ .
