4.
(Total marks: 41)
A quantum harmonic oscillator's Hamiltonian can be expressed as H0 = ħω (a^†a + 1/2). The raising operator a^† = √(mω/2ħ) (x - (ip/mω)). The lowering operator a is the Hermitian conjugate of a^†. |n⟩ denotes the normalized eigenstate of H0 with eigenvalue (n + 1/2) ħω.
(a) Evaluate the commutation brackets [a, a^†], and [H0, a^†]. Your answers need to be expressed in their most simplified forms. (7 marks)
(b) Show that the raising operator transforms an energy eigenstate |n⟩ to another energy eigenstate |n+1⟩, i.e. a^†|n⟩ = √(n+1)|n+1⟩, an being some constant. (3 marks)
(c) Find the coefficient a in (b)
(6 marks)
(d) If, at initial time, the harmonic oscillator has the wavefunction Ψ(x,t = 0) = (c0|0⟩ + c1|1⟩ + c2|2⟩), find the expectation value (Ψ(x,t)|a^†Ψ(x,t)) at a general time t. (7 marks)
(e) If the harmonic oscillator is now subject to a weak perturbation: H' = 1x, so that the total Hamiltonian becomes H0 + H'. Use perturbation theory to find the lowest non-vanishing order correction to the energy and wavefunction of ground state. (8 marks)
(f) If Ψ(x,t) now denotes a most general solution to the Schrodinger equation of Harmonic oscillator (not limited to the initial condition stated in (d)), prove that (Ψ(x,t)|a^†Ψ(x,t))=eiωt(Ψ(x,0)|a^†+Ψ(x,0)) (10 marks)