Question

ONLY (e) (f) NEEDED THANK YOU :)

Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures we demonstrated that àt n)-An |n 1), where An is a constant. Explain why An -Vn+1 once a sensible phase convention is adopted. Use this result from now on. (c) Hence show that 0O (d) Write down the bra corresponding to |a). Hence show that for a) to be normalised to 1, it suffices to choose (e) Show that under time-evolution the state a) at t -0 becomes at time t, where -iwt [The notation | 〉 just means replace α by Φ in the definition, Eq.(1). You may assume the known result for the energy eigenvalues of the H.O.] (f) Explain why

Answer 1

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