- C A Charged Particle Of Mass M Moves In A One Dimensional Harmonic Oscillator Potential With Natural Frequency W In T 1 (283.74 KiB) Viewed 48 times
(c) A charged particle of mass m moves in a one-dimensional harmonic oscillator potential with natural frequency w. In terms of the raising and lowering operators ât and â the Hamiltonian is H = hw(â¹â+1/2). These operators satisfy â|n) = √n\n-1), â¹|n) = √n+1|n+1), where the quantum number n = 0, 1, 2,.... Show that â¹ân) = n/n). [2 marks] (d) This particle experiences, in addition to the harmonic oscillator potential, a weak time-dependent electric field. The field acts between the times 0 and T, and can cause a transition from the harmonic- oscillator eigenstate with quantum number n; to that with quantum number nf. The perturbation is -q E (t), where E(t) is the strength of the field and q the charge. Express this perturbation in terms of the raising and lowering operators. Hence determine the values of n, and ny for which the ip transition probability is non-zero. [Note that â = mw 2ħ (ê + J mw [4 marks] (e) Determine the non-zero transition probabilities explicitly for the case where E(t) = En between the times 0 and T (and zero otherwise). Sketch the dependence of these probabilities on the natural frequency of the oscillator w. Discuss the physics responsible for (i) the oscillatory behaviour of these probabilities, (ii) their overall decrease as w increases, and (iii) their general dependence on the quantum number of the initial state. [Note that fetit dt = 2etiwT/2 sin(wT/2)/w.] [9 marks] (f) Suppose the perturbing potential was, instead of being proportional to the displacement , proportional to , for some integer p > 1. Identify those final states (values of nf) for which the transition probability is non-zero in this case, explaining your reasoning. If there is more than one possible final state for a given initial state, which of them would you expect to have the greatest probability and why? [4 marks]