( 4. The quantum harmonic oscillator and its coherent states Consider a quantum harmonic oscillator with mass m, frequen

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4 The Quantum Harmonic Oscillator And Its Coherent States Consider A Quantum Harmonic Oscillator With Mass M Frequen 1 (79.59 KiB) Viewed 22 times

( 4. The quantum harmonic oscillator and its coherent states Consider a quantum harmonic oscillator with mass m, frequency w, and lengthscale h/mw. Take as given the formulae at the foot of the question. Define the coherent state with parameter o to be the cx-eigenstate of the annihilation operator: à la) = aa). (a) Show that the coherent state can be written as 00 la) e-P/2 Varilne In), where \n) is the eigenstate of the Hamiltonian with n quanta of energy, and n=0,1,.... (5 marks) (b) Hence, show that the coherent-state wavefunction is (5 marks) e-le/2 Va(s) --2/2 **/2+V203 = /a. 1/4q1/2 (c) Show that [5 marks) (1) = V2a Real(a). (d) Suppose that a depends on time. Show that for a harmonic oscillator, 2 (7)a = - (r), and hence for some constants (A., B.), show that (5 marks) (1)a= An sin(wt) + B, cos(wt), You may assume the following results: • The normalised eigenfunctions of the quantum harmonic oscillator: 1 1 m() 4.= n!21/271/441/2 11, (se s={/a The generating function for Hermite polynomials: 9(r,t) := e =+*+2 :=Ž 11( 212, TI n=0 Ehrenfest's theorem for one-dimensional particles: d d -(P) e mbn {7 à (P) = -(U'()), where the expectation values are taken with respect to an arbitrary state. 772