Only part c needs answering (The Last Question)

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Only part c needs answering (The Last Question)

Only Part C Needs Answering The Last Question 1 (60.59 KiB) Viewed 29 times

4. The quantum harmonic oscillator and its coherent states Consider a quantum harmonic oscillator with mass m, frequency w, and lengthscale a = Vh/mw. Take as given the formulae at the foot of the question. Define the coherent state with parameter av to be the ar-eigenstate of the annihilation operator: à la) = |a). (a) Show that the coherent state can be written as a' la) = e tef/2) In), where In) is the eigenstate of the Hamiltonian with n quanta of energy, and n=0,1,.... [6 marks) (6) Hence, show that the coherent state wavefunction is e-let/2 -/2 > /2+V24 71/441/2" s=0/a. [6 marks) (c) Show the following overcompleteness identity for the coherent states: (ala) = (x[+]#"12-2° °)/2 (9) ته [8 marks) You may assume the following results: • The normalised eigenfunctions of the quantum harmonic oscillator: 1 1 Va(z) – L(5)e=a/2 Vn!2n/271/4q1/2 1,s s={/a The generating function for Hermite polynomials: 12 +21 ST, ):=t- ΣΠ.(α) το! T=0