Question

Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on the X basis. (ã) (6 points) Suppose at /-0, a particle starts out in /pro) V /O) t v 11) + 12). Find ly(t)), (X(t) , and the average energy at time t (e) (6 points) Coherent states le) are the eigenstates of the lowering operator: ala-ala) , where α c be any compley number. Calculate (X) 、(x)、()、(P) in the state 1α) and show that AxAP-h/2. (3(4 points) A coherent state can be expanded in terms of energy eigenstates: an P") in the state |a> and show that b/a. 1α)-Eeln). Show that the expansion coefficient is c,-(nla) 6 e-pra by the normalization condition (rie)-1. on co, and then determine n«Ο /2

Answer 1

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Dimensional Harmonic Oscillator Linear Harmonic Oscillator: - for L.H.theta, is V = 1/2 Kx^2 In terms of angular freq V = 1/2 mw^2 x^2 w = squareroot K/m Hamiltonian, H = p^2/2m + 1/2 mw^2 x^2 = -h^2/2m d^2/dx^2 + 1/2 mw^2 x^2 Schrodinger eqn, H psi = E psi [-h^2/2m d^2/dx^2 + 1/2 mw^2 x^2]psi(x) = E psi d^2 psi/dx^2 + 2m/h^2 [E - 1/2 mw^2 x^2]psi = 0 K is not constant, it is variable. Hamiltonian can be written as p^^& x^^H^^= p^^^2/2m + 1/2 mw^2 x^^^2 H^^/hw = (1/2 mw/h)x^^^2 + (1/2mhw)p^^^2 H^^/hw = (squareroot mw/2h)^2 x^^^2 + (1/squareroot 2 mhw)^2 p^^^2 Define a^^= squareroot mw/2h x^^+ i p^^/squareroot 2hmw a^+^^= squareroot mw/2h x^^- i p^^/squareroot 2 hmw Raising & lowering o/p are Hermitian conjugate of each other. a^+^^a^^= mw/2h x^^x^^+ i 1/2h x^^p^^- i 1/2h p^^x^^+ 1/2h mw p^^p^^