soln(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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Physics 742 – Graduate Quantum Mechanics 2 Solution Set N 1. [15] A particle of mass m is initially in the ground state of a harmonic oscillator with frequency ω . At time t = 0, a perturbation is suddenly turned on of the form () t Wt AQ e λ = . At late times ( t →∞ ), the quantum state is measured again. (a) [10] Calculate, to second order in A , the amplitude 0 n S that it ends up in the state n , for all n (most of them will be zero). We will need matrix elements of the form ( ) , t nW t m AQe = , which are given by ,1 2 2 tt t nm mn Ae nQm n a a m m Ae m n m λλ δδ −− == + =+ = = To second order, then, our amplitudes are 00 00 00 2 0 2 2 2 22 0 2 11 0 0 1 2 1 24 2 1 4 nn t it n t ti t t i t t SW t d t W t n e d t n W t e d t i i A e e dt e e dt m AA ee e d t mi m i i Ai m S ωω λω δ ωλω λλω ωλλ ω ∞∞ ′′ + ⎛⎞ =− ⎜⎟ ⎝⎠ ⎡⎤ ⎢⎥ ⎣⎦ −+ + + ∫∫ = = = = = = = 10 1 0 10 10 2 0 0 10 1 0 2 2 t i t n it t Wt e d t Wt ne d t nWt t i i ed t im i ωλ ωλ ω + = = = = =
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() 20 2 0 20 20 2 00 0 2 2 2 2 0 22 2 0 11 20 2 0 21 2 2 2 2 0, nn t it n t it t n S W t e
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soln(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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