By Levi A.F.J.

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**Example text**

The change in 〈 A ( t ) 〉 in time interval ∆t is the slope ∆t = ∆ A ⁄ d 〈 A〉 dt i ˆ , Bˆ ]〉 for time independent (a) Use the generalized uncertainty relation ∆A ∆B ≥ -- 〈 [ A 2 ˆ and Bˆ to show that ∆E∆ t ≥ h --- . operators A 2 (b) Show that the spread in photon number ∆n and phase ∆φ for light of frequency ω is 1 ∆n∆φ ≥ -2 and that for a Poisson distribution of such photons 1 ∆φ ≥ ---------------2 〈 n〉 2 PROBLEM 4 A particle of charge e, mass m, and momentum p oscillates in a one-dimensional har2 2 monic potential V ( xˆ ) = mω 0 xˆ ⁄ 2 and is subject to an oscillating electric field E x cos ( ω t ) .

The constants A and δ are found from the initial contitions. If ω 0 = ω the amplitude of the undamped oscillator grows to infinity with the particular solution increasing linearly in time as 8 Fx t sin ( ω 0 t ) 〈 xˆ 〉 p ( t ) = --------------------------2mω 0 When ω is close in value to ω 0 then Fx ( cos ( ωt ) – cos ( ω 0 t ) ) 〈 xˆ 〉 ( t ) = --------------------------2 2 m ( ω0 – ω ) which, since 2 sin ( x ) sin ( y ) = cos ( x – y ) – cos ( x + y ) , may be written as ( ω0 + ω ) (ω 0 – ω ) 2 Fx sin ---------------------t sin --------------------- t 〈 xˆ 〉 ( t ) = --------------------------2 2 2 2 m ( ω0 – ω ) The sum frequency ( ω 0 + ω ) is modulated by the difference frequency ( ω 0 – ω ) so that the amplitude of 〈 xˆ 〉 beats at the difference frequency ( ω 0 – ω ) .

0359976 ( 50 ) 4 πε 0 hc α – 1 = ----------------e2 Applied quantum mechanics 1 PROBLEM 1 In first-order time-dependent perturbation theory a particle initially in eigenstate | n〉 of the unperturbed Hamiltonian scatters into state | m〉 with probability a m ( t ) 2 after the perturbation V is applied. (a) Show that if the perturbation is applied at time t = 0 then the time dependent coefficient a m ( t ) is t′ = t iω mn t ′ 1 a m ( t ) = ----- ∫ W mn e d t′ ih t ′ = 0 where the matrix element W mn = 〈m | V |n〉 and hω mn = E m – En is the difference in eigenenergies of the states |m〉 and | n〉 .