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410ahw11soln

# 410ahw11soln - HW#11Solution Phys410Fall 2011...

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HW#11—Solution —Phys410—Fall 2011 Prof. Ted Jacobson Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/410a/ S11.1 Harmonic oscillator using complex phase space coordiantes (a) H = p 2 / 2 + ω 2 x 2 / 2, with a = p ω/ 2( x + ip/ω ), is H = ωa * a . (b) { a,a * } = ( ω/ 2) { x + ip/ω,x - ip/ω } = - i , so { a,H } = ω { a,a * a } = - iωa (using the product rule for Poisson brackets and { a,a } = 0). The complex conjugate yields { a * ,H } = iωa * . (c) ˙ a = { a,H } = - iωa , and so ˙ a * = iωa * . The solutions are a = e - iωt a 0 and a * = e iωt a * 0 . S11.2 Particle in a box with moving wall A particle of mass m moves in one dimension x between rigid walls at x = 0 and x = . (a) The magnitude of the momentum change in one collision with the wall at x = is Δ p = 2 mv , and the time between collisions is Δ t = 2 ‘/v . The average force is the rate of momentum change applied by the wall, F = Δ p/ Δ t = mv 2 /‘ = 2 E/‘ . (b) If the wall is moved adiabatically, the work done on the particle is

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410ahw11soln - HW#11Solution Phys410Fall 2011...

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