410ahw11 - HW#11 —Phys410—Fall 2011 Prof. Ted Jacobson...

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Unformatted text preview: HW#11 —Phys410—Fall 2011 Prof. Ted Jacobson Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/410a/ [email protected] S11.1 Harmonic oscillator using complex phase space coordiantes The Hamiltonian for a simple harmonic oscillator is H = p 2 / 2 + ω 2 x 2 / 2 in units where the mass m = 1. Let a be the complex phase space coordinate a = p ω/ 2( x + ip/ω ), and let a * be its complex conjugate. (a) Express H in terms of a and a * . (b) Evaluate the Poisson bracket { a,a * } , and use that to evaluate { a,H } and { a * ,H } . (c) Write and solve the equations of motion for a and a * using the Poisson bracket form of Hamilton’s equations. 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) Using elementary mechanics, show that the average (outward) force on one of the walls is 2 E/‘ , where E is the (kinetic) energy of the particle. (b) Suppose now that the wall at x = ‘ is moved adiabatically. The energy of the particle then changes as a result of its collisions with the moving wall. Find the relation between δE and δ‘ , and use this to show that E‘ 2 is an adiabatic invariant. (c) Derive the same result instead using adiabatic invariance if H pdx . (Note that you could also find the invariant by dimensional analysis:....
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This note was uploaded on 12/29/2011 for the course PHYSICS 410 taught by Professor Jacobson during the Fall '11 term at Maryland.

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410ahw11 - HW#11 —Phys410—Fall 2011 Prof. Ted Jacobson...

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