Quantum Mechanics I - Solution Manual for Questions Set 5 | PHYS 3810, Assignments of Quantum Mechanics

Download Quantum Mechanics I - Solution Manual for Questions Set 5 | PHYS 3810 and more Assignments Quantum Mechanics in PDF only on Docsity! Phys 3810, Spring 2009 Problem Set #5, Hint-o-licious Hints 1. Griffiths, 4.19 Part (a) is a simple application of the basic commutation relations [x, px] = xpx − pxx = ih̄, [x, py] = 0, etc. For (b), pull the commuting factors out in front to simplify things, and use the results in (a) get the answers. For (c), write out the string of operators and use, e.g. Lzx = xLz + ih̄y Lzy = yLz − ih̄x to switch the order of the operators and get cancellation. 2. Griffiths, 4.20 You need to use d dt 〈L〉 = i h̄ 〈[H,L]〉 where H = p2x + p 2 y + p 2 z 2m + V (r) to get the result. Just choose any component of this equation, say z. Show that 1 2m [p2x, xpy − ypx] = −ih̄ m pxpy 1 2m [p2y, xpy − ypx] = +ih̄ m pxpy so these terms will cancel. Then show [V (r), ypy − ypx] = ih̄ ( x ∂V ∂y − y∂V ∂x ) = ih̄(r ×∇V )z This will give d dt 〈Lz〉 = 〈Nz〉 3. Griffiths, 4.23 Apply the (analytic) raising operator to Y 12 (θ, φ) but also show from (4.120) and (4.121) L+Y 1 2 = 2h̄Y 2 2 4. Griffiths, 4.26 Multiply a lot of little matrices. Show that if j 6= k then σjσk = i ∑ l jklσl and if j = k then σjσk = σ 2 j = 1. But results are contained in σjσk = δjk + i ∑ l jklσl 1