P H = · + V 2m 1. Starting with the canonical commutation relations for position and momentum [Âi, Âj] = [Îi, Îj] = 0 [Îi‚Âj] = −[ƒ‚ Îi] = −iħdij work out the following commutators, use the definition of L₂ below (Q2). [L, x] = iħy, [L, y] = − iħx, [L, z] = 0, [LPx] = iħpy, [L₂,Py] = − ihpx, [L„P₂] = 0 2. Use these results to obtain [L, L] = iħL, directly from - Lx = YPz− ZPy, Ly = ZPx − xp² L₁ = xPy — YPx 3. Use the results from Q1 to evaluate the commutators [L₂,r²] & [L₂,p²] where: 2 ༡ r² = x² + y² + z² & p² = p²+p²+p² & [L₂,x²] = x[L₂,x] + [L₂, x]x 4. Show that the Hamiltonian (H, see top right) commutes with Lz and L, provided V depends only on r and not 0 or Q. For L², first show that Lx and p2 commute, you can assume the same for Ly. This is similar to Q3 for Lz. h Lx = [ д д - sin op ᏧᎾ cos cot 0- дф Ly = д ħ д h d cos . - sin & cot 0. L₂ = ᎧᎾ дф і дф