237module8 - Quantum Mechanics I Module 8 Getting Started 1...

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Unformatted text preview: Quantum Mechanics I - Module 8 Getting Started 1. Commutator craziness! Define the following: L2 = L2 + L2 + L2 x y z L = Lx iLy Work out the following commutators. Try to use as many shortcuts as possible! (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) [Lz , x] [Lz , y] [Lz , z] [Lz , px ] [Lz , py ] [Lz , pz ] [Lx , Ly ] [L2 , Lx ] [Lz , L ] [L2 , L ] [Lz , r2 ] [Lz , p2 ] 2. Use the previous question to show that the Hamiltonian p2 + V (r) 2m commutes with all three components of L. provided that V depends only on r. (What does it mean to say that two things commute?) H= 3. Considering the hydrogen atom, what do you think the following equation means? H|211 = E2 |211 Write this equation as you are used to seeing it. Discussion Questions 1. How do predictions of the Bohr and Schroedinger treatments of the hydrogen atom (ignoring spin and other relativistic effects) compare with regard to the location of the electron, its total energy, and its orbital angular momentum? 2. Exactly why do we say that for a hydrogen atom in free space the orbital angular momentum vector can be located with equal probability anywhere on a cone symmetrical about the z axis? 1 ...
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