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13_580_hw_9 - Physics 580 October 2010 Quantum Mechanics I...

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Unformatted text preview: Physics 580 Handout 13 26 October 2010 Quantum Mechanics I webusers.physics.illinois.edu/ ∼ goldbart/ Homework 9 Prof. P. M. Goldbart 3135 (& 2115) ESB University of Illinois 1) The algebra of angular momentum – optional : a) By using the definition L = R × P , and the canonical commutation relations [ R i , P j ] = i ¯ hδ ij , establish the following results: a.i) [ L i , L j ] = i ¯ hϵ ijk L k ; a.ii) [ L · L , L i ] = 0; a.iii) [ R , a · L ] = i ¯ h a × R for c-numbers a ; a.iv) [ R i , L j ] = i ¯ hϵ ijk R k ; a.v) [ P i , L j ] = i ¯ hϵ ijk P k ; a.vi) [ R · R , L i ] = 0; a.vii) [ R · R P i , L j ] = i ¯ hϵ ijk R · R P k . b) The raising and lowering operators, L ± , are defined via L ± = L x ± iL y . Use them to establish the following results: b.i) [ L ± , L 2 ] = 0, where L 2 ≡ L · L ; b.ii) [ L z , L ± ] = ± ¯ hL ± ; b.iii) [ L + , L- ] = 2¯ hL z ; b.iv) L 2 = L + L- + L 2 z − ¯ hL z . c) For the Hilbert space of functions on a sphere, discuss why the set { L 2 , L z } forms a complete set of commuting observables (CSCO), i.e. , argue that inclusion of L x or L y violates the commuting property. Would { L 2 , L x } also form a CSCO? d) Show that the eigenvalues of L 2 are positive or zero. Can they always be written in the form ¯ h 2 l ( l + 1) with l dimensionless and greater than or equal to zero? e) Denote the set of simultaneous eigenstates of L 2 and L z by | l, m ⟩ , where L 2 | l, m ⟩ = ¯ h 2 l ( l + 1) | l, m ⟩ ; L z | l, m ⟩ = ¯ hm | l, m ⟩ ....
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13_580_hw_9 - Physics 580 October 2010 Quantum Mechanics I...

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