HW10-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 10...

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PHYSICS 4455 — QUANTUM MECHANICS Problem Set 10 — Solutions. 1. Angular momentum and its commutators – part of every physicist’s rite of passage! Confirm Liboff Eq. (9.8) and (9.13). We worked out Eq. (9.8) in class, using the Levi-Civita tensor. As for (9.13), we can use the results from (9.8): L x , L 2 = L x L x 2 + L y 2 + L z 2 L x 2 + L y 2 + L z 2 L x = L x L y 2 + L x L z 2 L y L x L y + L z L x L z + L y L x L y + L z L x L z L y 2 L x + L z 2 L x Note that we have added and subtracted an appropriate term here. Continuing, we can now combine terms and write them as commutators of individual components: L x , L 2 = ... = L x L y 2 L y L x L y + L y L x L y L y 2 L x + L x L z 2 L z L x L z + L z L x L z L z 2 L x = L x , L y L y + L y L x , L y + L x , L z L z + L z L x , L z = i L z L y + L y L z L y L z L z L y = 0 Note how we have used the results L x , L y = i L z and L x , L z =− i L y (watch the signs!). Clearly, the other two relations follow in a completely analogous fashion. Liboff Problem 9.4 Given  , L x =  , L y =  , L z = 0, evaluate  2 , L 2 .  2 , L 2 =  2 , L x 2 + L y 2 + L z 2 =  2 , L x 2 +  2 , L y 2 +  2 , L z 2 Let’s look at one of these terms, e.g.,  2 , L x 2 and use the fact that ÂL x L x  = 0 whence ÂL x = L x  :  2 , L x 2 =  2 L x 2 L x 2  2 =  ÂL x L x L x L x  =  L x  L x L x L x  = ÂL x ÂL x L x L x  = L x  L x  L x L x  = L x ÂL x  L x L x  = L x L x   L x L x  = L x L x  L x L x  = 0 The other two terms follow the same procedure, whence  2 , L 2 = 0 Liboff Problem 9.7 You already saw quite a few of these in class. Here are a few, without using the Levi-Civita tensor: L x , x = [ ŷ p z p y z , x ] = ( ŷ p z p y z ) x x ( ŷ p z p y z ) = 0 since all coordinates commute with one another, and [ x k , p l ] = i δ kl for different components of position and momentum operators.
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L x , ŷ = [( ŷ p z p y z ) , ŷ ] = [ ŷ p z , ŷ ] − [ p y z , ŷ ] = ŷ p z ŷ ŷŷ p z − ( p y z ŷ ŷ p y z ) = ŷŷ p z ŷŷ p z − ( p y ŷ z ŷ p y z ) = 0 − [ p y , ŷ ] z = i z (c) and (d) are analogous. (e) is now trivial: ŷ , L z = [ ŷ , ( x p y ŷ
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This note was uploaded on 11/09/2009 for the course PHY 4604 taught by Professor Mucciolo during the Spring '09 term at University of Central Florida.

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HW10-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 10...

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