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Unformatted text preview: Quantum Mechanics: Commutation Relation Proofs 5th April 2010 I. Proof for Non-Commutativity of Indivdual Quantum Angular Momentum Operators In this section, we will show that the operators L x , L y , L z do not commute with one another, and hence cannot be known simultaneously. The relations are (reiterating from previous lectures): L x =- i h y z- z y L y =- i h z x- x z L z =- i h x y- y x We would like to proove the following commutation relations: [ L x , L y ] = i h L z , [ L y , L z ] = i h L x , [ L z , L x ] = i h L y . We will use the first relation for our proof; the second and third follow analo- gously. Lets also consider a function, f ( x, y, z ) that we will have the opera- tors act upon in our discussion. The expanded version of [ L x , L y ] = i h L z is: [ L x , L y ] = L x L y- L y L x 1 = y z- z y z x- x z...
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