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lecture18 - 5.61 Angular Momentum Page 1 Angular Momentum...

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Unformatted text preview: 5.61 Angular Momentum Page 1 Angular Momentum Since L 2 and L commute, they share common eigenfunctions. These functions are z extremely important for the description of angular momentum problems they determine the allowed values of angular momentum and, for systems like the Rigid Rotor, the energies available to the system. The first things we would like to know are the eigenvalues associated with these eigenfunctions. We will denote the eigenvalues of L 2 and L z by and , respectively so that: 2 ( , ) = Y , ) ) = Y ( , ) L Y ( L Y ( , z For brevity, in what follows we will omit the dependence of the eigenstates on and so that the above equations become 2 L Y = Y L Y = Y z It is convenient to define the raising and lowering operators (note the similarity to the Harmonic oscillator!): L L iL x y Which satisfy the commutation relations: L , L = 2 L L , L = L L , L 2 = + z z These relations are relatively easy to prove using the commutation relations weve already derived: 2 = L x , L y = i L z L y , L z = i L x L z , L x = i L y L , L z 0 For example: L , L = L , L i L , L z z x z y = i L y i ( x ) = ( L x y ) i L...
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This note was uploaded on 09/24/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.

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lecture18 - 5.61 Angular Momentum Page 1 Angular Momentum...

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