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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 ˆ 2 and L ˆ 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 ˆ 2 and L ˆ z by α and β , respectively so that: 2 β ( , ) = α Y β , ) ˆ β θ φ ) = β Y β ( , ) L ˆ Y θ φ ( θ φ L Y ( , θ φ α α z α α For brevity, in what follows we will omit the dependence of the eigenstates on θ and φ so that the above equations become ˆ 2 β β ˆ β β L Y = α Y L Y = β Y α α z α α It is convenient to define the raising and lowering operators (note the similarity to the Harmonic oscillator!): L ˆ ≡ L ˆ ± iL ˆ ± x y Which satisfy the commutation relations: ⎡ L ˆ , L ˆ ⎤ = 2 L ˆ ⎡ L ˆ , L ˆ ⎤ = ± L ˆ ⎡ L ˆ , L ˆ 2 ⎤ = ⎣ + − ⎦ z ⎣ z ± ⎦ ± ⎣ ± ⎦ These relations are relatively easy to prove using the commutation relations we’ve already derived: ⎡ ˆ ˆ ˆ ⎡ ˆ ˆ ˆ ⎡ ˆ ˆ ˆ ⎡ ˆ 2 ˆ ⎤ = ⎣ 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 ˆ , L ˆ ⎤ = ⎡ L ˆ , L ˆ ⎤ ± i ⎡ L ˆ , L ˆ ⎤ ⎣ z ± ⎦ ⎣ z x ⎦ ⎣ z y ⎦ = i L y ± i (− x ) = ± ( L x y ) i L ±...
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