7.2: Representation of Angular Momentum
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Now, we saw earlier, in Section [s7.2], that the operators, pi, which represent the Cartesian components of linear momentum in quantum mechanics, can be represented as the spatial differential operators −iℏ∂/∂xi. Let us now investigate whether angular momentum operators can similarly be represented as spatial differential operators.
It is most convenient to perform our investigation using conventional spherical polar coordinates: that is, r, θ, and ϕ. These are defined with respect to our usual Cartesian coordinates as follows: x=rsinθcosϕ,y=rsinθsinϕ,z=rcosθ.
We deduce, after some tedious analysis, that ∂∂x=sinθcosϕ∂∂r+cosθcosϕr∂∂θ−sinϕrsinθ∂∂ϕ,∂∂y=sinθsinϕ∂∂r+cosθsinϕr∂∂θ+cosϕrsinθ∂∂ϕ,∂∂z=cosθ∂∂r−sinθr∂∂θ.
Making use of the definitions ([e8.1])–([e8.3]), ([e8.9]), and ([e8.13]), the fundamental representation ([e6.12])–([e6.14]) of the pi operators as spatial differential operators, Equations ([e8.21])–([e8zz]), and a great deal of tedious analysis, we finally obtain Lx=−iℏ(−sinϕ∂∂θ−cosϕcotθ∂∂ϕ),Ly=−iℏ(cosϕ∂∂θ−sinϕcotθ∂∂ϕ),Lz=−iℏ∂∂ϕ,
as well as L2=−ℏ2[1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ∂2∂ϕ2],
and L±=ℏe±iϕ(±∂∂θ+icotθ∂∂ϕ).
We, thus, conclude that all of our angular momentum operators can be represented as differential operators involving the angular spherical coordinates, θ and ϕ, but not involving the radial coordinate, r.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)