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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 i/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=sinθcosϕr+cosθcosϕrθsinϕrsinθϕ,y=sinθsinϕr+cosθsinϕrθ+cosϕrsinθϕ,z=cosθrsinθrθ.
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=i(sinϕθcosϕcotθϕ),Ly=i(cosϕθsinϕcotθϕ),Lz=iϕ,
as well as L2=2[1sinθθ(sinθθ)+1sin2θ2ϕ2],
and L±=e±iϕ(±θ+icotθϕ).
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)


This page titled 7.2: Representation of Angular Momentum is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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