25 - Physics 6210/Spring 2007/Lecture 25 Lecture 25...

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Physics 6210/Spring 2007/Lecture 25 Lecture 25 Relevant sections in text: § 3.6, 3.7 Position representation of angular momentum operators We have seen that the position operators act on position wave functions by multipli- cation and the momentum operators act by differentiation. We can combine these two results and, using spherical polar coordinates ( r,θ,φ ), get a useful position wave function representation for the angular momentum operators. We have L x ψ ( r,θ,φ ) = ¯ h i ( - sin φ∂ θ - cot θ cos φ∂ φ ) ψ ( r,θ,φ ) , L y ψ ( r,θ,φ ) = ¯ h i ( - cos φ∂ θ - cot θ sin φ∂ φ ) ψ ( r,θ,φ ) L z ψ ( r,θ,φ ) = ¯ h i φ ψ ( r,θ,φ ) . You can see that L z is particularly simple – it clearly generates “translations” in φ , which are rotations about the z axis, of course. The other two components of ~ L also generate rotations about their respective axes. They do not take such a simple form because spherical polar coordinates give the z axis special treatment. Combining these results we have, in addition, L 2 ψ ( r,θ,φ ) = - ¯ h 2 ± 1 sin 2 θ 2 φ + 1 sin θ θ (sin θ∂ θ ) ² ψ ( r,θ,φ ) . You may recognize that this last result is, up to a factor of - ¯ h 2 r 2 , the angular part of the Laplacian. This result arises from the identity (see text) L 2 = r 2 P 2 - ( ~ X · ~ P ) 2 + i ¯ h ~ X · ~ P, where r 2 = X 2 + Y 2 + Z 2 , so that (exercise) P 2 ψ ( r,θ,φ ) = - ¯ h 2 2 ψ ( r,θ,φ ) = - ¯ h 2 ( 1 ¯ h 2 r 2 L 2 + 2 r + 2 r r ) ψ ( r,θ,φ ) . Thus we get, in operator form, the familiar decomposition of kinetic energy into a radial part and an angular part. Orbital angular momentum eigenvalues and eigenfunctions; spherical harmon- ics A good way to see what is the physical content of the orbital angular momentum eigenvectors is to study the position probability distributions in these states. Thus we consider the position wave functions ψ lm = h ~x | l,m i 1
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.

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25 - Physics 6210/Spring 2007/Lecture 25 Lecture 25...

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