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Lecture 10 - Angular momentum Algebraic theory We consider...

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Angular momentum Algebraic theory
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Eigenfunctions of angular momentum: zero momentum We consider several examples of the eigenfunctions. We begin with a wave function, which depends only on the distance from a center of the coordinate system. 0,0 ( ) ( ) R r ψ = r Apply operator of the orbital momentum to this function ( ) i R r - ×∇ r h ( ) ( ) 0 r r dR dR R r i R r i dr dr = ⇒ - ×∇ = - × = e r r e h h Thus, all components of the angular momentum in this state have definite value of zero. Probability distribution in this case is fully angle independent (isotropic) and can be presented as a sphere
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Eigenfunctions of angular momentum: l=1 Consider the following functions ( 29 ( 29 1,0 1,1 1, 1 ( ); ( ); ( ) zR r x iy R r x iy R r ψ ψ ψ - = = + = - And check how components of the operator of the angular momentum acts on them ( 29 ( 29 1,0 1,1 1, 1 ( ) 0 0 ( ) ( ) ( ) ( ) ( ) 1 z z z dR r dR r L i x y zR r i z x y y x dr y dr x dR r r dR y x i z x y i z x y m dr y x dr r r L i x y xR r iyR r i yR r xR r x iy R r m y x L i x y y x ψ ψ ψ - = - - = - - = - - = - - = = = - - + = + = + = = - - h h h h h h h h h ( 29 ( 29 ( ) ( ) ( ) ( ) ( ) 1 xR r iyR r i yR r xR r x iy R r m - = - = - - = - h h h Probability distributions are in this case ( 29 2 2 2 2 2 2 2 2 2 2 2 1,0 1, 1 cos ; sin P z R r R P x y R r R θ θ ± = = = + =
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Eigenfunctions of angular momentum: l=1 Again consider the following functions 1,0 ( ) zR r ψ = 2 2 1,0 x yz y L i y z zR i yR R z R i yR z y r r ψ = - - = - + - = - h h h As expected this function is not an eigenfunction of the x-component of momentum At the same it is the eigefunction of the square of this component ( 29 2 2 2 2 2 2 1,0 x z y L y z yR y zR z zR zR z y r r ψ = - - = - - - = h h h Doing the same with y-components we have ( 29 2 2 1,0 2 2 2 2 2 2 1,0 y y xz x L i x z zR i xR R z R i xR z x r r z x L x z xR x R zR z zR z x r r ψ ψ = - - + = - - - + = = - + = - + + = h h h h h h Finally, considering the square of the momentum ( 29 ( 29 ( 29 2 2 2 2 2 2 1,0 0 2 x y z L L L zR zR zR ψ + + = + + = h h h Which corresponds to l=1
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Momentum operator in spherical coordinates Spherical coordinates are related to Cartesian coordinates by 2 2 2 2 2 2 sin cos ; sin sin ; cos ;cos ;tan x r y r z r z y r x y z x x y z θ ϕ θ ϕ θ θ ϕ = = = = + + = = + + Using these relations we can express components and the square of the angular momentum operator in the spherical coordinates 2 2 2 2 2 ; sin cos cot ; 1 1 ˆ cos sin cot ; sin sin sin z x y L i x y i L i y x L i L ϕ ϕ θ ϕ θ ϕ ϕ ϕ θ θ θ ϕ θ θ θ θ ϕ = - - = - = +
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