ang mom generator formulas
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A one-page style formula sheet in a Word file, initialed PhL and dated 1.23.08, so it appears to be Phil's own notes. It lists angular momentum commutation relations, raising and lowering operators, spherical harmonics, tensor operators, Cartesian and spherical vector operators and their relations, and L operators in spherical coordinates. Many symbols are lost in extraction.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
Angular Momentum Formulas PhL 1.23.08
= 1 everywhere
[Ji, Jk] = ijklJl [Jz, J] = J J = Jx iJy J∓J = J2 - Jz2 ∓ Jz
J x J = iJ Jx = ( J+ + J-) Jy = ( J+ J-) J|m> = |m 1>
< |lm> = [ , < | lm> ] // equation in C , not space of functions, so < |m> 1 = 0
< |lm> = < | L | lm> = < | | l m 1> = < | l m 1>
Ylm(,) = < |lm> // spherical harmonics
ab= azbz + (1/2) [ a- b+ + a+ b-] given that a ax iay and b bx iby , and a b vectors
Tensor Operators
[ J , A,m ] = A,m1 [ J3 , A,m ] = m A,m
Cartesian Vector Operators
J x V = iV
R = R() exp(-iJ) RVR-1 = R-1 V [Ji, Vj ] = iijkVk
R = R() exp(-iJ) [Ji]jk = -i ijk [J, Vj ] = [ ixjk ∓ yjk]Vk
Spherical Vector Operators and relation to Cartesian
[J+ , V1,m ] = V1,m+1 [J, Vx ] = V1,0
[J- , V1,m ] = V1,m-1 [J, Vy ] = +i V1,0
[Jz , V1,m ] = m V1,m [J, Vz ] = + V1,1
[J+, V1,1] = 0 [J+, V1,-1] = V1,0 [J+, V1,0] = V1,1
[J-, V1,1] = V1,0 [J-, V1,-1] = 0 [J-, V1,0] = V1,-1
V1,1 = (1/)(Vx + iVy) Vx = - (1/) [ V1,1 V1,-1 ]
V1,-1 = (1/)(Vx iVy) Vy = - (1/i) [ V1,1 + V1,-1 ]
V1,0 = Vz Vz = V1,0
Angular Momentum differential operators in Spherical Coordinates
Lx = i [ S + cot C ] L = ei [ + i cot ]
Ly = i [ C cot S ] L2 = [ 2 + cot + (1/sin)2 2]
Lz = -i L2 = [ (1/S) [ S ] + (1/sin)2 2 ] = – r22Ω