Explicit vector spherical harmonics Y_JLM, all cases for J=0,1,2. (in spherical polar coords and basis vectors) ============================== NEW J ============================== J = , 0 ==================== NEW L ==================== j,l,m = , 0, 1, 0 Y_jlm = , 1, * [] 1 [] = , - -------, * e_r 1/2 2 Pi ___________________________________________________________________ ============================== NEW J ============================== J = , 1 ==================== NEW L ==================== j,l,m = , 1, 0, 1 Y_jlm = , exp(phi~ I), * [] 1/2 1/2 2 sin(theta~) 2 cos(theta~) [] = , -1/4 ----------------, * e_r +, -1/4 ----------------, * e_th +, 1/2 1/2 Pi Pi 1/2 -1/4 I 2 -----------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 1, 0, 0 Y_jlm = , 1, * [] cos(theta~) sin(theta~) [] = , 1/2 -----------, * e_r +, -1/2 -----------, * e_th 1/2 1/2 Pi Pi ___________________________________________________________________ j,l,m = , 1, 0, -1 Y_jlm = , exp(-I phi~), * [] 1/2 1/2 2 sin(theta~) 2 cos(theta~) [] = , 1/4 ----------------, * e_r +, 1/4 ----------------, * e_th +, 1/2 1/2 Pi Pi 1/2 -1/4 I 2 -----------, * e_phi 1/2 Pi ___________________________________________________________________ ==================== NEW L ==================== j,l,m = , 1, 1, 1 Y_jlm = , exp(phi~ I), * [] 1/2 1/2 3 1/4 I 3 cos(theta~) [] = , -------, * e_th +, ----------------------, * e_phi 1/2 1/2 4 Pi Pi ___________________________________________________________________ j,l,m = , 1, 1, 0 Y_jlm = , 1, * [] 1/2 1/4 I 6 sin(theta~) [] = , ----------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 1, 1, -1 Y_jlm = , exp(-I phi~), * [] 1/2 1/2 3 -1/4 I 3 cos(theta~) [] = , -------, * e_th +, -----------------------, * e_phi 1/2 1/2 4 Pi Pi ___________________________________________________________________ ==================== NEW L ==================== j,l,m = , 1, 2, 1 Y_jlm = , exp(phi~ I), * [] sin(theta~) cos(theta~) -1/4 I [] = , 1/2 -----------, * e_r +, -1/4 -----------, * e_th +, ------, 1/2 1/2 1/2 Pi Pi Pi * e_phi ___________________________________________________________________ j,l,m = , 1, 2, 0 Y_jlm = , 1, * [] 1/2 1/2 2 cos(theta~) 2 sin(theta~) [] = , -1/2 ----------------, * e_r +, -1/4 ----------------, * e_th 1/2 1/2 Pi Pi ___________________________________________________________________ j,l,m = , 1, 2, -1 Y_jlm = , exp(-I phi~), * [] sin(theta~) cos(theta~) -1/4 I [] = , -1/2 -----------, * e_r +, 1/4 -----------, * e_th +, ------, 1/2 1/2 1/2 Pi Pi Pi * e_phi ___________________________________________________________________ ============================== NEW J ============================== J = , 2 ==================== NEW L ==================== j,l,m = , 2, 1, 2 Y_jlm = , exp(2 I phi~), * [] 1/2 1/2 1/2 3 cos(2 theta~) 3 3 sin(2 theta~) [] = , -1/8 ------------------ + -------, * e_r +, 1/8 ------------------, 1/2 1/2 1/2 Pi 8 Pi Pi 1/2 1/4 I 3 sin(theta~) * e_th +, ----------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 2, 1, 1 Y_jlm = , exp(phi~ I), * [] 1/2 1/2 3 sin(2 theta~) 3 cos(2 theta~) [] = , -1/4 ------------------, * e_r +, -1/4 ------------------, * e_th +, 1/2 1/2 Pi Pi 1/2 -1/4 I 3 cos(theta~) -----------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 2, 1, 0 Y_jlm = , 1, * [] 1/2 1/2 1/2 2 cos(2 theta~) 2 2 sin(2 theta~) [] = , 3/8 ------------------ + -------, * e_r +, -3/8 ------------------, 1/2 1/2 1/2 Pi 8 Pi Pi * e_th ___________________________________________________________________ j,l,m = , 2, 1, -1 Y_jlm = , exp(-I phi~), * [] 1/2 1/2 3 sin(2 theta~) 3 cos(2 theta~) [] = , 1/4 ------------------, * e_r +, 1/4 ------------------, * e_th +, 1/2 1/2 Pi Pi 1/2 -1/4 I 3 cos(theta~) -----------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 2, 1, -2 Y_jlm = , exp(-2 I phi~), * [] 1/2 1/2 1/2 3 cos(2 theta~) 3 3 sin(2 theta~) [] = , -1/8 ------------------ + -------, * e_r +, 1/8 ------------------, 1/2 1/2 1/2 Pi 8 Pi Pi 1/2 -1/4 I 3 sin(theta~) * e_th +, -----------------------, * e_phi 1/2 Pi ___________________________________________________________________ ==================== NEW L ==================== j,l,m = , 2, 2, 2 Y_jlm = , exp(2 I phi~), * [] 1/2 1/2 5 sin(theta~) -1/8 I 5 sin(2 theta~) [] = , -1/4 ----------------, * e_th +, -------------------------, * e_phi 1/2 1/2 Pi Pi ___________________________________________________________________ j,l,m = , 2, 2, 1 Y_jlm = , exp(phi~ I), * [] 1/2 1/2 5 cos(theta~) 1/4 I 5 cos(2 theta~) [] = , 1/4 ----------------, * e_th +, ------------------------, * e_phi 1/2 1/2 Pi Pi ___________________________________________________________________ j,l,m = , 2, 2, 0 Y_jlm = , 1, * [] 1/2 1/8 I 30 sin(2 theta~) [] = , -------------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 2, 2, -1 Y_jlm = , exp(-I phi~), * [] 1/2 1/2 5 cos(theta~) -1/4 I 5 cos(2 theta~) [] = , 1/4 ----------------, * e_th +, -------------------------, * e_phi 1/2 1/2 Pi Pi ___________________________________________________________________ j,l,m = , 2, 2, -2 Y_jlm = , exp(-2 I phi~), * [] 1/2 1/2 5 sin(theta~) -1/8 I 5 sin(2 theta~) [] = , 1/4 ----------------, * e_th +, -------------------------, * e_phi 1/2 1/2 Pi Pi ___________________________________________________________________ ==================== NEW L ==================== j,l,m = , 2, 3, 2 Y_jlm = , exp(2 I phi~), * [] 1/2 1/2 1/2 2 cos(2 theta~) 3 2 2 sin(2 theta~) [] = , 3/16 ------------------ - --------, * e_r +, 1/8 ------------------, 1/2 1/2 1/2 Pi 16 Pi Pi 1/2 1/4 I 2 sin(theta~) * e_th +, ----------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 2, 3, 1 Y_jlm = , exp(phi~ I), * [] 1/2 1/2 2 sin(2 theta~) 2 cos(2 theta~) [] = , 3/8 ------------------, * e_r +, -1/4 ------------------, * e_th +, 1/2 1/2 Pi Pi 1/2 -1/4 I 2 cos(theta~) -----------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 2, 3, 0 Y_jlm = , 1, * [] 1/2 1/2 1/2 3 cos(2 theta~) 3 3 sin(2 theta~) [] = , -3/8 ------------------ - -------, * e_r +, -1/4 ------------------, 1/2 1/2 1/2 Pi 8 Pi Pi * e_th ___________________________________________________________________ j,l,m = , 2, 3, -1 Y_jlm = , exp(-I phi~), * [] 1/2 1/2 2 sin(2 theta~) 2 cos(2 theta~) [] = , -3/8 ------------------, * e_r +, 1/4 ------------------, * e_th +, 1/2 1/2 Pi Pi 1/2 -1/4 I 2 cos(theta~) -----------------------, * e_phi 1/2 Pi ___________________________________________________________________ j,l,m = , 2, 3, -2 Y_jlm = , exp(-2 I phi~), * [] 1/2 1/2 1/2 2 cos(2 theta~) 3 2 2 sin(2 theta~) [] = , 3/16 ------------------ - --------, * e_r +, 1/8 ------------------, 1/2 1/2 1/2 Pi 16 Pi Pi 1/2 -1/4 I 2 sin(theta~) * e_th +, -----------------------, * e_phi 1/2 Pi ___________________________________________________________________