Demo

code

from sympy import pprint
from sympy import Matrix
from sympy import I
from sympy import sqrt

jx_cartesian = Matrix([[0, 0, 0],
                       [0, 0, I],
                       [0, -I, 0]])
jy_cartesian = Matrix([[0, 0, I],
                       [0, 0, 0],
                       [-I, 0, 0]])
jz_cartesian = Matrix([[0, I, 0],
                       [-I, 0, 0],
                       [0, 0, 0]])

print('==== SO(3) generators in cartesian representation is ===')
print('jx_cartesian:')
pprint(jx_cartesian)
print('jy_cartesian:')
pprint(jy_cartesian)
print('jz_cartesian:')
pprint(jz_cartesian)
print('========================================================')
print('we want to change bisis to a representation which jz are diagonalized.')
print('the eigenvaluses and eigenvectors of jz are:')
pprint(jz_cartesian.eigenvects())
print(('note: we need to choose proper coefficients and order of eigenvectors'
       + ' to ensure the results is the right form which we are familiar'
       + ' with in quantum mechanics.'))

u = Matrix([[-1/sqrt(2), 0, 1/sqrt(2)],
            [I/sqrt(2), 0, I/sqrt(2)],
            [0, 1, 0]])

print('so use transorm u**(-1) j u go to spherical representation:')
pprint(u)
print('the results are:')
pprint('jx_spherical:')
pprint(u**(-1) * jx_cartesian * u)
pprint('jy_spherical:')
pprint(u**(-1) * jy_cartesian * u)
pprint('jz_spherical:')
pprint(u**(-1) * jz_cartesian * u)
print('========================================================')

# p, d = jz_cartesian.diagonalize()

==== SO(3) generators in cartesian representation is ===
jx_cartesian:
⎡0  0   0⎤
⎢        ⎥
⎢0  0   ⅈ⎥
⎢        ⎥
⎣0  -ⅈ  0⎦
jy_cartesian:
⎡0   0  ⅈ⎤
⎢        ⎥
⎢0   0  0⎥
⎢        ⎥
⎣-ⅈ  0  0⎦
jz_cartesian:
⎡0   ⅈ  0⎤
⎢        ⎥
⎢-ⅈ  0  0⎥
⎢        ⎥
⎣0   0  0⎦
========================================================
we want to change bisis to a representation which jz are diagonalized.
the eigenvaluses and eigenvectors of jz are:
⎡⎛       ⎡⎡-ⅈ⎤⎤⎞  ⎛      ⎡⎡0⎤⎤⎞  ⎛      ⎡⎡ⅈ⎤⎤⎞⎤
⎢⎜       ⎢⎢  ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟⎥
⎢⎜-1, 1, ⎢⎢1 ⎥⎥⎟, ⎜0, 1, ⎢⎢0⎥⎥⎟, ⎜1, 1, ⎢⎢1⎥⎥⎟⎥
⎢⎜       ⎢⎢  ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟⎥
⎣⎝       ⎣⎣0 ⎦⎦⎠  ⎝      ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣0⎦⎦⎠⎦
note: we need to choose proper coefficients and order of eigenvectors to ensure the results is the right form which we are familiar with in quantum mechanics.
so use transorm u**(-1) j u go to spherical representation:
⎡-√2       √2 ⎤
⎢────  0   ── ⎥
⎢ 2        2  ⎥
⎢             ⎥
⎢√2⋅ⅈ     √2⋅ⅈ⎥
⎢────  0  ────⎥
⎢ 2        2  ⎥
⎢             ⎥
⎣ 0    1   0  ⎦
the results are:
jx_spherical:
⎡    √2    ⎤
⎢0   ──  0 ⎥
⎢    2     ⎥
⎢          ⎥
⎢√2      √2⎥
⎢──  0   ──⎥
⎢2       2 ⎥
⎢          ⎥
⎢    √2    ⎥
⎢0   ──  0 ⎥
⎣    2     ⎦
jy_spherical:
⎡      -√2⋅ⅈ         ⎤
⎢ 0    ──────    0   ⎥
⎢        2           ⎥
⎢                    ⎥
⎢√2⋅ⅈ          -√2⋅ⅈ ⎥
⎢────    0     ──────⎥
⎢ 2              2   ⎥
⎢                    ⎥
⎢       √2⋅ⅈ         ⎥
⎢ 0     ────     0   ⎥
⎣        2           ⎦
jz_spherical:
⎡1  0  0 ⎤
⎢        ⎥
⎢0  0  0 ⎥
⎢        ⎥
⎣0  0  -1⎦
========================================================
>>>

Reference