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"## Spherical harmonics and angular part\n",
"\n",
"### Complex representations\n",
"\n",
"We start with the simultaneous eigenstates of both $\\hat{L}^2$ and $\\hat{L}_z$, the spherical harmonics $Y_l^m(\\theta,\\phi)$. These are the solution of the angular part of the central-field Schrödinger equation. Note that the central field is rotationally invariant w.r. to rotations about any axis, such that all the $m$ states are degenerate. "
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"Now we look at the 'real' orbitals in which we sacrifce the m quantum number in favor of a purely real representation of the eigenstates. Because all the $m$ are degenerate, this is still a nice complete basis for the degenerate eigenstates of with a given $l$. Those orbitals are no longer eigenstates of $\\hat{L}_z$."
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