軌道角運動量
L 2 ^ ′ x ± = ( L ^ x ′ − i L ^ y ′ ) ( L ^ x ′ + i L ^ y ′ ) x ± + i L ^ y ′ L ^ x ′ x ± − i L ^ x ′ L ^ y ′ x ± + L ^ z ′ 2 x ± = L ^ ∓ ′ L ^ ± ′ x ± − i [ L ^ x ′ , L ^ y ′ ] x ± + L ^ z ′ 2 x ± {\displaystyle {\hat {{\boldsymbol {L}}^{2}}}'x_{\pm }=({\hat {L}}'_{x}-i{\hat {L}}'_{y})({\hat {L}}'_{x}+i{\hat {L}}'_{y})x_{\pm }+i{\hat {L}}'_{y}{\hat {L}}'_{x}x_{\pm }-i{\hat {L}}'_{x}{\hat {L}}'_{y}x_{\pm }+{\hat {L}}'_{z}{}^{2}x_{\pm }={\hat {L}}'_{\mp }{\hat {L}}'_{\pm }x_{\pm }-i[{\hat {L}}'_{x},{\hat {L}}'_{y}]x_{\pm }+{\hat {L}}_{z}'{}^{2}x_{\pm }} = L ^ ∓ ′ L ^ ± ′ x ± − i L ^ z ′ x ± + L ^ z ′ 2 x ± {\displaystyle ={\hat {L}}_{\mp }'{\hat {L}}_{\pm }'x_{\pm }-i{\hat {L}}'_{z}x_{\pm }+{\hat {L}}'_{z}{}^{2}x_{\pm }} 、 ここで L ^ ∓ ′ L ^ ± ′ x ± = x ± L ^ ∓ ′ L ^ ± ′ + L ^ ∓ ′ [ L ^ ± ′ , x ± ] + [ L ^ ∓ ′ , x ± ] L ^ ± ′ {\displaystyle {\hat {L}}'_{\mp }{\hat {L}}'_{\pm }x_{\pm }=x_{\pm }{\hat {L}}'_{\mp }{\hat {L}}'_{\pm }+{\hat {L}}'_{\mp }[{\hat {L}}'_{\pm },x_{\pm }]+[{\hat {L}}'_{\mp },x_{\pm }]{\hat {L}}'_{\pm }} = x ± L ^ ∓ ′ L ^ ± ′ ∓ 2 z L ^ ± ′ {\displaystyle =x_{\pm }{\hat {L}}'_{\mp }{\hat {L}}'_{\pm }\mp 2z{\hat {L}}'_{\pm }} − i L ^ z ′ x ± = − i x ± L ^ z ′ − i [ L ^ z ′ , x ± ] {\displaystyle -i{\hat {L}}'_{z}x_{\pm }=-ix_{\pm }{\hat {L}}'_{z}-i[{\hat {L}}'_{z},x_{\pm }]} = − i x ± L ^ z ′ ∓ i x ± {\displaystyle =-ix_{\pm }{\hat {L}}'_{z}\mp ix_{\pm }} L ^ z ′ 2 x ± = x ± L ^ z ′ 2 + L ^ z ′ [ L ^ z , x ± ] + [ L ^ z ′ , x ± ] L ^ z ′ {\displaystyle {\hat {L}}'_{z}{}^{2}x_{\pm }=x_{\pm }{\hat {L}}'_{z}{}^{2}+{\hat {L}}'_{z}[{\hat {L}}_{z},x_{\pm }]+[{\hat {L}}'_{z},x_{\pm }]{\hat {L}}'_{z}} = x ± L ^ z ′ 2 + 2 [ L ^ z ′ , x ± ] L ^ z ′ + [ L ^ z ′ , [ L ^ z ′ , x ± ] ] {\displaystyle =x_{\pm }{\hat {L}}'_{z}{}^{2}+2[{\hat {L}}'_{z},x_{\pm }]{\hat {L}}'_{z}+[{\hat {L}}'_{z},[{\hat {L}}'_{z},x_{\pm }]]} = x ± L ^ z ′ 2 ± 2 x ± L ^ z ′ + x ± {\displaystyle =x_{\pm }{\hat {L}}'_{z}{}^{2}\pm 2x_{\pm }{\hat {L}}'_{z}+x_{\pm }}
これまでの記述から分かるように
L 2 ^ ψ l , m = l ( l + 1 ) ħ 2 ψ l , m {\displaystyle {\hat {{\boldsymbol {L}}^{2}}}\psi _{\ell ,m}=\ell (\ell +1)\hbar ^{2}\psi _{\ell ,m}}
L ^ z ψ l , m = m ħ ψ l , m {\displaystyle {\hat {L}}_{z}\psi _{\ell ,m}=m\hbar \psi _{\ell ,m}}
を満たす ψ l , m {\displaystyle \psi _{\ell ,m}} が存在し、必要なら ψ l , m {\displaystyle \psi _{\ell ,m}} を定数倍すれば、
ψ l , m = Y l , m {\displaystyle \psi _{\ell ,m}=Y_{\ell ,m}}
が成立する。
l {\displaystyle \ell } を軌道角運動量量子数(方位量子数)、m は軌道磁気量子数という。前節で述べたように、
l = 0 , 1 , 2 , ... {\displaystyle \ell =0,1,2,\ldots }
m = 0 , ± 1 , ± 2 , ... , ± l {\displaystyle m=0,\pm 1,\pm 2,\ldots ,\pm \ell }
を満たす。