ボーア＝ゾンマーフェルトの量子化条件
であることを用いると
∮ p φ d φ = 2 π p φ = 2 π M z ∮ p θ d θ = 2 ∫ θ 1 θ 2 M 2 − M z 2 sin 2  θ d θ = 2 π ( M − | M z | ) ∮ p r d r = 2 ∫ r 1 r 2 2 m e E + 2 m e e 2 r − M 2 r 2 d r = − 2 π M + π e 2 2 m e − E {\displaystyle {\begin{aligned}\oint p_{\phi }\,\mathrm {d} \phi &=2\pi p_{\phi }=2\pi M_{z}\\\oint p_{\theta }\,\mathrm {d} \theta &=2\int _{\theta _{1}}^{\theta _{2}}{\sqrt {M^{2}-{\frac {M_{z}^{2}}{\sin ^{2}{\theta }}}}}\,\mathrm {d} \theta =2\pi (M-|M_{z}|)\\\oint p_{r}\,\mathrm {d} r&=2\int _{r_{1}}^{r_{2}}{\sqrt {2m_{\mathrm {e} }E+{\frac {2m_{\mathrm {e} }e^{2}}{r}}-{\frac {M^{2}}{r^{2}}}}}\,\mathrm {d} r=-2\pi M+\pi e^{2}{\sqrt {\frac {2m_{\mathrm {e} }}{-E}}}\end{aligned}}}