A primeira condição dá p = n h 2 π {\displaystyle p={\frac {nh}{2\pi }}} .
Quanto á segunda, tem-se
J = ∫ ( r ) A + 2 B r + C r 2 d r = n 1 h {\displaystyle J=\int _{(r)}{\sqrt {A+2{\frac {B}{r}}+{\frac {C}{r^{2}}}}}dr=n_{1}h}
em que
A = 2 m W + W 2 c 2 , B = m e E + e E W c 2 − m c 2 γ 2 , C = n 3 h 2 4 π 2 + e 2 E 2 c 2 {\displaystyle A=2mW+{\frac {W^{2}}{c^{2}}},\quad B=meE+{\frac {eEW}{c^{2}}}-{\frac {mc^{2}\gamma }{2}},\quad C={\frac {n^{3}h^{2}}{4\pi ^{2}}}+{\frac {e^{2}E^{2}}{c^{2}}}}
As integraes do typo de J {\displaystyle J} já foram calculadas por Sommerfeld. Encontra-se J = − 2 π i ( C − B A ) = n 1 h {\displaystyle J=-2\pi i\left(C-{\frac {B}{\sqrt {A}}}\right)=n_{1}h}
− 2 π i C = − h n 2 − u 2 , a = 2 π e 2 h c {\displaystyle -2\pi iC=-h{\sqrt {n^{2}-u^{2}}},\quad a={\frac {2\pi e^{2}}{hc}}}
B A = e E ( 1 + W m c 2 ) − c 2 γ 2 c [ ( 1 + W m c 2 ) 2 − 1 ] 1 / 2 , u = a E e {\displaystyle {\frac {B}{\sqrt {A}}}={\frac {eE\left(1+{\frac {W}{mc^{2}}}\right)-{\frac {c^{2}\gamma }{2}}}{c\left[\left(1+{\frac {W}{mc^{2}}}\right)^{2}-1\right]^{1/2}}},\quad u=a{\frac {E}{e}}}
Obtem-se a relação approximada
[ 4 π 2 e 2 E 2 + c 2 h 2 ( n 1 + n 2 − u 2 ) 2 ] ( 1 + W m c 2 ) 2 − 4 π 2 c 2 γ e E ( 1 + W m c 2 ) − c 2 h 2 ( n 1 + n 2 − u 2 ) 2 = 0 {\displaystyle \left[4\pi ^{2}e^{2}E^{2}+c^{2}h^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{2}\right]\left(1+{\frac {W}{mc^{2}}}\right)^{2}-4\pi ^{2}c^{2}\gamma eE\left(1+{\frac {W}{mc^{2}}}\right)-c^{2}h^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{2}=0}
Com grande aproximação póde-se escrever
1 + W m c 2 = [ 1 + u 2 ( n 1 + n 2 − u 2 ) 2 ] − 1 / 2 × [ 1 + π 2 γ 2 c 2 2 h 2 u 2 ( n 1 + n 2 − u 2 ) − 4 u 2 ( n 1 + n 2 − u 2 ) − 2 + 1 ] + π γ c h u ( n 1 + n 2 − u 2 ) − 2 u 2 ( n 1 + n 2 − u 2 ) − 2 + 1 {\displaystyle 1+{\frac {W}{mc^{2}}}=\left[1+{\frac {u^{2}}{(n_{1}+{\sqrt {n^{2}-u^{2}}})^{2}}}\right]^{-1/2}\times \left[1+{\frac {{\dfrac {\pi ^{2}\gamma ^{2}c^{2}}{2h^{2}}}u^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-4}}{u^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-2}+1}}\right]+{\frac {{\dfrac {\pi \gamma c}{h}}u(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-2}}{u^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-2}+1}}}
Ann. da Acad. Brasileira de Sciencias
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já foram calculadas por Sommerfeld. Encontra-se 
![{\displaystyle {\frac {B}{\sqrt {A}}}={\frac {eE\left(1+{\frac {W}{mc^{2}}}\right)-{\frac {c^{2}\gamma }{2}}}{c\left[\left(1+{\frac {W}{mc^{2}}}\right)^{2}-1\right]^{1/2}}},\quad u=a{\frac {E}{e}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30bc9f73c0fb899648d3b76244d4f73ffe3dfcba)
![{\displaystyle \left[4\pi ^{2}e^{2}E^{2}+c^{2}h^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{2}\right]\left(1+{\frac {W}{mc^{2}}}\right)^{2}-4\pi ^{2}c^{2}\gamma eE\left(1+{\frac {W}{mc^{2}}}\right)-c^{2}h^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9792bca745bcfedaf747293a93f6649b1a8e19)
![{\displaystyle 1+{\frac {W}{mc^{2}}}=\left[1+{\frac {u^{2}}{(n_{1}+{\sqrt {n^{2}-u^{2}}})^{2}}}\right]^{-1/2}\times \left[1+{\frac {{\dfrac {\pi ^{2}\gamma ^{2}c^{2}}{2h^{2}}}u^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-4}}{u^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-2}+1}}\right]+{\frac {{\dfrac {\pi \gamma c}{h}}u(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-2}}{u^{2}(n_{1}+{\sqrt {n^{2}-u^{2}}})^{-2}+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45159e76cfe38cedfe4dd40df3c8d33618209b29)
