# energy diff

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NoetherTheorem
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g = 1, I = 1/2 , B_0 = 2 T, |m_J| = 1, mu_B = 9.2741e-24 J/T

photons absorbed in the case without hyperfine interaction: E (m_J) = – g * mu_B * B_0 * m_J
E (m_J) = – 1 * 9.2741e-24 J/T * 2 T * 1 = – 18.5482e-24 J = – 18.5482e-24 * 6.24150913e+18 eV = 115.7687e-6 eV = 115.7687 microelectron volts = 0.1157687 meV

switching on the hyperfine interactions:
E (m_I) = – g * mu_N * B_0 * m_I + g * mu_N * B_hf /J * m_I * m_J = g * mu_N * m_I * ( – B_0 + (B_hf * m_J)/J )

Then substituting:
m_J = -3/2; E (m_I = 1/2) = 1 * 5.0508e-27 J/T * 1/2 * ( – 2 T + (10 T * -3/2)/3/2 ) = – 30.3048e-27 J = – 30.3048e-27 * 6.24150913e+18 eV = – 189.147 neV
m_J = -1/2; E (m_I = 1/2) = 1 * 5.0508e-27 J/T * 1/2 * ( – 2 T + (10 T * -1/2)/3/2 ) = – 84.065 neV

transition: – 84.065 neV – ( – 189.147 neV) = 105.082 neV

m_J = -1/2; E (m_I = – 1/2) = 1 * 5.0508e-27 J/T * (- 1/2) * ( – 2 T + (10 T * -1/2)/3/2 ) = 84.065 neV
m_J = 1/2; E (m_I = – 1/2) = 1 * 5.0508e-27 J/T * (- 1/2) * ( – 2 T + (10 T * 1/2)/3/2 ) = -21.016 neV

transition: 84.065 neV – ( – 21.016 neV) = 105.081 neV

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