A hydrogen atom is placed in a uniform magnetic field B0=B0z^ (the Hamiltonian can be written as in Equation 4.230). Use the Feynman–Hellmann theorem (Problem 7.38) to show that
∂B0∂En=−⟨ψn∣μz∣ψn⟩(7.114)
where the electron's magnetic dipole moment (orbital plus spin) is
μ=γoLmechanical+γS.
The mechanical angular momentum is defined in Equation 4.231.
*Note:* From Equation 7.114 it follows that the magnetic susceptibility of N atoms in a volume V and at 0K (when they're all in the ground state) is
χ=μ0∂B0∂M=−VNμ0∂B02∂2E0,(7.115)
where E0 is the ground-state energy. Although we derived Equation 7.114 for a hydrogen atom, the expression applies to multi-electron atoms as well—even when electron–electron interactions are included.