Application of the VFM to the Zeeman Effect in Hydrogen

Abstract

Let us consider a hydrogen atom of nuclear charge Z, placed in an external uniform magnetic field along the x₃≡z direction. The Hamiltonian operator describing this system in the non-relativistic approximation, and with suitable units (Appendix H), is:

$${{H}_{e}}=-\frac{1}{2}({{\frac{\partial }{\partial {{\rho }^{2}}}}^{2}}+\frac{1}{\rho }\frac{\partial }{\partial \rho }+{{\frac{\partial }{\partial {{x}_{3}}^{2}}}^{2}})+{{\frac{m}{2{{\rho }^{2}}}}^{2}}+{{\frac{\lambda }{8}}^{2}}{{\rho }^{2+}}\frac{\lambda }{2}({{L}_{{{x}_{3}}}}+{{g}_{s}}+{{S}_{{{x}_{3}}}})-\frac{z}{r}$$

(33.1)

Wherer

$${{r}^{2}}=x\frac{2}{1}+x\frac{2}{2}+x\frac{2}{3}={{\rho }^{2}}+x\frac{2}{3}$$

(33.2)

Let E (Z, λ) denote the set of eigenvalues of that portion of H excluding the paramagnetic field terms (Eq. (31.1)), i.e.:

$$E(Z,\lambda )=<H(Z,\lambda )>=<{{H}_{e}}>-\frac{\lambda }{}(m+{{g}_{s}}{{m}_{s}})m=0,\pm 1,\pm 2\ldots {{m}_{s}}=\pm 1/2$$

(33.3)

The importance of this problem was already widely discussed in §.30.As pointed out, the eigenvalue problem has no analytic solution due to the coupling of the two coulombic degrees of freedom (p and x₃). Our purpose is to apply the VFM to derive valid approximate expressions for E(Z,A), VX>0 /1,2/.