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The first-order energy shift is not well defined, since there is no unique way to choose a basis of eigenstates for the unperturbed system. The various eigenstates for a given energy will perturb with different energies, or may well possess no continuous family of perturbations at all.
The first-order perturbation matrix on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This gives shifts and splittings in the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent electric dipole moment of the symmetric top molecule.
The downward shift in the confined energy level discussed in the above equation is referred to as the Franz-Keldysh effect. The approximations made so far are quite crude, nonetheless the energy shift does show experimentally a square law dependence from the applied electric field, [5] as predicted.
Using perturbation theory, the first-order energy shift can be calculated as = >, which requires the knowledge of accurate many-electron wave function. Due to the 1 / M N {\displaystyle 1/M_{N}} term in the expression, the specific mass shift also decrease as 1 / M N 2 {\displaystyle 1/M_{N}^{2}} as mass of nucleus increase, same as normal mass ...
The good quantum numbers are n, ℓ, j and m j, and in this basis, the first order energy correction can be shown to be given by =, where = / is called the Bohr Magneton. Thus, depending on the value of m j {\displaystyle m_{j}} , each degenerate energy level splits into several levels.
In the position representation, this is the first-order differential equation (+) =, whose solution is easily found to be the Gaussian [nb 1] =. Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the ...
This equation is known as the Breit–Rabi formula and is useful for systems with one valence electron in an (= /) level. [ 9 ] [ 10 ] Note that index F {\displaystyle F} in Δ E F = I ± 1 / 2 {\displaystyle \Delta E_{F=I\pm 1/2}} should be considered not as total angular momentum of the atom but as asymptotic total angular momentum .
Fine and hyperfine structure in hydrogen (not to scale). This section presents a relatively simple and quantitative description of the spin–orbit interaction for an electron bound to a hydrogen-like atom, up to first order in perturbation theory, using some semiclassical electrodynamics and non-relativistic quantum mechanics.