Hydrogen atom Stark effect

The advent of tunable lasers created a radically new situation as to the possibility for selective excitation of high Rydberg states and for making precise measurements on their properties. Highly excited atoms are very sensitive to external fields, and currently used field ionization methods are very powerful for detecting Rydberg states. As a consequence of these circumstances, such an old problem as the Stark effect in atomic hydrogen attracted a renewed interest. 

Stark effect of a hydrogen-like atom, using the Schrodinger wave mechanics. Their equation, obtained independently and by different methods, is... 

The foundation of our approach is the analytic calculations of the perturbed wave-functions for a hydrogenic atom in the presence of a constant and uniform electric field. The resolution into parabolic coordinates is derived from the early quantum calculation of the Stark effect (29). Let us recall that for an atom, in a given Stark eigenstate, we have ... 

In many problems for which no direct solution can be obtained, there is a wave equation which differs but slightly from one that can be solved analytically. As an example, consider die hydrogen atom, a problem that was resolved in Section 6.6. Suppose now that an electric field is applied to the atom. The energy levels of the atom are affected by the field, an example of the Stark effect. If die field (due to the potential difference between two electrodes, for example) is gradually reduced, the system approaches that of the unperturbed hydrogen atom. 

Example 7. Stark effect of the hydrogen-like atom. The Hamiltonian is given by... 

The Stark effect. The conditions l)v, 2)v just stated are rather complicated and may appear very restrictive at first sight. But in reality this is not so, although their rigorous verification is often rather tedious. We shall illustrate what these conditions mean in the case of the Stark effect of the hydrogen-like atom (Ex. 7, 12). Here we have... 

Figure 2.10 -trajectories for the resonance energy (Stark effect in the hydrogen atom), where N is the number of basis functions per /-value, for more details see Ref. [67]. Taken from Reinhardt [67] with permission of IJQC. 

M. Hehenberger, H.V. McIntosh, E. Brandas, Weyl s Theory Applied to the Stark Effect in the Hydrogen Atom, Phys. Rev. A10 (1974) 1494. 

For an accurate data analysis, a detailed understanding of systematic effects is necessary. Although they are significantly reduced with the improved spectroscopy techniques described above, they still broaden the absorption line profile and shift the center frequency. In particular, the second order Doppler shift and the ac-Stark shift introduce a displacement of the line center. To correct for the second order Doppler shift, a theoretical line shape model has been developed which takes into account the geometry of the apparatus as well as parameters concerning the hydrogen atom flow. The model is described in more detail in Ref. [13]. 

The last term in the first equation (4) provides the initial non-adiabatical condition at t = 0 for 2s-metastable state of hydrogen atom. The corresponding terms describing the 2p to 2s spontaneous transitions and the Is Stark effect have been neglected as small. 

The study of the hydrogen atom in a uniform magnetic field (HAMF) is considerably more complex than the LoSurdo-Stark effect (see Cizek and Vrscay, 1982, and references therein). The Hamiltonian is not separable and reducible to a one-dimensional problem. For a field along the z axis the problem is inherently two-dimensional. Thus, the methods mentioned above which rely on the one-dimensional aspect of the LoSurdo-Stark effect and its separability in parabolic coordinates are special and not directly extendable to the Zeeman effect. 

---