S.I.Vavilov State Optical Institute

Tuchkov per. 1, St.Petersburg, 199034 Russia

e-mail sergeev@soi.spb.su

In a "circular" Rydberg state with large magnetic quantum number m and small value of n- |m|, an electron performs small oscillations about fixed position (r0,z0) corresponding to the minimum of an effective potential Veff(r,z)=(m2- 1/4)/2r2+V(r,z). Its energy is

E(n1,n2)=V(r0,z0)+(n1+1/2)w1+(< I>n2+1/2)*2+(anharmonicterms).

Here, we investigate the case when w1/w2 is a rational number and so, two or more excited states with different (n1,n2) may have approximately the same energy. In a molecular vibration theory, the similar near-degenerate states are known as Fermi resonances (e.g., in CO2 molecule, where w1* 2w2).

The motivation for our study stems from large-dimensional approach. Within its framework, the quantum- mechanical problem reduces to a classical electrostatic problem and a subsequent vibrational analysis. Earlier, it was shown [1] that while the lowest orders of 1/D-expansion yield qualitatively correct results for a ground state of helium, it is no more the case for an excited 1s2s1S state where the 1/D-series strikingly diverges. Here, we explain the origin of the phenomenon discussed in the paper [1]. We prove that 1s2s1S state (n1=0, n2=1, n3=0) is strongly mixed with (200) state having near the same energy, and the energy levels of two states represent the branches of a single multivalued analytical function with a branching point at 1/D=-0.0114. This near-degenerate state is highly sensitive to a perturbation caused by anharmonic terms in a potential, and so the 1/D-expansion diverges. Suitable modification of the method is proposed that avoids the troubles related to energy quasi-crossing.

Apart from the helium problem, we study in detail a hydrogen atom in a uniform magnetic field (circular states) as a simple instructive example. The relation w1/w2=k1/k2*R holds, when the field reaches the strength H[k1/k2]=(R2- 1)1/2(R2+3)3/2/8m3, in atomic units. Particularly, H[1/2]*4.010m-3. The branching points connecting (10) and (02) levels are located closely to the real axis, at H=H[2/1]*ic-1/2m-7/2, where c=21*71/2. By means of 1/n-expansion [2] we calculate the energy spectrum and investigate the level quasi-crossings.


[1] D.Z.Goodson, D.K.Watson, Phys.Rev.A 48, 2668 (1993).

[2] V.M.Vainberg, V.S.Popov, A.V.Sergeev, JETP 71, 470 (1990).

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