Parabolic quantum numbers

In Ref, [2] relationships among alternative notation for parabolic quantum numbers are presented. Expression (12) is preferred for the form of the coefficient rather than (lm iim) =... 

The summation over usually poses great difficulties, especially when there is an infinite summation (continuous spectrum). Although the introduction of parabolic quantum numbers allows one to evaluate the sum in some cases, the calculations are still very complicated. 

Ionization of highly excited atoms is considered in detail in ref. 15. In particular, in the case of the ionization of highly excited H atoms in the state with the parabolic quantum number which is equal to the principal quantum number n, the authors have obtained the expression... 

The classic way of determining the energies of hydrogenic levels in a field is to solve the zero field problem in parabolic coordinates and calculate the effect of the field using perturbation theory. The zero field parabolic wavefunctions obtained by solving Eqs. (6.8a) and (6.8b) have, in addition to the quantum numbers n and m, the parabolic quantum numbers n, and n2, which are nonnegative integers.1 and n2 are the numbers of nodes in the iq and u2 wavefunctions and are related to n and m by... 

Fig. 6.7 Typical energy dependence of the square root of the density of states for the states = 7, m = 0 in the field E = 1.5 x 10 5 au. Vb is the energy of the peak in the potential V(t ). Each resonance is characterized by the parabolic quantum number and the width r 2 which varies rapidly with n2. For W Vb the continuum exhibits oscillations which damp out with increasing n2. The energies and widths of the prominent resonances are.

This result coincides with the exact ground-state energy in each azimuthal eigenspace, as is explicitly verified by writing the exact Coulomb level spectrum in terms of parabolic quantum numbers as... 

In turn, substitution of this frequency in Equation (67) reproduces the familiar expression for the energy eigenvalues and the identification of the principal quantum number in terms of the parabolic quantum numbers ... 

The reader can be convinced that the number of independent states with the same principal quantum number n is the same as that of Equation (40) by counting the different combinations of the parabolic quantum numbers in Equation (73). 

In Ref, [2] relationships among alternative notation for parabolic quantum numbers are presented. 

An equivalent form is given by Englefield.11 It is possible to find quite a variety of phases for the transformation coefficients of Eq. (6.18).10-13 The phase depends on the phase conventions established for the spherical and parabolic states. The choice of phase in Eq. (6.18) is for spherical functions with an /, as opposed to (-r)e, dependence at the origin and the spherical harmonic functions of Bethe and Salpeter. A few examples of the spherical harmonics are given in Table 2.2. The parabolic functions are assumed to have an ( n) ml/2 behavior at the origin and an e m angular dependence. This convention means, for example, that for all Stark states with the quantum number m, the transformation coefficient (nni>i2m nmm) is positive. To the extent that the Stark effect is linear, i.e. to the extent that the wavefunctions are the zero field parabolic wavefunctions, the transformation of Eqs. (6.17) and (6.18) allows us to decompose a parabolic Stark state in a field into its zero field components, or vice versa. 

Fig. 8.11 (a) Calculated density ofm = Ostates C, above the zero field ionization threshold for final states with quantum numbers nj = 25,26 at E = 5714 V/cm. Positions Z = 0 and 1 are indicated by the arrows, (b) Calculated oscillator strengths for excitation from the n = 2 parabolic states 210 (blue) and 200 (red) into the channel with quantum number n = 26 at 5714 V/cm in the energy region W s 0. Curve I, dfw [02io /dW curve II, dftymo,20o/dW... 

To account for quantum mechanical effects, an approximate quantum model that reproduces the findings of the two classical spin-based approaches was constructed in a next step.37 One foundation of this model was the finding that several (nonfmstrated) molecular antiferromagnets of N spin centers 5 (which can be decomposed into two sublattices) have as their lowest excitations the rotation of the Neel vector, that is, a series of states characterized by a total spin quantum number S that runs from 0 to N x 5. In plots of these magnetic levels as a function of S, these lowest S states form rotational (parabolic) bands with eigenvalues proportional to S(S +1). While this feature is most evident for nonfmstrated systems, the idea of rotational bands can be... 

In parabolic coordinates the hydrogen states are known as the Stark states. They are classified with the help of three quantum numbers. 

The potential energy curve shown in Figure 30, for the hydrogen molecule has been obtained experimentally it is, however, desirable that such curves be represented by a mathematical equation. We have already seen that the parabolic curve only approximates to the potential energy curve at low values of the quantum number and the expression for the variation of energy with intemuclcar distance obtained by Heitler and London is also a poor approximation, and moreover, has only been derived for the case of the hydrogen molecule. Recourse has therefore to be made to the derivation of empirical equations. The attraction energy may be represented by an... 

The particle is now considered in a one-dimensional potential well given by V — l2)kx (which is a parabolic function), like a pendulum. There is again only one quantum number, n, and the energies obtained by solving the wave equation are given by... 

By contrast, the strong electric field problem has appeared (perhaps prematurely) to be well understood. This view is reinforced by the fact that the Schrodinger equation for an atom in a strong electric field, although nonseparable in spherical polar coordinates (n and i are not good quantum numbers) does turn out to be separable in parabolic cylinder coordinates, given by... 

The stationary motions in a weak electric field are, however, essentially different from those in a spherically symmetrical field differing only slightly from a Coulomb field. In the latter (for which the separation variables are polar co-ordinates) the path is plane it is an ellipse with a slow rotation of the perihelion. In the former (separable in parabolic co-ordinates) it is likewise approximately an ellipse, but this ellipse performs a complicated motion in space. If then, in the limiting case of a pure Coulomb field, k or nf be introduced as second quantum number, altogether different motions would be obtained in the two cases. The degenerate action variable has therefore no significance for the quantisation. 

The Hamiltonian of the electron-photon interaction will be used in a very simplified form taking into account only the simplest band structure of a semiconductor with parabolic electron and hole bands without complications related to heavy and light holes, spin-orbit splitted hole band or with the Dirac model of the band structure in the case of small band gap semiconductors. In the case of simple parabolic band after their size quantization in a spherical symmetry quantum dots the electrons and holes are characterized by envelope wave functions with the quantum numbers I, n, m. An essential simplification of the future calculations is the fact that in the selected simple model the band-to-band transitions under the influence of the electron-photon interaction Hamiltonian take place with the creation of an e-h pair with exactly the same quantum numbers for electron and for hole as follows e l,n,m), h l,n,m). ... 

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