Parabolic coordinates

The electrons within the atom are actually not quantised in parabolic coordinates, but instead, on account of the central field of the atom core, in polar co-ordinates. It would, then, not be logical to attempt to select favoured values of m and n3. Instead, we shall calculate the quantity... 

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 ... 

Show that in parabolic coordinates f, >], tp defined hy the equations... 

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

Let us now consider the overlap between the spherical and the Stark basis. For the latter, the momentum space eigenfimctions, which in configuration space correspond to variable separation in parabolic coordinates, are similarly related to alternative hyperspherical harmonics [2]. The connecting coefficient between spherical and 5 torA basis is formally identical to a usual vector coupling coefficient (from now on n is omitted from the notation) ... 

Calculation of the pre-exponential factor in eqn. (7) is connected with the analysis of electron motion in parabolic coordinates. The first time such calculations were conducted was by Lanczos [12]. The formulae he obtained were cumbersome and we shall not give them here. The simple formula for the probability of ionization of a slightly excited atom is given in ref. 13 as... 

The most straightforward way to treat the Stark effect is to use parabolic coordinates, for in parabolic coordinates the problem remains separable even with the electric field.1-3 The parabolic coordinates are defined in terms of the familiar Cartesian and spherical coordinates by... 

In parabolic coordinates the Schroedinger equation for an electron orbiting a singly charged ion with an external field in the z direction, i.e. in the potential given by Eq. (6.1), is written using Eqs. (6.5) as1-3... 

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... 

Taking the squared absolute value of Eq. (6.12) and using the definition of the parabolic coordinates given in Eq. (6.4), we can write an expression for the electron probability distribution in spherical coordinates,1... 

In parabolic coordinates the volume element dr = +rj)d drjd

general form of the wavefunction given in Eq. (6.21) and carrying out the angular integration, leads to... 

In parabolic coordinates the motion in the direction is bounded. Thus for ionization to occur the electron must escape to / = °°. Classically, ionization only occurs for energies above the peak Vb of the potential barrier in V(rj). If we ignore the short range Vrj1 term in Eq. (6.23b), an approximation good for low m, and set W = Vb we find the required field for ionization to be... 

As an example, we consider first the excitation of the n = 15 Stark states from the ground state in a field too low to cause significant ionization of n = 15 states. From Chapter 6 we know the energies of the Stark states, and we now wish to calculate the relative intensities of the transitions to these levels. One approach is to calculate them in parabolic coordinates. This approach is an efficient way to proceed for the excitation of H however, it is not easily generalized to other atoms. Another, which we adopt here, is to express the n = 15 nn n2m Stark states in terms of their nfm components using Eqs. (6.18) or (6.19) and express the transition dipole moments in terms of the more familiar spherical nim states.1,2... 

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. 

In Chapter 3 we investigated the development in time of a decaying state, expressed in terms of the time-independent eigenfunctions satisfying a system of two coupled differential equations, resulting from the separation of the Schrodinger equation in parabolic coordinates. In this analysis we obtained general expressions for the time-dependent wave function and the probability amplitude. 

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

Only one degree of freedom, /s, is needed to parameterize this curve, and we call it parabolic coordinate. Here, it is convenient to make this parameter equal to 2x3, such that the seam is defined parametrically as... 

Our new coordinates q[, qz, q t though dependent on the direction of the electric field, will not yet quite correspond to the parabolic coordinates used in the usual theory of the Stark effect. In order to obtain these, it would be necessary to go further in the approximation, by supposing that even the second order terms of the electric pertiurbation dominate over the relativistic terms. 

The probabilities of spontaneous transition differ from these expressions by a factor const.//. Hence, in spite of the infinite number of quantic states, the expression for the total probability of capture of an electron is convergent. As appears from Temple s work, it is of advantage to use in the treatment of problems involving capture of electrons parabolical coordinates. But we should like to point out that our expressions (10), (13), (14) are directly applicable, also, in the case of parabolical coordinates. They need only a change of notations to represent the functions, denoted by one of us by the letters M and AZ, and determining the matrices in this case. We propose to calculate the rates of recombination by this method in a later publication. 

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