Phase-integral approximation

Gutzwiller M C 1967 Phase-integral approximation in momentum space and the bound states of an atom J Math. Phys. 8 1979... 

We share the opinion expressed by Farrelly and Reinhardt (1983) that discrepancies between Stark effect results obtained by the use of the Carlini (JWKB) approximation and by accurate numerical calculations cannot be attributed to the break-down of the approximation, but are due to a failure to use the approximation in a correct way. An appropriate approach based on the phase-integral approximation of arbitrary order generated from an appropriately chosen base function is a still more efficient and often highly accurate method for the treatment of several problems, not only in quantum mechanics, but in various fields of theoretical physics. With... 

We shall first briefly describe the phase-integral approximation referred to in item (i). Then we collect connection formulas pertaining to a single transition point [first-order zero or first-order pole of Q2(z) and to a real potential barrier, which can be derived by... 

Phase-integral approximation generated from an unspecified base function... 

The corresponding phase-integral approximation is not valid close to z = 0, but the wave function that is regular and tends to zl+l, when z is a dimensionless variable that tends to zero, can be obtained... 

We emphasize that for the validity of (4.46) with the expressions (4.47a-c) for (2ra+1) the essential restriction is that d2Q2 z)/dz2 must not be too small at the top of the barrier, which means that close to its top the barrier is approximately parabolic, i.e., that the distance from the barrier to the transition points that are not associated with the barrier must be much larger than t" — t. However, when the energy is close to the top of the barrier, it is the slight deviation from parabolic shape close to the top that determines the values of the quantities K2n, n > 0, and one needs accurate values of these quantities for obtaining accurate values of in higher orders of the phase-integral approximation. 

In the present chapter we shall start from the results obtained in Chapter 3 and treat the Stark effect of a hydrogenic atom or ion with the use of the phase-integral approximation generated from an unspecified base function developed by the present authors and briefly described in Chapter 4 of this book. Phase-integral formulas for profiles, energies and half-widths of Stark levels are obtained. The profile has a Lorentzian shape when the level is narrow but a non-Lorentzian shape when the level is broad. A formula for the half-width is derived on the assumption that the level is not too broad. 

For the (2N + f)th-order phase-integral approximation we have according to (4.11) with A = f... 

In this chapter we collect and present, without derivation, in explicit, Anal form the relevant phase-integral quantities and their partial derivatives with respect to E and Z expressed in terms of complete elliptic integrals for the first, third and fifth order of the phase-integral approximation. For the first- and third-order approximations some of the formulas were first derived by means of analytical calculations, and then all formulas were obtained by means of a computer program. In practical calculations it is most convenient to work with real quantities. For the phase-integral quantities associated with the r -equation we therefore give different formulas for the sub-barrier and the super-barrier cases. As in Chapter 6 we use instead of L2 , L2n, K2n the notations LAn+1 >, L( 2n+1 KAn+l). 

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