The Phase Integral

Since is a distribution function, we obtain the value unity if 

This multiple integral (IX.2.2) is the famous phase integral of Gibbs and may be represented by the symbol Z. Thus 

Z is a function not of the coordinates of any of the particles but only of the limits of integration (wall boundaries for xi zn), the fields present, and the temperature. 

The average value of any quantity may be obtained by the usual procedures of averaging, and we have the following formula for the average value R of any quantity which depends on the variables 

By similar methods we can obtain for the other thermodynamic functions  

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An anharmonic correction for the density of states was also evaluated by solving the phase integral for the Cl-—CH3C1 intermolecular complex 39 i.e. ... 

According to equation (21) the systems of the canonical ensemble are conservative. Each system moves independently of all others and the phase integral exists for each of them. Each system therefore moves on a surface of constant energy and corresponds to a microeanonical ensemble. In this sense the canonical ensemble is built up from a multitude of microeanonical ensembles. Quantities defined for the microeanonical ensemble may therefore be averaged over the canonical ensemble. The original system which is represented by the canonical ensemble however, cannot be described, even approximately, as conservative. It is customary to denote the Hamiltonian of the systems of the canonical ensemble as the Hamiltonian of the original system, which is not justified. 

The phase integral s(r) has the derivative p(r) and it satisfies the secular equation... 

But no fine structure - yet - until in 1915 Bohr considered the effect of relativistic variation of mass with velocity in elliptical orbits under the inverse square law of binding, and pointed out that the consequential precessional motion of the ellipses would introduce new periodicities into the motion of the electron, whose consequences would be satellite lines in the spectra. The details of the dynamics were worked out independently by SOMMERFELD [38] and WILSON [39] in 1915/16 based on a generalisation of Bohr s quantization, namely, the quantization of action the values of the phase integrals Jf = fpj.d, - of classical mechanics should be constrained to assume only integral multiples of h. 

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... 

The phase-integral method for solving differential equations of the... 

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... 

Inserting (4.11) into (4.3a,b) along with (4.4) and putting A = 1, we get the phase-integral functions of the order 2N + 1, generated from the base function Q(z), which are approximate solutions of the differential equation (4.1). For N > 0 the function q(z) has poles at the transition zeros, i.e., the zeros of Q2(z), and simple zeros in the neighborhood of each transition zero (N. Froman 1970). 

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. 

With Q2 rj) given by (5.2) we recall (4.19) and (4.29) and normalize the physically acceptable solution g(m,ni,EmynuS rj) of the differential equation (2.32a,b) such that in the classically allowed region to the left of the barrier in Figs. 5.1b,c and Figs. 5.2b,c the phase-integral expression for this solution, with the use of the short-hand notation defined in (4.17), is... 

Procedure for Transformation of the Phase-Integral Formulas into Formulas Involving Complete Elliptic Integrals... 

The phase-integral quantities in the formulas obtained in Chapter 5 can be expressed in terms of complete elliptic integrals. One thereby achieves the result that well-known properties of complete elliptic integrals, such as for instance series expansions, can be exploited for analytic studies. Furthermore, complete elliptic integrals can be evaluated very rapidly by means of standard computer programs. 

In this chapter we shall describe the procedure for expressing the phase-integral formulas derived in Chapter 5 in terms of complete elliptic integrals. The integral in question is first expressed in terms of a Jacobian elliptic function and then in terms of complete elliptic integrals. Different elliptic functions are appropriate for different phase-integrals. For practical calculations it is most convenient to work with real quantities. For the phase-integrals associated with the r -equation it is therefore appropriate to use different formulas for the sub-barrier case and for the super-barrier case. We indicate in this chapter, where we use the notations L 2n+l L 2n+1 K< 2n+1 ) instead of the notations 2 , 2n used previously, the main... 

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). 

Values of the energy E and the half-width T for different states of a hydrogen atom in an electric field F[= F according to (2.17)] of various strengths, obtained both in previous work by other authors and in the present work with the use of the phase-integral formulas, are presented in the tables of the present chapter. We use atomic units (au), i.e., such units that p = e = h = 1. The positions E of the Stark levels were obtained from (5.33) except for the state with n = 30 in Table 8.7, where the more accurate formula (5.40) along with (5.42) has been used. The half-widths T were obtained from (5.54) along with (5.55) and are therefore accurate only when the barrier is sufficiently thick, which means that T is sufficiently small. 

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