2.5 Semi-classical approximation

For high radial number n or orbital momentum l, the Wentzel–Kramers–Brilloin (WKB) method is known to provide an excellent approximation. In fact, the WKB approximation works already surprisingly well for small values of n or l. We refer here to the review of Quigg and Rosner [17], who present a detailed study of the WKB approximation for confining potentials. Let us simply recall here some basic results. In the WKB approximation, the eigenenergies E_n,0 of S-waves are given by

(2.33)

where r_c is the classical turning point, defined by V (r_c) = E. This formula is exact for the harmonic oscillator. There exists a variant, appropriate for potentials which are singular at the origin, and this modified formula is exact in the Coulomb case [17]. For positive power-law potentials Br^β, Eq. (2.33) and its generalizations to orbital excitations result in a simple analytic expression (μ = B = 1) [17]

(2.34)

For the radial excitations with l = 0 in a linear potential (β = 1), one gets the results shown in Table 2.2, where a comparison is made with the exact eigenvalues.

Table 2.3 shows the WKB approximation vs.compared with the exact numerical value for the leading Regge trajectory (n = 0) of the linear potential. Also shown is the variational approximation obtained from the trial wave function r^l+1 exp(−1 2αr²) borrowed from the harmonic oscillator, with the result

(2.35)

The quality of the WKB approximation at large orbital momentum is good, but not impressive.