WKB to all orders and the accuracy of the semiclassical quantization

We perform a systematic WKB expansion to all orders for a one–dimensional system with potential V (x) = U0/ cos 2 (αx). We are able to sum the series to the exact energy spectrum. Then we show that any finite order WKB approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq Submitted to Journal of Physics A: Mathematical and General e–mail: [email protected] e–mail: [email protected] 1 In the last years many studies have been devoted to the transition from classical mechanics to quantum mechanics. These studies are motivated by the so–called quantum chaos (see Ozorio de Almeida 1990, Gutzwiller 1990, Casati and Chirikov 1995). An important aspect is the semiclassical quantization formula of the energy levels for integrable and quasi–integrable systems, i.e. the torus quantization initiated by Einstein (1917) and completed by Maslov (1972, 1981). As is well known, the torus quantization is just the first term of a certain h̄-expansion, the so–called WKB expansion, whose higher terms can be calculated with a recursion formula at least for one degree systems (Dunham 1932, Bender, Olaussen and Wang 1977, Voros 1983). Recently it has been observed by Prosen and Robnik (1993) and also Graffi, Manfredi and Salasnich (1994) that the leading–order semiclassical approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing. This result has been shown to be true also for the leading (torus) semiclassical approximation by Salasnich and Robnik (1996). In this paper we analyze a simple one–dimensional system for which we are able to perform a systematic WKB expansion to all orders resulting in a convergent series whose sum is identical to the exact spectrum. For this system we show that any finite order WKB (semiclassical) approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing. The Hamiltonian of the system is given by H = p 2m + V (x) , (1) where V (x) = U0 cos2 (αx) . (2) Of course, the Hamiltonian is a constant of motion, whose value is equal to the total energy E. To perform the torus quantization it is necessary to introduce the action variable I = 1 2π ∮ pdx = √ 2m α ( √ E − √ U0) . (3) The Hamiltonian as a function of the action reads H = α 2m I + 2α √ U0 2m I + U0 , (4) 2 and after the torus quantization I = (ν + 1 2 )h̄ , (5) where ν = 0, 1, 2, . . ., the energy spectrum is given by E ν = A[(ν + 1 2 ) + 1 2 B] , (6) where A = α2h̄/(2m) and B = √ 8mU0/(αh̄). The Schrödinger equation of the system [− h̄ 2 2m d dx2 + V (x)]ψ(x) = Eψ(x) , (7) can be solved analytically (as shown in Landau and Lifshitz 1973, Flügge 1971) and the exact energy spectrum is: E ν = A[(ν + 1 2 ) + 1 2 √ 1 +B2] , (8) where ν = 0, 1, 2, . . .. We see that the torus quantization does not give the correct energy spectrum, but it is well known that the torus quantization is just the first term of the WKB expansion. To calculate all the terms of the WKB expansion we observe that the wave function can always be written as ψ(x) = exp ( i h̄ σ(x)) , (9) where the phase σ(x) is a complex function that satisfies the differential equation σ′ 2 (x) + ( h̄ i )σ′′(x) = 2(E − V (x)) . (10) The WKB expansion for the phase is given by σ(x) = ∞