Wilson–Sommerfeld Quantization Rule Revisited S. MUKHOPADHYAY, K. BHATTACHARYYA, R. K. PATHAK Department of Chemistry, The University of Burdwan, Burdwan 713 104, India Received 25 July 2000; revised 9 October 2000; accepted 13 November 2000 ABSTRACT: A fresh look at the origin of the Wilson–Sommerfeld quantization rule has been pursued to gain new insight. The rule is shown to provide states that satisfy several well-known theorems of standard quantum mechanics. A few other useful results and scaling relations are also derived. They emerge to act as nice guiding rules of thumb in the course of rigorous computations. Certain features of true excited-state densities can be understood. Goodness of approximate densities can be assessed. Compressed systems can be studied profitably. A route is also sketched that allows one to retrieve classical trajectories from near-exact energy eigenfunctions for both bound and resonant states by exploiting this rule. Additionally, a discussion on semiclassical perturbation theory is presented emphasizing the asymptotic behavior. Pilot calculations demonstrate the success of the present endeavor under various circumstances. c  2001 John Wiley & Sons, Inc. Int J Quantum Chem 82: 113–125, 2001 Key words: semiclassical method; Wilson–Sommerfeld rule; classical trajectory; quantum classical correspondence Introduction T he Wilson–Sommerfeld (WS) quantization rule [1], represented succinctly by the formula J =  |p| dq = nh, (1) with two conjugate variables p and q and the in- teger n, was introduced primarily to account for certain experimental facts. Usually, it is viewed as a simplification [2 – 4] of the WKB approximation. Correspondence to: K. Bhattacharyya; e-mail: burchdsa@cal. vsnl.net.in; chemkbbu@yahoo.com. Contract grant sponsors: CSIR; DSA; UGC. However, a survey of the relevant literature reveals regrettably that the WS quantization rule (WSQR) has attracted little attention compared to its oft- quoted successor [5], the WKB formalism [6]. So, we feel obliged to study it in detail. As we shall see, it is possible to derive the WSQR without any reference to the WKB analysis. Its connection with the de Broglie hypothesis will also be of interest. Furthermore, we show that the rule is able to fur- nish states that satisfy the virial theorem [7] (VT), the Hellmann–Feynman theorem [4, 8] (HFT), and Ehrenfest’s theorem [3] (ET). All these features are characteristic of exact quantum mechanical energy eigenfunctions. Some other results and a few scal- ing relations that the WSQR yields are also found International Journal of Quantum Chemistry, Vol. 82, 113–125 (2001) c  2001 John Wiley & Sons, Inc.