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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 c 2001 John Wiley & success of the present endeavor under various circumstances. 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 I J = |p| dq = nh, (1) with two conjugate variables p and q and the integer 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; [email protected]. Contract grant sponsors: CSIR; DSA; UGC. International Journal of Quantum Chemistry, Vol. 82, 113–125 (2001) c 2001 John Wiley & Sons, Inc. However, a survey of the relevant literature reveals regrettably that the WS quantization rule (WSQR) has attracted little attention compared to its oftquoted 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 furnish 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 scaling relations that the WSQR yields are also found MUKHOPADHYAY, BHATTACHARYYA, AND PATHAK to be quite useful in the course of studying the same systems by exact quantum mechanics. Thus, the present endeavor is intended to supplement a rigorous analysis via the easier WS route. The general advantages of such a semiclassical approach are obvious. One quickly arrives at some exact results, and a few approximate ones that are otherwise significant, the more so in the large-n limit. Moreover, our understanding of the quantum classical correspondence becomes transparent. Predicting the behavior of maxima of near-exact probability functions, studying compressed systems, and regaining classical trajectories from quantal stationary wave functions, to be discussed below, are a few clear cases in point in this regard. Additionally interesting is the behavior of resonant states. Finally, we can see the advantages of developing a general semiclassical perturbation theory based on WSQR. The quantum classical correspondence is manifested in a number of approaches. Here, we modestly restrict ourselves to isolated stationary states and thus carefully bypass, for example, problems that are more general such as the arrival [9] at the Schrödinger’s equation from Newton’s laws, connection [10] between density of quantum eigenstates and classical periodic orbits, and the like. Near-exact results reported and employed in this work, wherever necessary, are obtained through a coupled variational strategy [11] by employing Fourier-like expansions [12], discussed elsewhere [13] in detail and need not be reiterated. Our organization is as follows. In the following section, we establish the connection of WSQR with the de Broglie relation and quantum mechanics proper, avoiding the more standard WKB route. The third section concentrates on two existing variants of WSQR and clarifies our stand in this regard, based on the “derivations” presented in the earlier section. Our major findings are put together in the fourth section. We then summarize the outcome of the whole endeavor, including possible future developments. Theoretical Analysis We present now two alternative ways of arriving at the WSQR. It is hoped that the heuristic arguments provided here will be of importance elsewhere as well. First, let us sketch the simplest route to arrive at (1). To this end, we confine attention to one dimension, import the traditional de Broglie relation, valid for fixed, positive momentum, and the 114 associated condition for periodic motion: I λ = h/p, nλ = dx. (2) These two relations can be coupled to yield I nh = p dx. (3) Consider now a conservative system with energy E in a potential field V that is a function of the coordinates. The momentum p is then well defined as long as E ≥ V {p = ±[2m(E − V)]1/2 }; but here it becomes a function. If we restrict ourselves again to motion in one dimension and wish to extend the applicability of (3) to include such cases of varying momentum, a very sensible choice will be to replace (3) by I nh = |p| dx, (4) keeping it in mind that (2) or (3) involves only the absolute value of p. Thus, (4) acts as a generalization of (3), reducing to the latter when p is a constant. However, it is the same as the WSQR given by (1), since x is conjugate to p now. Note also that the transition from (3) to (4) requires small |dp/dx|, a condition reminiscent of the applicability of WKB theory and is hence considered reasonable in semiclassical contexts. A kinship of WSQR with de Broglie hypothesis is thus unveiled. One further observes the following two definitions of wavelength (λ) and time period (τ ), or equivalently the frequency (ν), which complete the characterization of the de Broglie wave associated with particle motion under such a condition: H I dx dx 1 H . (5) , τ = =m λ=h ν |p| |p| dx The first of these follows from our discussion and appears to be new; the second is a standard prescription to estimate τ . Traditionally, the phase velocity is given by λν while h/(λm) defines the socalled particle velocity. It is important to also point out at this stage that neither (2) nor (3) accepts n equal to zero. This implies that condition (4) should involve only positive integers. Indeed, later we shall see that this is more reasonable than the original WSQR proposition that n can take values 0, 1, 2, . . . . Let us also notice that one cannot go back from (4) to the couple of equations in (2), or duality. Now we concentrate on the other aspect of WSQR. Since probability is inversely related to speed, one infers naturally that the WSQR should VOL. 82, NO. 3 WILSON–SOMMERFELD QUANTIZATION RULE correspond, along with the quantization condition (4), also to the probability distribution function P(x) given by P(x) = N/p(x), (6) with some constant α. We now go back to (9) and equate the real parts. The result is 2 ∇P 2 2 ∇ P 2 2 −h̄ − − α p = p2 , (12) 2P 2P where N refers to the normalization constant and x is in [XL , XR ], the left and right turning points. The inference is legitimate. Result (6) for stationary states follows from the equation of continuity whenever a real potential is considered. Each nondegenerate eigenenergy En thus describes a unique state via (6). We shall call it a WS state. The above route to WSQR rests on the de Broglie hypothesis. Basically, one quantization condition is derived from another ad hoc scheme. This may seem rather odd. Therefore, now we explore the other, presumably more convincing, origin. Here we start from a standard form for wave function in coordinate representation √ (7) ψ(x) = P eiθ , using (11). The first two terms at the left side of (12) are independent of mass. There is thus already a clear hint that α = ±1/h̄. A better route is to rearrange (12) as follows: 3(∇|p|)2 ∇ 2 |p| 1 − (13) = 2. α2 − 4 3 4p 2|p| h̄ where θ is a real function so that P would turn out to be |ψ|2 , the probability function. Noting that WSQR involves the momentum function, we now insist that ψ offer us the very function p through the defining relation −h̄2 ∇ 2 ψ = 2m(E − V) = p2 . ψ (8) Clearly, there is no better and more direct way to identify some momentum function in quantum mechanics than the satisfaction of (8). However, relation (8) reveals that p may be singular at the nodes. This is commonly interpreted as the “particle” motion with infinite speed. We shall see its manifestation later. Now, putting (7) in (8), one finds 2 ∇P 2 2 ∇ P − − (∇θ )2 −h̄ 2P 2P ∇θ ∇P = p2 . (9) + i ∇ 2θ + P The imaginary part at the left side of (9) must vanish. This leads us to ∇P ∇|p| ∇ 2θ =− = , ∇θ P |p| (10) where we employ (6) for the last equality. Note that (10) then demands ∇θ = α|p| (11) It justifies now beyond doubt the association α = ±1/h̄ since the bracketted term at the left side in (13) has to have, in general, a spatial dependence. We therefore obtain ∇θ = ±|p|/h̄, (14) where either sign may work. Thus, we now know the two functions P and θ in (7), subject to the restriction that is obvious from (13), viz. 2 2 h̄∇|p|/p2 34 − 12 |p|∇ 2 |p| ∇|p| 1. (15) It is remarkable that this relation is exactly the one found in the WKB context [14]. Thus, satisfaction of (15) is sufficient to ensure the validity of the whole endeavor. Needless to say, (15) fails to hold generally near the classical turning points and thus limits the validity of such WKB-type theories. This is also easily evident in the course of our transition from (12) to (13). At the turning points, |p| vanishes and hence the division by p2 is not permissible. Nevertheless, the quantization condition may now be seen to follow from choice (7) for the wave function. Confining attention to a particular coordinate, say x, and its conjugate momentum p, we write from (14) Z 1 x |p| dx, (16) θ (x) = h̄ taking the positive sign, for example. It then also follows that I I 1 |p| dx. (17) θ x + dx = θ (x) + h̄ As ψ should take the same value after completing a period, we now insist that [15] I ψ(x) = ψ x + dx . (18) This leads us to conclude, that the second term in (17) must be equal to 2nπ in order that (18) could be satisfied. Once again, we arrive at the WSQR (1). Note that here the quantum number n cannot take INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 115 MUKHOPADHYAY, BHATTACHARYYA, AND PATHAK negative integers on a physical ground: The term concerned in (17) is never negative. This completes the proof of the quantization condition (1) and the description of WS states via (6), which, we believe, are the two chief elements of the WSQR. We shall find occasions to deal with both of them or, in some cases, only one. Variants of WSQR Having explored the origin, we now focus attention on two prevalent variants of WSQR. This is necessary before proceeding to extract some useful features of the rule. As stated before, some authors [2, 16 – 18] use WSQR in the spirit of WKB results. Thus in the quantization condition (1), nh is often replaced by (n+ 12 )h in favor of the zero-point energy. Let us clarify the situation. The essential point is, if n in WSQR can take the value zero, it will violate the uncertainty principle. This is true. By putting n = 0, one deliberately assigns the state as a classical state of rest, which is known to defy any quantum description. Undoubtedly, a semiclassical technique cannot afford all the features of a rigorous theory. So, to be wise after the event, it seems apt to say that, in case of translational motion, the state obtained by putting n = 0 in WSQR does not correspond to any quantum state. In other words, we should employ n = 1 to find an approximation to the ground state, and so on. Our earlier observation via the de Broglie relation points to a similar inference. Rarely, however, this view is adopted (but see [5, 19]). Primarily it is due to an uncritical look at WSQR as being equivalent to the WKB result, J = (n + 12 )h, where n = 0 can be put, quickening a routine bypass to the aforesaid problem. Table I displays a few results to highlight the actual situation. Here we choose a family of oscillators described by the Hamiltonian H = −∇ 2 + x2N (19) TABLE I Comparative ground-state energies for V = x2N . N WSQR I WSQR II Exact 1 2 3 4 2 2.1851 2.2651 2.3098 1 0.8671 0.8008 0.7619 1 1.0604 1.1448 1.2258 ∞ π 2 /4 π 2 /16 π 2 /4 and display the energy eigenvalues for the ground state. In the table, WSQR I refers to our view that J = h and WSQR II takes J = h/2 to define the ground state. Note that we employ the standard convention [h = 2π, 2m = 1] to estimate the energies. One immediately observes that WSQR II leads to a wrong variation of E with N. On the other hand, results provided by WSQR I are initially far off, but increase rightly toward exactness in a gradual manner. Indeed, the agreement of WSQR II with exactness for the special case of N = 2 is usually highlighted in its favor, but the observed opposite trend is never pointed out. Very recently [20], however, this particular observation has been given the due importance to explore further modifications of WSQR II with respect to quantization of the action integral J, introducing ideas of complex phase-space trajectories and consequent complex turning points. We should also remark that the disturbing feature with WSQR II, especially in the N → ∞ limit, has attracted considerable attention. Addition of a fictitious potential to remedy the oddness for these kinds of problems in finite or semi-infinite domain is sometimes advocated [17, 21, 22]. In another version [23, 24], one attributes the defect to a phase loss and recovers the correct expression by introducing a “master index” in course of quantizing J. To avoid any confusion, we choose throughout the convention of WSQR I, and refer to it plainly as WSQR. Table I reveals its worth. Also, calling the WSQR the Bohr–Sommerfeld rule is no less widespread in the literature than its association with WKB. However, here we refrain from any discussion on this historical aspect because our sole objective is to explore various facets of the rule itself. Results and Discussion SOME GENERAL RESULTS AND THEOREMS Let us now step forward to derive some useful results. For convenience, here we restrict ourselves mainly to the one-dimensional context, but most of the findings are easily extendable to higher dimensions. To this end, the constant N in (6) is determined first by Z XR dx = 1. (20) N |p| XL However, from the basic WSQR we find Z XR p 2m(E − V) dx = nh/2. (21) XL 116 VOL. 82, NO. 3 WILSON–SOMMERFELD QUANTIZATION RULE Differentiating (21) with respect to n via the Liebniz rule [25] and using (20), one obtains N neatly as 2m dE (22) h dn and notes, in passing, also its connection with the time period defined in (5): τ = 2m/N. The significance of N is thus quite apparent. Now, the average value of a physical quantity O(x) in a WS state should be obtained by Z XL O(x)P dx, (23) hOi = N= XR where P(x) is the normalized WS probability function defined by (6). One then immediately finds that hpi = 0. (24) The reason is that p(x) is a double-valued function of x and hence can take either a positive value or an equally probable negative value, having the same magnitude, at a given x. To find hdV/dxi, we first note that |p| d|p| dV =− (25) dx m dx and so N X dV = − |p| XR = 0 (26) L dx m as p is zero at the turning points. Relations (24) and (26) imply, respectively, that the average momentum and force vanish for stationary states. They just ensure satisfaction of the ET for WS states. The average kinetic energy hTi is likewise found to be hTi = N n dE nh = , 4m 2 dn (27) where the last equality made use of (22). In semiclassical contexts, this result is known [16], but its importance has never been adequately emphasized. We expect that, as it is a semiclassical prescription, it should be valid primarily in the large-n limit. Accordingly, we approximate the differential by a difference and recast (27) as hTi 1 ≈ , (n 1En ) 2 (28) with 1En = En+1 − En . This result is very significant. We can estimate the kinetic energy of the nth state from spectroscopic data. A look at (22) reveals something similar with the normalization constant. Having recourse to the Bohr correspondence principle where the energy gap has been related to the frequency of revolution, we may have an explanation of such observations. This frequency, in turn, is the inverse of τ , and we already noted how τ is connected with N, and N with hTi. In Table II, we display some results for various potentials to show how far (28) is satisfied even in the small-n regime. The kinetic part of the Hamiltonian is of the same form as (19), but the potential is different for the various cases as shown in the table. One happily notes that near-exact calculational data exhibit a rapid approach to the WSQR result in all the cases, although a relation such as (28) [or (27)] cannot be had by clinging solely to quantum mechanics proper. This numerical evidence suggests that we may employ (28) even for low-lying states without incurring much error. In certain cases, it is also possible to provide an analytical justification of the parent form (27). To this end, we rearrange TABLE II Verification of the semiclassical relation (28) for low-lying states of various oscillators using near-exact calculational data. n V = x4 V = x6 V = x8 V = x2 + x4 V = x2 + x6 V = x2 + x8 1 2 3 4 5 6 7 8 9 0.25806 0.34643 0.39551 0.42035 0.43569 0.44608 0.45358 0.45925 0.46369 0.26883 0.34364 0.38694 0.41309 0.42939 0.44055 0.44867 0.45484 0.45969 0.27780 0.34657 0.38489 0.40972 0.42620 0.43764 0.44602 0.45242 0.45747 0.25374 0.35346 0.39873 0.42249 0.43721 0.44723 0.45449 0.45999 0.46430 0.26105 0.34854 0.38929 0.41431 0.43018 0.44110 0.44907 0.45515 0.45993 0.26831 0.34984 0.38663 0.41057 0.42669 0.43797 0.44625 0.45259 0.45760 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 117 MUKHOPADHYAY, BHATTACHARYYA, AND PATHAK (27) as 1 hTi = n(dE/dn) 2 (29) and find quickly that it is exact for the particle-in-abox model. For the harmonic oscillator, the left-hand side (lhs) of (29) becomes (2n + 1)/4n, leading to an error of O(1/n) that decays rapidly. The anharmonic oscillator defined by H = − 21 ∇ 2 + 12 x2 + λx4 (30) has been studied in detail [26] elsewhere. One finds an asymptotic expansion for energy of the form 4/3 1 2/3 −2/3 δ 1 + β n + λ E = λ1/3 α n + + 2 n + 12 2 −4/3 +γλ + · · · . (31) Here, values for constants α, β, etc. are also available. Employing (31), we obtain for the lhs of (29) an estimate 12 (1 + O((nλ)−2/3 ), which again leads to the desired value, albeit a bit slowly now, but surely. Going back to (25), it is readily found that Z N XR d|p| dV =− dx. (32) x x dx m XL dx An integration by parts leads us from (32) to Nnh dV = = 2hTi, x dx 2m (33) where the last equality follows by virtue of (27). It shows nicely that the VT is obeyed as well by the WS states. Consider next a Hamiltonian of the form h̄2 2 X ∇ + ki Vi (x), (34) H=− 2m i for which WSQR reads as v Z XR u X u t2m E − ki Vi dx = nh/2. XL (35) i (36) which is precisely the HFT. Thus, we have shown that quite a few standard quantum mechanical theorems are obeyed by WS states. SCALING RELATIONS A number of important scaling relations emerge from the strategy presented here. First, (22) shows 118 H(λ) = −∇ 2 + λx2N (37) and scale the coordinate as x → µx. An appropriate choice for µ then yields H(λ) = λ1/(N+1) H(I). (38) This means that energy must go as λ1/(N+1) . From the VT and HFT proved above, we obtain exactly the same scaling property of E via WSQR in this case. Such a scaling law is often useful in asymptotic perturbation theory. For example, the potential part of H in (30) for large λ goes practically as λx4 . Therefore, we expect E ∼ λ1/3 in the large-λ regime, and expansion (31) reveals that this is indeed the real situation. Finally, we shall uncover the mass dependence of energy. Differentiating (21) with respect to m, we are led to m dE = hVi − E = −hTi, dm (39) which is a known and correct result [28]. Thus, one can quickly gain insight via WSQR about the correct scaling behavior of eigenenergy. BEHAVIOR OF EXCITED-STATE DENSITIES Differentiating [25] (35) with respect to ki , we obtain straightforwardly ∂E = hVj i, ∂kj that dE/dn is always positive. This tells us that constants α, β should have the same signs if E is written in the form E = αnβ . Second, for a positive semidefinite V, we have hVi ≥ 0. Hence, we infer from (27) that 2E/n ≥ dE/dn, leading to the inequality β ≤ 2. It puts a limit to how E can scale with n. Third, the VT, HFT, and equation (27) yield a value for β in case of power-law potentials x2N : β = 2N/(N + 1). No exception to any of these results is known, as far as exactly solvable models are concerned. In a few other circumstances, however, one can justify beyond doubt that WSQR offers the correct scaling behavior. Fourth, to this end, we import Symanzik’s argument [27]. Let us choose the Hamiltonian It may appear that the WSQR is based on too gross an approximation to be practically useful in near-exact theoretical or computational work. We intend to provide here a clue to justify the converse. Let us pose a problem as follows. Consider an excited, bound, stationary state for a given quantum system. The probability density will show several maxima and minima (nodes) in general. Nevertheless, unless we solve the problem, it is neither possible to locate the positions of the nodes nor is one in a position to estimate the relative heights of VOL. 82, NO. 3 WILSON–SOMMERFELD QUANTIZATION RULE the peaks. Keeping in mind that only a handful of systems are exactly solvable, it will be extremely useful if there is any reasonable guideline with regard to any of the previously mentioned issues. The WS density really provides one such guideline. This refers to the relative peak heights. It should be remembered that all the oscillations in the density for a given state occur within the classical turning points. Beyond these points, one can observe only a monotonic behavior. Therefore, it seems likely that this problem is within the scope of WSQR. Further, as a semiclassical theory cannot offer the concept of nodes because of its conflict with infinite speed at a classical level of thought, the other problem of identifying the positions of nodes is surely beyond its reach. Now, confining attention to the problem proper, we designate by h(xi ) the height of a peak at position xi of an accurate density |ψ|2 and presume that it should be proportional to the WS probability func- tion at that point. This means s E − V(xj ) h(xi ) h(i, j) = = h(xj ) E − V(xi ) (40) should hold, with h(xi ) = (|ψ(xi )|2 )max . Surprisingly, (40) is obeyed very satisfactorily in all the cases we studied, though the lhs is derived from very accurate calculations while the right hand side (rhs) is merely of WS origin. Table III displays a few data to support our view. Here, we order the positions of maxima in P(x) as x1 < x2 < x3 , . . . , etc. Thus, x1 is the maximum closest to the origin. The chosen states in the table are such that x1 = 0. Owing to symmetry, further, we take the maxima along the positive real axis only. As regards the level of calculations reported throughout this work, we may remark that our data for the quartic anharmonic oscillator problem agree exactly with those quoted TABLE III Comparison of near-exact h(m, l) with the Wilson–Sommerfeld prediction (40) for bound and resonant states of some oscillators. h(m, l) Potential State (n) x2 11 x4 Energy m Exact WS 21.0 2 3 4 5 6 1.010 1.049 1.129 1.305 2.019 1.011 1.050 1.131 1.310 2.196 D0 50.2562545167 2 3 4 5 6 1.000 1.005 1.032 1.124 1.601 1.000 1.006 1.034 1.129 1.741 x2 − 0.01x4 D0 18.9688727458 2 3 4 5 6 1.013 1.061 1.159 1.378 2.263 1.014 1.061 1.161 1.384 2.515 D0 19 26.3995067837 2 3 4 5 6 7 8 9 10 1.004 1.024 1.063 1.134 1.251 1.449 1.917 4.103 1.420 1.007 1.030 1.071 1.141 1.252 1.459 1.932 4.225 1.804 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 119 MUKHOPADHYAY, BHATTACHARYYA, AND PATHAK recently [29] for ψ point wise. For others, results are of comparable status. Thus, the validity of (40) is truly startling. We may emphasize that relation (40) is not just meant for a matching. The nontrivial character of the problem may be appreciated from the following discussion. Consider the second excited state of the harmonic oscillator. We observe here that the maximum at x = 0 of the actual probability function is less pronounced than the other two on both sides of it, situated symmetrically. However, this we know only after solving the problem, and even then, we cannot physically explain why it is so. In fact, as the potential is lower toward the origin, one should have expected the reverse trend! This disturbing feature can be explained by (40) at the least. To assess the goodness of approximate densities, one can also check calculated h(m, l) against the WS predictions (40). Except for the maxima farthest from the origin, relation (40) is almost exactly obeyed for nearexact densities. Table III reveals it immediately. This is why h(m, l) shows the maximum deviation for largest xm . The departures in cases of maxima closest to the turning points are understandable. Opposite trends of the two P(x) functions considered here are responsible for the behavior. The near-exact one starts to exhibit an exponential fall-off just beyond this point; the WS function, on the contrary, rises very steeply to soon become singular. Another striking feature of (40) is that one can extend the applicability of such a relation to resonant states as well. To this end, we take classical turning points exist. As a result, P(x) is quite delocalized. Still, for both these states we find (40) to work reasonably well. This is noteworthy. In passing, one happily realizes the strength lying in WSQR with respect to the issue raised. H = −∇ 2 + x2 − λx4 so as to confine the motion in [−a, a]. WSQR will accordingly read as Z ap 2m(E − V) dx = nh/2 (43) (41) for which no bound eigenenergy state exists. Thus, the WSQR given by (1) cannot be employed, though (6) retains its validity. Here we try to find bound quasi-eigenstates of energy by following the stabilization method [30]. While the spirit behind the search for such long-lived states is different [31], here too we can get several orthogonal states at sufficiently small λ. We thus take λ = 0.01 and choose two cases. Table III includes our observation on the adequacy of (40) in these situations. For n = 11, one is sure to deal with a resonance that is computationally detectable beyond doubt. The estimated energy reveals also two distinct classical turning points within which the major part of the true density lies. However, the other, much higher energy state is very different in character. It is a state found to be orthogonal to all lower states; yet, it does not admit any “stabilization” from a computational standpoint. More remarkable is that no 120 COMPRESSED SYSTEMS Considerable recent attention has been focused on compressed systems [32 – 37] of which some are dedicated exclusively to atoms [35, 36] and some concentrate on oscillators with a view to unraveling thermal properties of solids [32] and phase transitions [33]. Imposition of special boundary conditions renders most solvable quantum mechanical problems analytically intractable. Standard theorems like the VT, in the traditional form, also fail to hold [37], thus adding more to the problem. This is understandable in the WS context too. In going from (32) to (33), and hence establishing a proof of the VT, we have taken advantage of the fact that |p| vanishes at either turning point. If this natural boundary is replaced by something else, |p| will not vanish and so form (33) would not show up. However, here we refrain from further discussion on this aspect; instead, our objective is now to investigate how far WSQR fares under such situations in a general manner. To make things simpler, let us choose the case of oscillators and consider a symmetric potential, V(x) = V(−x) so that the turning points now satisfy XL = −XR . Compression now dictates that V = V(x), |x| ≤ a; V = ∞, |x| > a (42) −a and will work if a ≤ XR . Obviously, (43) satisfies the HFT if V contains an embedded parameter that can be varied. Differentiating [25] (43) with respect to a and n, respectively, we arrive after a few steps at |p(a)| dE dE = −4 , (44) da h dn which clearly shows a desirable result that (dE/da) < 0. If E is large and a is small so that the inequality E V(x) for any x in [−a, a] is obeyed, (44) may be approximated to yield [with h = 2π, 2m = 1]: E dE = −2 . (45) da a This simplification is achieved after solving (43) to find an expression for E that is next put in (44). We VOL. 82, NO. 3 WILSON–SOMMERFELD QUANTIZATION RULE thus obtain a neat result applicable to any potential, provided the said conditions are maintained. Casting (45) in a finite-difference form that looks like − a E(a + 1a) − E(a) = 2, E(a) 1a (46) one can easily check its validity. Table IV displays some near-exact calculational data for H in (37) at λ = 1 and N = 1, 2. The success is certainly worth mentioning though strictly (45) applies to a box model. From the table, it is seen that results reported for the lhs of (46) improve with n and N. A glance at the form of V(x) explains these in a transparent manner and further elaboration is not needed. By lowering a, results could be made still better. In order to analyze the error and hence to have a closer look into the problem, we may try to solve (43) via WSQR in a specific case. Here an asymptotic analysis for the harmonic oscillator is presented. Putting V = x2 in (43), along with h = 2π and 2m = 1, we obtain s # " a a2 −1 a (47) 1− nπ = E sin √ + √ E E E as the exact WSQR. It simplifies to a 5 1 a 3 a √ +O √ − (48) nπ ≈ E 2 √ 3 E E E when a E1/2 , showing quickly the order of magnitude of the neglected terms. Taking the first two terms at the rhs of (48), one can now check that the TABLE IV Testing the validity of semiclassical relation (46) for compressed oscillator states at a = 1. energy will assume the form 6 2 a a2 nπ +O 2 . + E= 2a 3 n (49) In arriving at (49), the tacit assumption a2 /nπ 1 is made. Notably, the first term in (49) corresponds to the box model. Thus, one realizes that the primary correction term to energy is a2 /3. Moreover, the leading energy term in (49) shows that when a E1/2 is satisfied, a2 /nπ 1 will be automatically obeyed. Hence, the overall analysis is based on the former requirement only. The question now is whether the above WS analysis has anything to do with rigorous studies. In this context, we have two points to mention. First, it is not so easy to uncover an analytic dependence of E on a through a sophisticated procedure. Second, and more importantly, the WSQR should work much better in this situation and so any inference based on it should correspond to near exactness. This is because normally quantum mechanics takes care of the contribution of V(x) beyond the turning points that WSQR cannot. It is a major reason to make them different. However, artificial boundaries prevent quantum mechanics from adding such extra contributions. As a result, they are likely to behave more closely. Indeed, Table V justifies this conjecture beyond doubt. We choose here H in (37) at λ = 1, N = 1. Several states are taken and the WS energies obtained through (47) are compared with nearly exact results. The observed agreement is very striking now as compared to data in Table I. Additionally, one can check that the contribution of the first two terms in (49) is indeed dominant here, enhancing our faith on the practical use of such asymptotic analysis even in a rigorous formulation. The moral is clear: studies on 1a V(x) n 0.1 0.01 0.001 x2 1 3 5 7 9 11 1.547 1.683 1.715 1.725 1.729 1.731 1.774 1.915 1.949 1.960 1.964 1.966 1.799 1.942 1.976 1.986 1.990 1.993 1 3 5 7 9 11 1.633 1.691 1.717 1.725 1.729 1.731 1.873 1.928 1.952 1.961 1.965 1.966 1.900 1.955 1.979 1.987 1.991 1.993 x4 TABLE V Performance of WSQR for V(x) = x2 in [−1, 1]. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY Energy n WS Exact 1 3 5 7 9 11 13 15 2.8101 22.5409 62.0187 121.2362 200.1929 298.8889 417.3242 555.4986 2.5969 22.5177 62.0105 121.2320 200.1904 298.8873 417.3230 555.4977 121 MUKHOPADHYAY, BHATTACHARYYA, AND PATHAK TABLE VI Adequacy of relation (50) in providing estimates of the virial ratio for compressed systems. V = x4 V = x2 + x4 a lhs rhs lhs rhs 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.5 1.32×10−9 8.43×10−8 5.40×10−6 6.15×10−5 3.45×10−4 1.32×10−3 3.93×10−3 1.50×10−2 1.35×10−9 8.61×10−8 5.51×10−6 6.28×10−5 3.53×10−4 1.35×10−3 4.03×10−3 1.55×10−2 1.12×10−7 1.86×10−6 3.38×10−5 2.05×10−4 8.00×10−4 2.43×10−3 6.22×10−3 2.05×10−2 1.13×10−7 1.88×10−6 3.42×10−5 2.08×10−4 8.12×10−4 2.47×10−3 6.37×10−3 2.13×10−2 compressed systems via WSQR are more worthwhile than the normal ones. To strengthen the above view further, we finally consider here the problem with the VT. Following the standard route [see Eqs. (32) and (33)], if we employ (25) and (43), WSQR yields dV x dx 2a =1− |p(a)|. (50) 2hTi nπ In spite of its semiclassical origin, this simple relation possesses a number of virtues. Since the second term at the rhs of (50) is always positive, it indicates that the virial ratio, given by the lhs, is less than unity under compression. Indeed, this should be so because contribution of the kinetic part has to dominate in such situations. As a → 0, |p(a)| ≈ E1/2 and one basically deals then with a box model. Putting the corresponding expression for E, we find that the rhs vanishes. From quantum mechanics, one infers the same. We shall now see that (50) is also approximately valid for ultracompressed states even when a rigorous formulation is adopted. To this end, we compute both the lhs and the rhs of (50) by employing near-exact wave functions and energies and test their agreement. Table VI displays relevant data for the state n = 11 in two cases. That (50) stands as an excellent guide is certainly evident here. strict quantum descriptions. The idea involved is very simple. We start with a pointwise precise P(x) for some excited quantum stationary state of the system concerned with eigenenergy E. Analytical form of P(x) need not be known. In view of the inverse WS relationship (6), we may say that ±1/P(x) should correspond to some momentum function. Now, we rescale it and call the derived “classical” momentum σ p(c) = ± , (51) P(x) where σ has to be found by insisting p(c) to satisfy the classical requirement p(c)2 + V(x) = E. (52) 2m The satisfaction of (52) at any one point will suffice. Note that, for reasons already discussed, it is apt to choose any minimum point of (51). We have here chosen the minimum closest to the origin to estimate the scale factor. In all cases of our concern, it is situated at x = 0. Now, joining all the minima, one obtains the classical trajectory. Stated otherwise, locus of the minima of p(c) describes exactly the trajectory of classical motion. Let us remark that the maxima of p(c) lie at infinity, corresponding to the nodes in P(x). Figure 1 displays the harmonic oscillator case where the trajectory is basically a circle with our choice of constants mentioned before. Figures 2–4 show how such trajectories are nicely constructed for the problems discussed in Table III. Comparing Figure 4 with Figure 3, one also clearly sees that resonances can be meaningful only if the corresponding classical trajectory is closed. Thus, one extracts benefit out of WSQR in interpreting a CLASSICAL TRAJECTORIES FROM NEAR-EXACT DENSITIES Variations of the peak heights of accurate P(x) have already been found to follow the WS wisdom. It highlights the importance of entry of a naïve semiclassical route in the quantum domain. Now, we venture whether one can restore classicality from 122 FIGURE 1. Plot of the derived classical momentum p(c) [see Eq. (51)] vs. x to construct the classical trajectory by following the locus of its minima. The state n = 11 of the harmonic oscillator is chosen here. VOL. 82, NO. 3 WILSON–SOMMERFELD QUANTIZATION RULE FIGURE 2. Same plot as in Figure 1, now highlighting FIGURE 4. Same plot as in Figure 1, here the state n = 11 of the quartic oscillator. concentrating on a state n = 19 of the potential V(x) = x2 − 0.01x4 . It does not provide a closed trajectory and hence is not a detectable resonant state, though orthogonal to all lower-energy resonant states. purely quantum mechanical observation. In these last two plots, we have displayed the full classical phase-space behavior to see how far our proposition works. It is obvious that these pathological cases are much more complicated to handle. Still, we are not far from the truth. This justifies the endeavor. Finally, we should remark that we have been careful to deliberately bypass any reference to time in this context because our concern is time-independent states. So, p(x) has been viewed as a function of x only, as it should be. Nevertheless, the conventional timeaveraged estimate of an observable O(x), defined by Z 1 τ O dt, (53) O= τ 0 leads one also to (23) after minor manipulations, establishing the desired equivalence. PERTURBATION THEORY The WSQR can guide us in a true quantum mechanical perturbative context also. This has been noted before [14] in dealing with anharmonic oscillators. So, here we shall present a brief but straightforward analysis and seek a few more interesting results. Choosing h = 2π and m = 12 , we find from (22) that dE/dn = 2πN. (54) For a given state n, this relation is primary for subsequent development. Let us consider now the Hamiltonian H = −∇ 2 + x2 + λx4 as an illustration. For this problem, we find Z XR 1 1 =2 dx, 2 − λx4 )1/2 N (E − x n 0 (55) (56) which simplifies to π 1 1 1 2 = , ; 1, κ F , N (1 + 4En λ)1/4 2 2 (57) where F stands for the standard hypergeometric function, admitting a known expansion [25] in κ 2 , and En is the nth state energy. Here κ 2 reads as (58) κ 2 = 12 1 − (1 + 4En λ)−1/2 . From (54) and (57), we obtain after rearrangement FIGURE 3. Same plot as in Figure 1 but now for the resonant state n = 11 of the potential V(x) = x2 − 0.01x4 . Note that here a closed classical trajectory has been found. Actually, an open trajectory is also associated with it as shown, but unrelated to any bound state. F(dEn /dn) = 2(1 + 4En λ)1/4 . (59) Now, expanding En in powers of λ and equating coefficients of each power, we obtain the n dependence of each energy correction term. In the large-n INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 123 MUKHOPADHYAY, BHATTACHARYYA, AND PATHAK regime, where (54) really applies, the integration constant for each such term should be disregarded while solving. Thus, we finally obtain En = 2n + 3n2 λ/2 − 17n3 λ2 /8 + · · · , (60) which agrees exactly with the true expansion [26, 38] for large n. Using the VT and HFT for H in (55), it is easy to derive from the above expansion similar ones for hx2 i and −h∇ 2 i: 2 x = n − 3n2 λ/2 + 85n3 λ2 /16 + · · · , (61) −∇ 2 = n + 3n2 λ/2 − 51n3 λ2 /16 + · · · . In addition, one can show that the following asymptotic behaviors (λ → ∞) hold: 2 2 ∞ −1/3 1/3 , x = x λ , En = E∞ n λ (62) 2 2 ∞ 1/3 −∇ = −∇ λ . Casting (60) and (61) in the form of polynomials raised to appropriate powers such that the leading λ dependences given by (62) are obeyed, we can estimate the asymptotic values of the coefficients in (62). For example, we rewrite (60) as 9nλ 1/3 + O λ2 En = 2n 1 + 4 9nλ 33n2 λ2 1/6 + + O λ3 (63) = 2n 1 + 2 16 to extract, respectively, the first and second approximations to E∞ n . Table VII shows these estimates along with the exact values [26]. Apart from the correct n dependence, the results derived via WSQR are seen to be very close to the exact ones. 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