Download Wilson-Sommerfeld quantization rule revisited

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; chemkbbu@yahoo.com.
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
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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. This
is remarkable since we employed merely a weakcoupling expansion and that too up to second-order
term only. The procedure can be extended to higher
orders. Other problems can also be treated likewise.
Conclusion
We tried here to demonstrate the usefulness of
WSQR and, in general, the WS route to explain, interpret, and analyze various quantum mechanical
observations. That it is able to supplement a rigorous analysis in a simple way should now be clear.
For brevity, we did not consider here a few other
aspects of WSQR, especially the multidimensional
extension, but hope to report them elsewhere in
near future.
ACKNOWLEDGMENT
One of us (K.B.) is grateful to Prof. D. Mukherjee,
and Dr. A. K. Bhattacharya for discussion. S.M. acknowledges CSIR, India, for a research fellowship.
K.B. acknowledges partial financial assistance from
DSA, UGC.
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TABLE VII
Asymptotic values of coefficients in (62) for total
energy, mean square displacement, and kinetic
energy from low-order WS expansions.a
Order of
approximation
1
2
a Exact
124
4/3
E∞
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