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Complex dynamics in diatomic molecules. Part I: Fine structure of internuclear potential
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Chaos, Solitons and Fractals 37 (2008) 962–976 www.elsevier.com/locate/chaos
Complex dynamics in diatomic molecules. Part I: Fine structure of internuclear potential Ciann-Dong Yang
*
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan, ROC Accepted 30 March 2007
Abstract The current understanding of internuclear potential of diatomic molecules is limited to electronic states. In this paper, we point out that every electronic-state internuclear potential is actually accompanied by a fine structure, which characterizes the detailed internuclear potential for each vibrational substate belonging to the same electronic state. This fine structure, which governs the bond length, the force constant, the vibration frequency, and the quantum motion for each vibrational state, is obtained by extending Bohm’s quantum potential [Bohm D. A suggested interpretation of the quantum theory in terms of hidden variable. Phys Rev 1952;85:166–79] to complex domain. It is shown that the fine structure of the internuclear potential can be determined exactly by the vibrational wavefunctions so that its predictions about the internuclear forces (Part I) and the vibrational nuclear quantum motion (Part II) are fully consistent with the Max Born’s probability interpretation of the vibrational wavefunctions, and are in line with El Naschie’s E-infinity [El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36; El Naschie MS. E-infinity theory – some recent results and new interpretations. Chaos, Solitons & Fractals 2006;29:845–53; El Naschie MS. The concepts of E-infinity. An elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22:495–511], Nottale’s scale relativity [Nottale L. Fractal space-time and microphysics: Towards a theory of scale relativity. Singapore: World Scientific; 1993], and Ord’s random walk [Ord G. Fractal space time and the statistical mechanics of random works. Chaos, Soiltons & Fractals 1996;7:821–43] approaches to quantum mechanics. 2007 Elsevier Ltd. All rights reserved.
1. Introduction The knowledge of internuclear potentials of diatomic molecules is very important in a wide variety of fields ranging from gas kinetics to astrophysics. A great deal of information about the structure of a molecule is summarized in its potential energy curve. Among the numerous methods proposed to determine the potential energy curve, perhaps the most satisfactory is the calculation of the curve from the experimental energy levels using the Rydberg–Klein–Rees (RKR [7]) method. The use of the RKR method is inevitably limited to the region where sufficient spectroscopic data are available. An alternative approach is to fit the potential curve by a certain form of algebraic expression when the *
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C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
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parameters in the expression are evaluated from the known spectroscopic constants. Many attempts to find the suitable empirical functions have been made, such as Morse [8], Hulburt–Hirschfelder [9], Rosen–Morse [10], Rydberg [7], Po¨schl–Teller [11], Linnett [12], Frost–Musulin [13], Varshni [14], Lippincott [15], and Hua [16], etc. The performance comparisons among these empirical potential functions with respect to their ability to reproduce the experimental data were studied in the review articles [14,17]. Up to now, the internuclear potentials determined either by quantum-mechanical computation or by experimental curve fitting are exclusively established for electronic states; no internuclear potential specific to vibrational states has been reported before. Consequently, all the vibrational substates belonging to the same electronic state are presumed to be governed by the same form of potential energy, sharing the same equilibrium bond length and the same force constant. However, the probability predictions using the vibrational wavefunctions solved from the Schro¨dinger equation indicate that each vibrational substate may have its own equilibrium bond length and its unique distribution of the possible bond lengths. It is evident that the single internuclear potential developed for an electronic state is not detailed enough to explain the diverse vibrational dynamics existing in the many vibrational substates belonging to the same electronic state. This paper aims to show that in addition to the common electronic-state potential energy, each vibrational substate does have its own potential energy, which can be determined by solving the Schro¨dinger equation with the electronic-state potential served as the input potential. By using the complex-space formulation of fractal spacetime [2–6], we demonstrate that a fine structure of internuclear potential arises naturally in the framework of complex-extended Hamilton mechanics. The existence and the correctness of this vibrational-state fine structure of internuclear potential are justified by its strong consistency with the probability predictions made from the vibrational wavefunctions. Knowing the fine structure of internuclear potential permits us to obtain the bond length, the force constant, the vibration frequency, and the equations of motion for each vibrational state, which otherwise cannot be afforded by the usual electronic-state internuclear potential. Moreover, in addition to the well-known quantization of the vibrational energy levels, theR fine structure further manifests the quantization of the vibration frequency and the quantization of the action variable p dq in each vibrational state.
2. Inconsistencies between Morse potential and probability interpretation To demonstrate the determination of the fine structure accompanying an internuclear potential, we will employ the Morse function [8] as representative model for an electronic-state potential energy. Although the Morse function is not the best choice to fit the experimental data [14,17], its exact solvability of the related Schro¨dinger equation allows us to derive a closed-form expression of the fine structure accompanying the internuclear potential it represents. Indeed, there is no loss of generality by adopting the Morse function to represent the internuclear potential, since the RKR potentials for various molecules have been shown to coalesce into a single curve, which can be represented by the Morse potential with a properly scaled internuclear distance [18]. The standard Morse potential is a 3-parameter function expressed in the form of V Morse ðrÞ ¼ ED ½1 ebðrr0 Þ 2 ;
ð2:1Þ
where r0 is the equilibrium bond length, ED is the potential energy for bond formation and b is a parameter controlling the width of the potential well. The three parameters r0, ED, and b are determined from the molecular spectroscopic data. We first raise several severe inconsistencies between the predictions of the Morse potential and the predictions of the vibrational wavefunctions, and then justify how these inconsistencies can be removed by considering the hidden fine structure behind the Morse potential. The time-independent Schro¨dinger equation with the Morse potential V Morse ðrÞ reads
h2 d2 w þ ED ½1 ebðrr0 Þ 2 w ¼ Ew; 2l dr2
ð2:2Þ
where l ¼ m1 m2 =ðm1 þ m2 Þ is the reduced mass of the equivalent single-particle model of a diatomic molecule with nuclear masses m1 and m2. By introducing the dimensionless parameters x ¼ bðr r0 Þ ¼ bx;
e¼
E ; h b =2l 2 2
k2 ¼
ED 2 2
h b =2l
:
ð2:3Þ
Eq. (2.2) can be recast into to a dimensionless form d2 w þ ½e k2 ð1 ex Þ2 w ¼ 0 dx2
ð2:4Þ
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from which the normalized eigenstates can be found as [19] z=2 kn1=2 2k2n1 wðkÞ xÞ ¼ N ðkÞ z Ln ðzÞ; n e n ð
where
z ¼ 2kex ;
n ¼ 0; 1; . . . ; ½k 1=2;
Lqp ðzÞ N ðkÞ n
is the associated Laguerre function. The normalization factor 1=2 ð2k 2n 1ÞCðn þ 1Þ ¼ : Cð2k nÞ
N ðkÞ n
ð2:5Þ
is given by ð2:6Þ
The quantum number n specifies the vibrational energy level via the relation 2 eðkÞ n e ¼ 2kðn þ 1=2Þ ðn þ 1=2Þ ;
n ¼ 0; 1; . . . ; ½k 1=2:
For the convenience of later use, the first several eigenfunctions are listed below: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k 1 z=2 k1=2 ðkÞ e z ; z ¼ 2kex ; k > 1=2; w0 ðxÞ ¼ Cð2kÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k 3 z=2 k3=2 ðkÞ e z ð2k 2 zÞ; k > 3=2; w1 ðxÞ ¼ Cð2k 1Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2k 5Þ z=2 k5=2 2 ðkÞ e z ½z =2 þ ð3 2kÞz þ 2k2 7k þ 6; k > 5=2; w2 ðxÞ ¼ Cð2k 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ð2k 7Þ z=2 k7=2 z3 =6 þ ðk 2Þz2 þ kð2k þ 9Þz ðkÞ e z ; k > 7=2: w3 ðxÞ ¼ Cð2k 3Þ 10z þ 4k3 =3 10k2 þ 74k=3 20
ð2:7Þ
ð2:8aÞ ð2:8bÞ ð2:8cÞ ð2:8dÞ
The eigenfunction wðkÞ xÞ provides us with the probability P ðkÞ xÞ ¼ wðkÞ xÞwðkÞ xÞ of finding the radial distance between n ð n ð n ð n ð the two nuclei at the location x ¼ bðr r0 Þ. The number of the vibrational states trapped in the internuclear potential VMorse depends on ED, the depth of the Morse potential. In general, for a given k in the range of ðm þ 1=2Þ < k 6 ðm þ 3=2Þ, there are m + 1 allowable energy levels eðkÞ n with the accompanying vibrational eigenfuncðkÞ ðkÞ ðkÞ tions denoted by w0 ; w1 ; w2 ; . . . ; wðkÞ . The allowed number of bound vibrational states and the associated energy levm els for four different depths of Morse potential are depicted in Fig. 1. It is observed that the only information the Morse potential can provide about a vibrational state is its displacement range. For example, at the ground-state energy level ðkÞ ðkÞ e0 in Fig. 1a, what V Morse can tell is the estimated vibrational range between A and B; but no detailed potential change ðkÞ
ðkÞ
between A and B at e0 can be read from V Morse . Although the use of the electronic-state potential VMorse leads to the accurate prediction of the vibrational energy levels in Eq. (2.7), its predictions about the quantum motion in each vibrational substate have severe inconsistency with those predicted by wðkÞ xÞ and P ðkÞ xÞ, as reflected in the following n ð n ð observations: • The prediction of the equilibrium bond length by the Morse potential is inconsistent with the prediction from the wavefunction wðkÞ xÞ. The action of the Morse potential ensures that once deviated from the equilibrium length n ð r0, the bound length r has a tendency to return to and remain at r0. This tendency leads to a maximum probability of finding the bond length at r ¼ r0 . Unfortunately, this prediction is inconsistent with that made from the probability density function P ðkÞ xÞ, which brings out an equilibrium bond length other than r ¼ r0 . Let us examine the n ð ðkÞ ðkÞ ðkÞ ground-state wavefunction w0 ðxÞ to raise this discrepancy. The probability density function P 0 ðxÞ ¼ w0 ðxÞ ðkÞ ðkÞ w0 ðxÞ with w0 ðxÞ given by Eq. (2.8a) ðkÞ
P 0 ðxÞ ¼
2k 1 z 2k1 e z ; Cð2kÞ
z ¼ 2kex ;
k > 1=2:
ð2:9Þ
ðkÞ ðkÞ The position xmax with the maximum value of P 0 ðxÞ is found from the condition dP 0 ðxÞ=dx ¼ 0, which yields
xmax ¼ ln
2k 1 2k > 0 ) rmax ¼ r0 þ ln > r0 : 2k 1 b 2k 1
ð2:10Þ
It can be seen that unless k ! 1, the maximum probability position rmax is distinct from the equilibrium position r0 ðkÞ ðkÞ predicted from Morse potential. The numerical distributions of P 0 ðxÞ and V Morse ðxÞ ¼ V Morse =ð h2 b2 =2lÞ ¼ k2 ð2Þ x 2 ð1 e Þ for k ¼ 2 are displayed in Fig. 2a. As predicted from Eq. (2.10), P 0 ðxÞ is stationary at xmax ¼ lnð4=3Þ 0:2877, where its maximum is achieved. The corresponding equilibrium bond length determined by ð2Þ ð2Þ P 0 ðxÞ is req ¼ rmax ¼ r0 þ ð1=bÞ lnð4=3Þ, which is different from req ¼ r0 as predicted from V Morse ðxÞ.
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
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Fig. 1. Electronic-state internuclear potentials represented by the Morse function with four different ranges of depth ðkÞ k ¼ ðED = h2 b2 =2lÞ1=2 . (a) In the range 1=2 < k < 3=2, only one bound vibrational state with energy level e0 is allowed. (b) In the ðkÞ ðkÞ range 3=2 < k < 5=2, two bound states with energy levels e0 and e1 are allowed. (c) In the range 5=2 < k < 7=2, three bound states ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ with energy levels e0 ; e1 and e2 are allowed. (d) In the range 7=2 < k < 9=2, four bound states with energy levels e0 ; e1 ; e2 and e3 are allowed. It is observed that the only information the Morse potential can provide about a vibrational state is its displacement range. ðkÞ ðkÞ For example, at the ground-state energy level e0 in (a), what V Morse can only tell is the vibrational range between A and B; the detailed ðkÞ ðkÞ change of potential between A and B at the energy level e0 cannot be read from V Morse .
ð2Þ
ð2Þ
Fig. 2. (a) The stationary positions of the probability density functions P 0 and P 1 do not coincide with the stationary position of the Morse potential at the origin. (b) The origin is a stable equilibrium position toward which the Morse potential always applies a ðkÞ restoring force f Morse even when k < 1=2. But the Morse potential cannot explain why a bound state is unable to develop when k < 1=2 ðkÞ by noting that the restoring force f Morse still exists in case of k < 1=2.
Besides the discrepancy in predicting the equilibrium bond length, a further inconsistency comes from the prediction of the number of the equilibrium bond lengths. The only possible equilibrium bond length predicted from Morse potential is req ¼ r0 for all the vibrational substates, but the probability density function P ðkÞ xÞ indicates that the number and the n ð location of the equilibrium bond length are different for different vibrational substates. To manifest this point, we conðkÞ sider the first excited vibrational state, whose probability density function w1 ðxÞ computed from Eq. (2.8b) is ðkÞ
P 1 ðxÞ ¼
2k 3 z 2k3 ð2k 2 zÞ2 ; e z Cð2k 1Þ
k > 3=2:
This function is stationary at two equilibrium positions:
ð2:11Þ
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xeq ¼ ln
4k 3
4k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 16k 15
ð2:12Þ ðkÞ
which yield two equilibrium bond lengths. The numerical of P 1 ðxÞ with k ¼ 2 is illustrated in Fig. 2a, pdistribution ffiffiffiffiffi ð2Þ showing the two equilibrium positions at xeq ¼ ln 8=ð5 17Þ 0:1314; 2:2108, where P 1 ðxÞ is stationary as local xÞ. It is hardly maximum. In general, there are n + 1 equilibrium bond lengths existing in the nth vibrational state wðkÞ n ð possible to explain the formation of such many different equilibrium bond lengths in the vibrational states wðkÞ xÞ, n ð ðkÞ n ¼ 0; 1; . . . ; ½k 1=2, by merely considering the action of the internuclear potential V Morse ðxÞ, since it is vibrationalstate-independent, and produces the only possible equilibrium point at xeq ¼ 0. ðkÞ • The Morse potential V Morse ðxÞ fails to explain the existence of nodes, where the probability P ðkÞ xÞ is zero. A node n ð xnode with P ðkÞ ð x Þ ¼ 0 is usually where the potential barrier approaches infinity; however, the Morse potential (2.1) node n is finite for any radial distance r P 0, which means that no radial distance between the two nuclei is absolutely prohibited under the action of Morse potential. Nevertheless, the probability P ðkÞ xÞ does become zero at the positions n ð ðkÞ of the roots of Ln2k2n1 ðzÞ ¼ 0, as can be seen from Eq. (2.5). Taking P 1 ðxÞ as an example, the node occurs at znode ¼ 2k 2, which amounts to xnode ¼ ln
k 1 k ) rnode ¼ r0 þ ln : k1 b k1 ð2Þ
ð2:13Þ ðkÞ
As can be seen from Fig. 2a, P 1 ðxnode Þ ¼ 0 at xnode ¼ ln 2, and notice that V Morse ðxnode Þ is a small finite value. In the nth eigenstate wðkÞ xÞ, we generally have n such nodes. If the heights of the Morse potential at all the nodes are finite, we n ð want to ask why the probability of reaching these nodes is zero and meanwhile why the probability of reaching the positions having much higher potential than xnode is not zero. Obviously, we cannot justify the existence of such nodes solely according to the action of the internuclear potential V Morse . • The action of the Morse potential is unable to clarify why bound vibrational state cannot exist in the range k 6 1=2. ðkÞ ð2=5Þ Inspecting the Morse potential V Morse with k ¼ 2=5 < 1=2 in Fig. 2b, we can see that the structure of V Morse is similar ðkÞ to those of V Morse with k > 1=2, and they all share the same properties that x ¼ 0 ðr ¼ r0 Þ is the equilibrium position ð2=5Þ ð2=5Þ ð2=5Þ ð2=5Þ with zero net force f Morse ð0Þ ¼ dV Morse ð0Þ=dx ¼ 0, and f Morse ðxÞ > 0, as x < 0, while f Morse ðxÞ < 0, as x > 0. This means that there is always a restoring force acting on the nuclei toward their equilibrium distance r ¼ r0 for the both cases of k 6 1=2 and k > 1=2. However, the Morse potential is unable to further distinguish the difference that the restoring force with k > 1=2 can form a bound state but that with k 6 1=2 cannot. ðkÞ
Above all, the vibrational-state-independent nature of the internuclear potential V Morse ðxÞ is incompatible with the vibrational-state-dependent nature of the probability density function P ðkÞ xÞ. How can a state-independent driving n ð ðkÞ force dV Morse ðxÞ=dx produce a state-dependent motion described by P ðkÞ xÞ? It is evident that the apparent internuclear n ð ðkÞ potential V Morse ðxÞ is not the sole driving potential within a diatomic molecule; it seems that there exists a state-dependent fine structure in the potential, which just produces the observed vibrational-state-dependent radial motion. In the following discussions, we will derive such a hidden fine structure from the Schro¨dinger equation and show that once the hidden fine structure is incorporated into the apparent internuclear potential, such as the Morse potential, to form a total potential, the aforementioned inconsistencies disappear automatically and the state-dependent quantum motion can be fully predicted and explained by the derived total potential.
3. The hidden potential in Schro¨dinger equation The hidden fine structure in the internuclear potential can be identified by treating the Schro¨dinger equation as an expression of energy conservation law. Firstly, we recall a classical result [20] that for a given classical Hamiltonian H c ðt; q; pÞ ¼ p2 =2m þ V ðt; qÞ; the classical Hamilton–Jacobi (H–J) equation reads oS c oS c 1 ¼ þ H c ðt; q; pÞ þ ðrS c Þ2 þ V ¼ 0; ot ot 2m p¼rS c
ð3:1Þ
ð3:2Þ
where Sc is the classical action function. By contrast, applying the transformation w ¼ expðiS=hÞ; to the Schro¨dinger equation
ð3:3Þ
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
ih
ow h2 2 r w þ V w; ¼ 2m ot
we obtain the quantum H–J equation oS 1 h 2 oS 1 2 h þ ðrSÞ2 þ V þ rS ¼ þ p þV þ rp ¼ 0: ot 2m 2mi ot 2m 2mi p¼rS If the above equation is recast into a form analogous to Eq. (3.2): oS þ H ðt; q; pÞ ¼ 0; ot p¼rS
967
ð3:4Þ
ð3:5Þ
ð3:6Þ
we find that the corresponding quantum Hamiltonian H turns out to be H ðt; q; pÞ ¼
1 2 p þ V ðt; qÞ þ Qðwðt; qÞÞ; 2m
where Qðwðt; qÞÞ is known as the quantum potential defined by h h 2 h2 2 r p r S¼ r ln wðt; qÞ: ¼ Qðwðt; qÞÞ ¼ 2mi 2mi 2m p¼rS
ð3:7Þ
ð3:8Þ
The quantum potential Q is just the aforementioned hidden potential, which, together with the externally applied potential V, completely determines a particle’s quantum motion. It is worth noting that the classical Hamiltonian Hc depends only on the externally applied potential V ðt; qÞ, whereas the quantum Hamiltonian H depends on V ðt; qÞ as well as on the internal state w where the particle lies. For a given state described by wðt; qÞ, the quantum Hamiltonian H defined in Eq. (3.7) is an explicit function of the canonical variables q and p that are regarded as independent variables. There are two roles played by the wavefunction w in the quantum Hamiltonian H. Firstly, as indicated in Eq. (3.5), it determines the canonical momentum according to pj ¼
oS o ln w ¼ ih : oqj oqj
ð3:9Þ
Secondly, it generates the quantum potential Q according to Eq. (3.8). The equations of motion for a particle moving in the quantum state w can be derived by applying the quantum Hamiltonian (3.7) to the Hamilton equations: dq oH ðwÞ 1 ¼ ¼ p; dt op m dp oHðwÞ o h2 2 r ln wðt; qÞ : ¼ ¼ V ðt; qÞ 2m dt oq oq
ð3:10aÞ ð3:10bÞ
Note that as in classical Hamilton mechanics, we have obtained oH =op and oH =oq by treating q and p as independent variables in the Hamiltonian (3.7). Each given wavefunction w represents a dynamic system described by the quantum Hamilton equations of motion (3.10), which are distinct from the classical ones in two aspects: the complex nature and the state-dependent nature. The state-dependent nature means that the quantum Hamilton equations of motion determine the quantum motion in the specific quantum state described by w. The complex nature is a consequence of the fact that the canonical variables solved from Eqs. (3.9) and (3.10) are, in general, complex variables. Complex canonical variable has a one-to-one correspondence with its associated quantum operator. For example, if we rewrite Eq. (3.9) as h ow ¼ pj w i oqj
ð3:11Þ
and compare it with the definition of the momentum operator ^ pj w ¼ pj w; we have b pj ¼
h o : i oqj
ð3:12Þ
The proof of the quantization axiom p ! ^p ¼ ihr thus becomes a one-line corollary of quantum Hamilton mechanics. Eq. (3.9) indicates that defining quantum momentum in complex domain is necessary to result in the correct momentum operator. A similar but different quantum momentum was proposed by Bohm [1] in the form of pB ¼ rS B , where SB is the phase of the wavefunction defined by w ¼ RB eiS B =h with RB and SB being real functions. If we follow the same procedures leading to Eq. (3.11) but employ the real quantum momentum pB ¼ rS B instead
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of the complex momentum p ¼ rS, we shall find that it is not possible to arrive at the correct momentum operator ^ p ¼ ihr. The other familiar example regards the operator of the angular momentum L ¼ q p. Evaluating the x component of L with p given by Eq. (3.9), we obtain h o ln w h o ln w i h o o z ¼ y z w: Lx ¼ ypz zpy ¼ y i oz i oy w oz oy Comparing the above equation to the definition, b L x w ¼ Lx w, gives b L x as o o b L x ¼ ih y z ¼ y^pz z^py : oz oy
ð3:13Þ
Following the same mechanism [21], we can identify any operator A from its related local observable Aðq; pÞ with p and q satisfying the Hamilton equations (3.10). The correctness of the quantum Hamiltonian (3.7) can be further jusb . Using Eq. (3.9) in Eq. (3.7), we have tified by showing that it leads to the correct Hamiltonian operator H h2 h2 2 1 h2 2 2 H ðt; q; pÞ ¼ ðr ln wÞ þ V r ln wðt; qÞ ¼ r wþVw ; ð3:14Þ w 2m 2m 2m which can be rewritten as h2 2 Hw ¼ r þ V w: 2m
ð3:15Þ
b w ¼ H w gives the desired operator The comparison with the definition H 2 b ¼ h r2 þ V : H 2m
ð3:16Þ
Thus the quantum Hamiltonian (3.7) is an equivalent expression of the Hamiltonian operator (3.16). Of significance in this equivalence is that the quantum Hamiltonian (3.7) and thus the Hamiltonian operator (3.16) reveal the existence of the quantum potential Q in the system’s total energy in addition to its classical component p2 =2m þ V . When the applied potential V is not an explicit function of time, the solution of the quantum H–J equation (3.6) can be expressed as Sðt; qÞ ¼ W ðqÞ Et;
ð3:17Þ
and the quantum H–J equation (3.6) is reduced to H ðt; q; pÞ ¼
1 2 oS p þ V ðqÞ þ QðwðqÞÞ ¼ ¼ E ¼ constant; 2m ot
ð3:18Þ
which is an energy conservation law for quantum systems. This energy conservation law is just an alternative expression of the time-independent Schro¨dinger equation, as can be verified by substituting p ¼ rS ¼ i hr ln w into Eq. (3.18) to obtain h2 2 r w þ ðE V Þw ¼ 0: 2m
ð3:19Þ
The equivalence between Eqs. (3.18) and (3.19) suggests that one can describe quantum motions either in terms of the probability density function w w solved from Eq. (3.19) or in terms of the canonical variables ðq; pÞ solved from the energy conservation law (3.18) and the quantum Hamilton equations of motion (3.10), which can be combined to form the quantum Newton equation: m
d2 qj oV total ¼ ¼ fj : dt2 oqj
ð3:20Þ
The total potential V Total ¼ V þ QðwÞ contains the applied potential V and the state-dependent quantum potential QðwÞ, which can be further simplified by using Eqs. (3.18) and (3.9) as V Total ¼ V þ QðwÞ ¼ E
1 2 h2 p ¼Eþ ðr ln wÞ2 : 2m 2m
ð3:21Þ
For a quantum state described by the wavefunction w, the total potential within the state w is determined from Eq. (3.21) by which the quantum equations of motion are then established according to Eq. (3.20).
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
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The discussions so far are valid for any potential V, and we have reached the general observation that for any quantum system subjected to an external potential V, there always exists an accompanying fine-structure potential QðwÞ for each quantum state w. We can employ Eq. (3.8) to determine this fine-structure potential behind the applied potential V for any quantum system. In the hydrogen atom, V represents the Coulomb potential between proton and electron, and the fine-structure potential QðwÞ forms the shell structure observed in the hydrogen atom [22]. In an atomic nucleus, we may regard V as the Yukawa potential between two nucleons and may employ the obtained fine-structure potential to explain the shell structure observed in a nucleus. When a diatomic molecular system is concerned, V represents the electronic-state internuclear potential described by the Morse function V Morse , and the accompanying fine-structure potential QðwÞ represents the vibrational-state potential for the vibrational state w. In the next section, we will clarify that it is the total potential V Total ¼ V Morse þ QðwÞ, rather than the Morse potential VMorse alone, that is responsible for the quantum motion predicted by the probability P ðkÞ xÞ. n ð
4. Refined internuclear potential In Section 2, we have pointed out three inconsistencies between the predictions of the Morse potential VMorse and the probability P ðkÞ xÞ. Now we go on to verify that the missing potential needed to bridge the gap between VMorse and n ð P ðkÞ ð x Þ is just the quantum potential QðwÞ derived in Eq. (3.8) whose existence, as shown previously, is not by assumpn tion but is inherent in the Schro¨dinger equation. We start with the quantum Hamilton equations of motion (3.10), where, for the present case, the canonical coordinate is q ¼ x ¼ r r0 and the canonical momentum p is given by Eqs. (3.10a) and (3.9) as p¼l
dx h d ¼ ln wðkÞ n ðxÞ: dt i dx
ð4:1Þ
The quantum Hamiltonian governing the motion of x and p is provided by Eq. (3.7): H¼
p2 p2 h2 d2 ðkÞ ðkÞ ðkÞ w ðxÞ: þ V Morse ðxÞ þ QðkÞ þ V Morse ðxÞ n ðxÞ ¼ 2l 2l 2l dx2 n
ð4:2Þ
Substituting the eigenfunction wðkÞ n ðxÞ from Eq. (2.5) into the above Hamiltonian, we arrive at the energy conservation law for diatomic molecules: H¼
p2 ðkÞ ðkÞ þ V Morse ðxÞ þ QðkÞ n ðxÞ ¼ En ¼ constant; 2l
ð4:3Þ
ðkÞ with the energy level EðkÞ n already given by Eq. (2.7). Without the participation of the quantum potential Qn , the value ðkÞ 2 of the classical Hamiltonian H c ¼ p =2l þ V Morse ðxÞ is uncertain and cannot guarantee the energy conservation in the state wðkÞ n . ðkÞ Now we are in a position to remedy the aforementioned inconsistencies by replacing the role of V Morse with the ðkÞ ðkÞ refined potential V Total ¼ V Morse þ Qn , which has a dimensionless form as 2 V Total ðkÞ ðkÞ ðxÞ ¼ k2 ð1 ex Þ2 d ln wðkÞ ðxÞ V ð x Þ þ Q ¼ Morse n n dx2 h2 b2 =2l d d z ln wðkÞ z ¼ 2kex ; ¼ ðk z=2Þ2 z n ðzÞ ; dz dz
V ðkÞ xÞ n ð
ð4:4Þ
where the over-bar symbols denote dimensionless quantities. By using the wavefunctions wðkÞ n ðzÞ obtained from Eq. (2.5), the total potential for every vibrational state can thus be determined uniquely from Eq. (4.4). Especially, with the wavefunctions wðkÞ n ðzÞ listed explicitly in Eqs. (2.8a)–(2.8d), the first three total potentials can be computed as ðkÞ
V 0 ðxÞ ¼ ðk z=2Þ2 þ z=2; ðkÞ
V 1 ðxÞ ¼ ðk z=2Þ2 þ ðkÞ
V 2 ðxÞ ¼ ðk z=2Þ2 þ þ
z ¼ 2kex ;
½z2 þ 4ð1 kÞz þ 4k2 4k 2ðz 2k þ 2Þ2
;
z ¼ 2kex ; ð4:5Þ
z½z4 þ ð12 8kÞz3 þ ð24k2 68k þ 48Þz2 ½z2
2
þ ð6 4kÞz þ 4k 14k þ 12
2
z½ð32k3 þ 128k2 152k þ 48Þz þ 16k4 80k3 þ 132k2 72k ½z2 þ ð6 4kÞz þ 4k2 14k þ 122
:
970
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976 ðkÞ
The first term in V ðkÞ xÞ is the Morse potential V Morse ðxÞ, and the remaining terms are due to the vibrational-state fine n ð structure. The first inconsistency mentioned previously concerns the prediction about the equilibrium bond length. Let us examine how the total potential VTotal yields the same equilibrium bond length as predicted by the probability P ðkÞ n . ðkÞ ðkÞ The equilibrium position is where V 0 has a local minimum for which the condition dV 0 ðzÞ=dz ¼ 0 leads to zeq ¼ 2k 1 ) xeq ¼ ln
2k ; 2k 1
ð4:6Þ ðkÞ P 0 ðxÞ
which is exactly the equilibrium position predicted by as given in Eq. (2.10). From dynamic consideration, an equilibrium position must be the one on which the net applied force is zero. To verify this point, we compute the total ðkÞ force f 0 from Eq. (3.20) as d ðkÞ d ðkÞ 2k 1 ðkÞ : ð4:7Þ f 0 ðxÞ ¼ V 0 ðxÞ ¼ z V 0 ðzÞ ¼ 2k2 ex ex dx dz 2k ðkÞ According to the sign of f 0 ðxÞ, three force regions can be classified as following: ðkÞ 2k , we have f 0 ðxeq Þ ¼ 0, indicating that the net force exerting the equilibrium • Equilibrium point: At x ¼ xeq ¼ ln 2k1 point is truly zero. ðkÞ • Repulsive region: In the region x < xeq , we have f 0 ðxÞ > 0, indicating that the two nuclei are subjected to a repulsive force such that their bond length is increased toward the equilibrium position xeq . ðkÞ • Attractive region: In the region x > xeq , we have f 0 ðxÞ < 0, indicating that the two nuclei are subjected to an attractive force such that their bond length is decreased toward the equilibrium position xeq .
Examining the force action near the equilibrium point, we find that there is always a restoring force exerting the two nuclei toward their equilibrium relative position xeq . Once they enter the equilibrium position, they will remain there because the net force applied at the equilibrium position is zero. Consequently, the probability of finding x at xeq is expected to be the maximum. This point has been verified in Eqs. (2.9) and (2.10) by showing that ðkÞ 2k xeq ¼ xmax ¼ ln 2k1 is just the position maximizing P 0 ðxÞ. A general property says that the local maximum of ðkÞ ðkÞ P n ðxÞ occurs at the local minimum of V n . This property can be proved from Eq. (3.21) whose dimensionless form becomes !2 1 1 dwðkÞ xÞ ðkÞ ðkÞ n ð V n ðxÞ ¼ en þ P eðkÞ ð4:8Þ n : 2 wðkÞ d x ð x Þ n ðkÞ The minimum value of V ðkÞ xÞ is eðkÞ xÞ=dx ¼ 0; but dwðkÞ xÞ=dx ¼ 0 is just the n ð n ð n , which occurs at the position with dwn ð ðkÞ ðkÞ ðkÞ condition that P n ðxÞ ¼ wn ðxÞwn ðxÞ achieves its maximum value by noting
dP ðkÞ xÞ d dwðkÞ ðxÞ n ð ¼ wðkÞ ¼ 0; xÞwðkÞ xÞ ¼ 2wðkÞ xÞ n n ð n ð n ð dx dx dx
ð4:9Þ
since wðkÞ xÞ is a real function of x. The total potential V ðkÞ n ð n at several vibrational states specified by n for four different depths of the Morse potential is demonstrated in Fig. 3, where it can be observed that the minimum value of V ðkÞ n occurs ðkÞ precisely at the energy level eðkÞ corresponding to the vibrational state w . n n From the viewpoint of dynamic motion, a diatomic molecule must be stationary at its equilibrium position with zero velocity and zero acceleration. The Hamilton equations of motion (3.10) are helpful to clarify this point. A dimensionless expression of the Hamilton equations reads dx i dwðkÞ xÞ n ð ¼ ðkÞ ; dt wn ðxÞ dx
ð4:10aÞ
dp dV ðkÞ ðxÞ 1 dwðkÞ xÞ d2 ln wðkÞ xÞ n ð n ð ¼ n ¼ ðkÞ ; dt dx dx2 wn ðxÞ dx
ð4:10bÞ
where t ¼ ðhb2 =lÞt and p ¼ p=ðbhÞ are the dimensionless time and momentum, respectively. It is evident that the condition dwðkÞ xÞ=dx ¼ 0 exactly implies the equilibrium condition dx=dt ¼ d p=dt ¼ 0. In short, we have shown that the n ð following four positions are coincident: (a) the dynamic equilibrium position determined from Eqs. (4.10), (b) the position with minimum total potential determined from Eq. (4.8), (c) the position with maximum probability determined from Eq. (4.9), and (d) the position with zero applied force determined from Eq. (4.7).
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
971
Fig. 3. The total potential V ðkÞ n at each vibrational state specified by n for four different depths of the Morse potential. It can be occurs precisely at the energy level eðkÞ corresponding to the vibrational state wðkÞ observed that the minimum value of V ðkÞ n n n . In ðkÞ comparison with Fig. 1, the Morse potential V Morse can only provide the displacement range at enðkÞ , while the total potential ðkÞ ðkÞ ðkÞ V ðkÞ xÞ=dx at every energy level eðkÞ n ¼ V Morse þ Qðwn Þ can further offer the detailed internuclear force V n ð n .
A numerical verification of k ¼ 2 is shown in Fig. 4, where the correspondence between the maximum of P ð2Þ xÞ and n ð the minimum of V ð2Þ ð x Þ are clearly demonstrated for n ¼ 0; 1. By contrast, the equilibrium position obtained from n ðkÞ dV Morse ðxÞdx ¼ 0 is always at x ¼ 0, which is nothing to do with the correct equilibrium position given by Eq. (2.12). ðkÞ The second inconsistency between V Morse and P ðkÞ xnode with P ðkÞ xnode Þ ¼ 0. Under n regards the prediction of the node n ð ðkÞ ðkÞ the action of the Morse potential V Morse , such a node does not exist since V Morse ðxÞ is everywhere finite and no location is xÞ, node does exist in that there are positions where inaccessible. However, under the action of the total potential V ðkÞ n ð V ðkÞ xÞ approaches infinity. The relation between P ðkÞ xnode Þ and V ðkÞ xnode Þ is best illustrated via Eq. (4.8) as n ð n ð n ð !2 1 dwðkÞ xÞ n ð V ðkÞ xÞ ¼ eðkÞ ; ð4:11Þ n ð n þ ðkÞ dx 2P n ðxÞ ðkÞ ðkÞ ðkÞ ðkÞ which indicates that the position having zero probability P ðkÞ n ¼ wn wn ¼ wn wn ¼ 0 is just where the total potential ðkÞ V ðkÞ ð x Þ approaches infinity. The inaccessibility to the position with w ¼ 0 is also reflected in the equations of motion n n (4.10), which give rise to infinite velocity and acceleration at the points where wðkÞ n ¼ 0.
ð2Þ
ð2Þ
ð2Þ
Fig. 4. (a) The local minimum of V 0 coincides with the local maximum of P 0 . (b) The two local minimums of V 1 coincide with the ð2Þ ð2Þ two local maximums of P 1 , and the node position xnode at which V 1 approaches infinity is inaccessible and thus has zero probability, ð2Þ ð2Þ P 1 ¼ 0. The action of the Morse function V Morse alone cannot explain the existence of the node and cannot predict the locations ð2Þ ð2Þ maximizing P 0 and P 1 .
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C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976 ðkÞ
It has been mentioned that the Morse potential V Morse cannot explain why bound state is unable to develop when ðkÞ k 6 1=2. If we consider the action of the total potential V ðkÞ xÞ instead of V Morse , the reason becomes apparent. Accordn ð ðkÞ ing to the prediction of V 0 ðxÞ, the equilibrium position xeq for the ground state is given by Eq. (4.6), from which we can see that when k ¼ 1=2, the equilibrium position xeq and the bond length approach infinity, resulting in the dissociation of the diatomic molecules. As k is reduced to lower than 1=2, Eq. (4.6) reveals that real equilibrium position no longer exists and no bound state can be developed. A deeper understanding about the annihilation of the bound state can be gained from the force distribution (4.7), which indicates that as k 6 1=2, the internuclear force is everywhere repulsive so that no attractive force is available to form a bond state. ð1=2Þ ð1=2Þ ðxÞ, the total Fig. 5 illustrates the actual distributions of the Morse potential V Morse ðxÞ, the quantum potential Q 0 ðkÞ potential V 0 ðxÞ, and their associated internuclear forces for the two cases of k ¼ 1=2 and k ¼ 1. In case of k ¼ 1=2, ð0:5Þ ð0:5Þ ð0:5Þ ð0:5Þ although the Morse potential contributes attractive internuclear force f Morse , the total force f 0 ¼ f Morse þ f Q0 is ð1Þ everywhere repulsive, making a bound state unable to establish. When k > 1=2, the attractive force f Morse afforded ð1Þ by the Morse potential becomes large enough to overbalance the repulsive quantum force f Q0 , and allows a bound state to exist as shown in Fig. 5c and d for k ¼ 1.
5. Quantizations in fine-structure potential Besides the energy quantization, further quantization rules can be discovered from the total potential V ðkÞ xÞ and n ð R the related Hamilton equations of motion. The first noticeable quantization is concerned with the action variable p dx. Using Eq. (3.9), we can express the action variable as Z Z Z Z h 1 dwðkÞ xÞ h 1 dwðkÞ zÞ n ð n ð d x ¼ dz; ð5:1Þ p dx ¼ h p dx ¼ ðkÞ ðkÞ i d x i d z xÞ zÞ c c c wn ð c wn ð ðkÞ where c is any closed trajectory in the vibrational state wðkÞ zÞ from Eq. (2.5) further yields n . The substitution of wn ð Z Z 2k2n1ðzÞ h dðLn Þ=dz dz: ð5:2Þ p dx ¼ 2k2n1 2pi ðzÞ L c c n
To evaluate the above contour integral, we quote a useful theorem from the calculus of residue:
Fig. 5. The comparisons of the internuclear potential and force between k ¼ 1=2 and k ¼ 1. (a, b) In case of k ¼ 1=2, although the ð0:5Þ ð0:5Þ ð0:5Þ ð0:5Þ Morse potential contributes attractive internuclear force f Morse , the total force f 0 ¼ f Morse þ f Q0 is everywhere repulsive, making a ð1Þ bound state unable to establish. (c, d) In case of k ¼ 1, the attractive force afforded by f Morse becomes large enough to overbalance the ð1Þ repulsive quantum force f Q0 , and allows a bound state to exist.
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
1 2pi
Z C
F 0ðzÞ dz ¼ N P ; F ðzÞ
973
ð5:3Þ
where N and P are, respectively, the number of zeros and poles of F ðzÞ enclosed by the contour C. The application of the above residue theorem to Eq. (5.2) leads to Z p dx ¼ mh; m ¼ 0; 1; . . . ; n ð5:4Þ c
by noting that Ln2k2n1 ðzÞ is an nth-order polynomial so that its number of zeros enclosed by c is at most n, while the number of poles is always zero. The quantization rule (5.4)R reveals that in spite of the infinitely many closed trajectory c in the complex plane of x, the value of the action variable c p dx is only allowed to be the integer multiple of the Planck constant h. The other new quantization rule to be noted is the quantization of the vibration frequency. Using the fine-structure potential (4.4), we can find the force constant in each vibrational state as h i2 d2 V kn 2 xeq 2 K ðkÞ ðxeq Þ ¼ 2 eðkÞ Þ ¼ 2½2kðn þ 1=2Þ ðn þ 1=2Þ2 k2 ð1 exeq Þ2 2 ; ð5:5Þ n ¼ n k ð1 e 2 dx where xeq is the equilibrium position in the nth vibrational state. In the above evaluation of d2 V kn =d2x, we have applied the equilibrium condition dV ðkÞ xeq Þ=dx ¼ 0 and the fact that the eigenfunction wðkÞ xÞ satisfies the Schro¨dinger equation n ð n ð ðkÞ ðkÞ with the eigenvalue en . When applied to the ground vibrational state where xeq is given by Eq. (2.10) and e0 ¼ k 1=4, Eq. (5.5) yields the force constant ðkÞ
K0 ¼
h2 b2 =2l ðkÞ h2 b4 ðk 1=2Þ2 ; K0 ¼ l 1=b2
ð5:6Þ
k > 1=2;
and the related ground-state frequency becomes sffiffiffiffiffiffiffiffi ðkÞ K0 hb2 ðkÞ ðk 1=2Þ: x0 ¼ ¼ l l
ð5:7Þ
Comparing calculated from the Morse potential xMorse ¼ ffi with the above results, we see that the frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ V 00Morse =l ¼ hb2 k=l is close to the ground-state frequency x0 only when k 1=2; but for excited vibrational states with n > 0, the Morse frequency xMorse is unable to provide any information about the actual vibration frequency xðkÞ n . ðkÞ To demonstrate the self-consistency of the fine-structure inter-nuclear potential, we can further compute x0 directly from the Hamilton equation of motion (4.10a), which gives by substituting w0 ðzÞ ¼ ez=2 zk1=2 dz i ¼ zðz 2k þ 1Þ; dt 2
ð5:8Þ
whose solution can be found as zðtÞ k xðtÞ ¼ ln ¼ ln þ C 1 eiðk1=2Þt : 2k k 1=2
ð5:9Þ ðkÞ
The time response of xðtÞ indicates a dimensionless vibration period t0 equal to 2p=ðk 1=2Þ and a corresponding frequency equal to ðkÞ
x0 ¼
2p ðkÞ
T0
¼
2p ðkÞ
T 0 l=ðhb2 Þ
¼
hb2 ðk 1=2Þ; l
ð5:10Þ
which is the same as Eq. (5.7) obtained from force constant. The third method to compute the vibration frequency is via the contour integral along the trajectories determined by Eq. (5.8), leading to the same result Z Z 1 2=ð2k 1Þ 1 2 2p ðkÞ T 0 ¼ dt ¼ dz ¼ ð2piÞ ¼ 8c; ð5:11Þ i c z ð2k 1Þ i 2k 1 k 1=2 where c is any closed trajectory enclosing the equilibrium point zeq ¼ 2k 1. The contour integral is solely determined by the enclosed equilibrium point and is independent of the actual trajectory. Only one frequency is allowed in the ground vibrational state, because there is only one equilibrium point there. ðjÞ In the nth vibrational state, we have n+1 equilibrium positions xeq , j ¼ 1; 2; . . . ; n þ 1, which are the roots of ðjÞ dwðkÞ ð x Þ=d x ¼ 0, as mentioned in Section 4. The vibration about a specific equilibrium position xeq has its own force n
974
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976 ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðjÞ constant K n;j determined by substituting xeq into Eq. (5.5). Therefore, n+1 force constants K n;1 ; K n;2 ; . . . ; K n;nþ1 , and n+1 related vibration frequencies can be identified in the nth vibrational state. For instance, we have two equilibrium positions for the first excited vibrational state as derived in Eq. (2.12), which, when substituted into Eq. (5.5), yields two force constants 2 h2 b4 1 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ ðkÞ K 1;1 ; K 1;2 ¼ ð5:12Þ ð16k 15Þ 16k 15 ; l 8 8
and each of the two equilibrium positions has its own vibration period as 1 2pl 1 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ ðkÞ 16k 15 : T 1;1 ; T 1;2 ¼ 2 ð16k 15Þ 8 hb 8
ð5:13Þ
ðkÞ
The determination of vibration frequency from the force constant K n;j is based on the underlying assumption that the vibrational motion takes place with respect to an equilibrium position. However, if the vibrational amplitude is so qffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ ðkÞ large that two or more equilibrium positions are enveloped within the amplitude, the relation xn;j ¼ K n;j =l becomes invalid. Such a large-amplitude vibration, which does occur in the first and other excited vibrational states, can only be detected by solving the equations of motion. As to the first excited state, the equation of motion is obtained by substiðkÞ tuting w1 into Eq. (4.10a) as dz z3 þ ð3 4kÞz2 þ ð2k 3Þð2k 2Þz ¼i ; dt 2ðz 2k þ 2Þ
z ¼ 2kex :
ð5:14Þ
The resulting trajectories in the complex plane xR xI can be classified into three sets as shown in Fig. 6a. The sets X1 and X2 contain the trajectories enclosing only one of the two equilibrium positions xð1Þ xð2Þ eq and eq listed in Eq. (2.12), and X3 contain the aforementioned large-amplitude trajectories enclosing both the two equilibrium points. The time responses of xR ðtÞ are depicted in Fig. 6b from which the vibration periods can be read. We can determine the vibration periods analytically by the residue theory without actually solving Eq. (5.14). The contour integral along a closed trajectory satisfying Eq. (5.14) gives Z Z 1 1 2ðz zk þ 2Þ ðkÞ dt ¼ dz: ð5:15Þ T1 ¼ i i c z3 þ ð3 4kÞz2 þ ð2k 3Þð2k 2Þz Note that the two equilibrium positions xð1Þ xð2Þ eq , eq listed in Eq. (2.12) are just the two poles of the above integrand. Depending on the poles enclosed by the contour c, the evaluation of the integral by the calculus of residue gives 8 2pl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðkÞ 1 > ð16k 15Þ þ 38 16k 15 ¼ T 1;1 8c 2 X1 ; > hb2 8 > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2pl ðkÞ ðkÞ ðkÞ T 1 ¼ 2 T 1 ¼ 2pl2 18 ð16k 15Þ 38 16k 15 ð5:16Þ ¼ T 1;2 8c 2 X2 ; hb > hb > > : 2pl ðkÞ ðkÞ ðkÞ =ðk 3=2Þ ¼ T 1;3 ¼ T 1;1 þ T 1;2 8c 2 X3 ; hb2
Fig. 6. The eigen trajectories solved from Eq. (5.14) for the first excited state with k ¼ 2. (a) The trajectories in the complex plane xR xI can be classified into three sets. The sets X1 and X2 contain the trajectories enclosing only one equilibrium point, and X3 contain the large-amplitude trajectories enclosing two equilibrium points. (b) The time responses of xR ðtÞ show that the vibration periods for all the periodic trajectories are quantized into three values T 1 , T 2 , and T 3 , corresponding to the three trajectory sets X1, X2, and X3. All the trajectories in X1(X2) have the same period T 1 ðT 2 Þ, and all the large-amplitude trajectories in the set X3 have the same period T 3 ¼ T 1 þ T 2.
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
975 ðkÞ
ðkÞ
ðkÞ
which shows that the vibration periods for all the periodic trajectories are quantized into three values T 1;1 , T 1;2 , and T 1;3 , ðkÞ ðkÞ corresponding to the three trajectory sets X1, X2, and X3 . All the trajectories in X1 (X2) have the same period T 1;1 ðT 1;2 Þ, ðkÞ ðkÞ ðkÞ and all the large-amplitude trajectories in the set X3 have the same period T 1;3 ¼ T 1;1 þ T 1;2 . Comparing Eq. (5.16) with ðkÞ ðkÞ Eq. (5.13), we can see that the force constants in Eq. (5.13) can only determine the periods T 1;1 and T 1;2 corresponding ðkÞ to the trajectories in the sets X1 and X2 enclosing only one equilibrium points, while the determination of the period T 1;3 corresponding to the large-amplitude trajectories in X3 needs the information of equation of motion (5.14). In general, for any vibrational state wðkÞ n we can determine the quantized vibration periods by evaluating the contour integral along the equation of motion (4.10a).
6. Conclusions In this paper we pointed out that an experimentally determined electronic-state internuclear potential described by ðkÞ V Morse is inadequate in describing the quantum motions in its vibrational substates and is even inconsistent with the ðkÞ probability predictions of the vibrational wavefunctions determined from V Morse . For a given electronic-state internuclear potential, we have verified that there exists an accompanying fine-structure potential V ðkÞ xÞ, which provides us n ð with all the information to determine the radial motion of a diatomic molecule in each vibrational state described by wðkÞ xÞ. We can employ V ðkÞ xÞ to derive the quantum equations of motion (4.10), to derive the internuclear forces n ð n ð ðkÞ ðkÞ f n ðxÞ ¼ V ðkÞ xÞ=dx, and to compute the force constant K n ð n for the nth vibration state. All the predictions made from ðkÞ ðxÞ, K ðkÞ , and the quantum equations of motion are consistent with the probability predictions made from V ðkÞ ð x Þ, f n n n P ðkÞ xÞ ¼ wðkÞ xÞwðkÞ xÞ. In other words, the vibrational wavefunction wðkÞ xÞ not only provides the probability distrin ð n ð n ð n ð bution for the bond length of diatomic molecules, but also provides the fine-structure internuclear potential V ðkÞ xÞ n ð via Eq. (4.4), which in turn furnishes a complete quantum dynamic model for a diatomic molecule. The proposed method of finding the hidden fine-structure potential accompanying an applied potential is equally applicable to other quantum systems. For example, we may replace the Morse potential V Morse ðrÞ with the Coulomb potential V Coulomb ðrÞ ¼ e2 =ð4pe0 rÞ between electron and proton, and by following the same procedures mentioned above we can obtain a fine-structure potential V Total ¼ V Coulomb þ QðwÞ; which correctly manifests the shell structure observed in the Hydrogen atom. We may also replace the Morse potential V Morse ðrÞ with the Yukawa potential V Yukawa ðrÞ ¼ c2 elr =r between two nucleons and reveal the fine structure potential existing in a nucleus.
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Chaos, Solitons and Fractals 37 (2008) 962–976 www.elsevier.com/locate/chaos
Complex dynamics in diatomic molecules. Part I: Fine structure of internuclear potential Ciann-Dong Yang
*
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan, ROC Accepted 30 March 2007
Abstract The current understanding of internuclear potential of diatomic molecules is limited to electronic states. In this paper, we point out that every electronic-state internuclear potential is actually accompanied by a fine structure, which characterizes the detailed internuclear potential for each vibrational substate belonging to the same electronic state. This fine structure, which governs the bond length, the force constant, the vibration frequency, and the quantum motion for each vibrational state, is obtained by extending Bohm’s quantum potential [Bohm D. A suggested interpretation of the quantum theory in terms of hidden variable. Phys Rev 1952;85:166–79] to complex domain. It is shown that the fine structure of the internuclear potential can be determined exactly by the vibrational wavefunctions so that its predictions about the internuclear forces (Part I) and the vibrational nuclear quantum motion (Part II) are fully consistent with the Max Born’s probability interpretation of the vibrational wavefunctions, and are in line with El Naschie’s E-infinity [El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36; El Naschie MS. E-infinity theory – some recent results and new interpretations. Chaos, Solitons & Fractals 2006;29:845–53; El Naschie MS. The concepts of E-infinity. An elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22:495–511], Nottale’s scale relativity [Nottale L. Fractal space-time and microphysics: Towards a theory of scale relativity. Singapore: World Scientific; 1993], and Ord’s random walk [Ord G. Fractal space time and the statistical mechanics of random works. Chaos, Soiltons & Fractals 1996;7:821–43] approaches to quantum mechanics. 2007 Elsevier Ltd. All rights reserved.
1. Introduction The knowledge of internuclear potentials of diatomic molecules is very important in a wide variety of fields ranging from gas kinetics to astrophysics. A great deal of information about the structure of a molecule is summarized in its potential energy curve. Among the numerous methods proposed to determine the potential energy curve, perhaps the most satisfactory is the calculation of the curve from the experimental energy levels using the Rydberg–Klein–Rees (RKR [7]) method. The use of the RKR method is inevitably limited to the region where sufficient spectroscopic data are available. An alternative approach is to fit the potential curve by a certain form of algebraic expression when the *
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C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
963
parameters in the expression are evaluated from the known spectroscopic constants. Many attempts to find the suitable empirical functions have been made, such as Morse [8], Hulburt–Hirschfelder [9], Rosen–Morse [10], Rydberg [7], Po¨schl–Teller [11], Linnett [12], Frost–Musulin [13], Varshni [14], Lippincott [15], and Hua [16], etc. The performance comparisons among these empirical potential functions with respect to their ability to reproduce the experimental data were studied in the review articles [14,17]. Up to now, the internuclear potentials determined either by quantum-mechanical computation or by experimental curve fitting are exclusively established for electronic states; no internuclear potential specific to vibrational states has been reported before. Consequently, all the vibrational substates belonging to the same electronic state are presumed to be governed by the same form of potential energy, sharing the same equilibrium bond length and the same force constant. However, the probability predictions using the vibrational wavefunctions solved from the Schro¨dinger equation indicate that each vibrational substate may have its own equilibrium bond length and its unique distribution of the possible bond lengths. It is evident that the single internuclear potential developed for an electronic state is not detailed enough to explain the diverse vibrational dynamics existing in the many vibrational substates belonging to the same electronic state. This paper aims to show that in addition to the common electronic-state potential energy, each vibrational substate does have its own potential energy, which can be determined by solving the Schro¨dinger equation with the electronic-state potential served as the input potential. By using the complex-space formulation of fractal spacetime [2–6], we demonstrate that a fine structure of internuclear potential arises naturally in the framework of complex-extended Hamilton mechanics. The existence and the correctness of this vibrational-state fine structure of internuclear potential are justified by its strong consistency with the probability predictions made from the vibrational wavefunctions. Knowing the fine structure of internuclear potential permits us to obtain the bond length, the force constant, the vibration frequency, and the equations of motion for each vibrational state, which otherwise cannot be afforded by the usual electronic-state internuclear potential. Moreover, in addition to the well-known quantization of the vibrational energy levels, theR fine structure further manifests the quantization of the vibration frequency and the quantization of the action variable p dq in each vibrational state.
2. Inconsistencies between Morse potential and probability interpretation To demonstrate the determination of the fine structure accompanying an internuclear potential, we will employ the Morse function [8] as representative model for an electronic-state potential energy. Although the Morse function is not the best choice to fit the experimental data [14,17], its exact solvability of the related Schro¨dinger equation allows us to derive a closed-form expression of the fine structure accompanying the internuclear potential it represents. Indeed, there is no loss of generality by adopting the Morse function to represent the internuclear potential, since the RKR potentials for various molecules have been shown to coalesce into a single curve, which can be represented by the Morse potential with a properly scaled internuclear distance [18]. The standard Morse potential is a 3-parameter function expressed in the form of V Morse ðrÞ ¼ ED ½1 ebðrr0 Þ 2 ;
ð2:1Þ
where r0 is the equilibrium bond length, ED is the potential energy for bond formation and b is a parameter controlling the width of the potential well. The three parameters r0, ED, and b are determined from the molecular spectroscopic data. We first raise several severe inconsistencies between the predictions of the Morse potential and the predictions of the vibrational wavefunctions, and then justify how these inconsistencies can be removed by considering the hidden fine structure behind the Morse potential. The time-independent Schro¨dinger equation with the Morse potential V Morse ðrÞ reads
h2 d2 w þ ED ½1 ebðrr0 Þ 2 w ¼ Ew; 2l dr2
ð2:2Þ
where l ¼ m1 m2 =ðm1 þ m2 Þ is the reduced mass of the equivalent single-particle model of a diatomic molecule with nuclear masses m1 and m2. By introducing the dimensionless parameters x ¼ bðr r0 Þ ¼ bx;
e¼
E ; h b =2l 2 2
k2 ¼
ED 2 2
h b =2l
:
ð2:3Þ
Eq. (2.2) can be recast into to a dimensionless form d2 w þ ½e k2 ð1 ex Þ2 w ¼ 0 dx2
ð2:4Þ
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from which the normalized eigenstates can be found as [19] z=2 kn1=2 2k2n1 wðkÞ xÞ ¼ N ðkÞ z Ln ðzÞ; n e n ð
where
z ¼ 2kex ;
n ¼ 0; 1; . . . ; ½k 1=2;
Lqp ðzÞ N ðkÞ n
is the associated Laguerre function. The normalization factor 1=2 ð2k 2n 1ÞCðn þ 1Þ ¼ : Cð2k nÞ
N ðkÞ n
ð2:5Þ
is given by ð2:6Þ
The quantum number n specifies the vibrational energy level via the relation 2 eðkÞ n e ¼ 2kðn þ 1=2Þ ðn þ 1=2Þ ;
n ¼ 0; 1; . . . ; ½k 1=2:
For the convenience of later use, the first several eigenfunctions are listed below: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k 1 z=2 k1=2 ðkÞ e z ; z ¼ 2kex ; k > 1=2; w0 ðxÞ ¼ Cð2kÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k 3 z=2 k3=2 ðkÞ e z ð2k 2 zÞ; k > 3=2; w1 ðxÞ ¼ Cð2k 1Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2k 5Þ z=2 k5=2 2 ðkÞ e z ½z =2 þ ð3 2kÞz þ 2k2 7k þ 6; k > 5=2; w2 ðxÞ ¼ Cð2k 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ð2k 7Þ z=2 k7=2 z3 =6 þ ðk 2Þz2 þ kð2k þ 9Þz ðkÞ e z ; k > 7=2: w3 ðxÞ ¼ Cð2k 3Þ 10z þ 4k3 =3 10k2 þ 74k=3 20
ð2:7Þ
ð2:8aÞ ð2:8bÞ ð2:8cÞ ð2:8dÞ
The eigenfunction wðkÞ xÞ provides us with the probability P ðkÞ xÞ ¼ wðkÞ xÞwðkÞ xÞ of finding the radial distance between n ð n ð n ð n ð the two nuclei at the location x ¼ bðr r0 Þ. The number of the vibrational states trapped in the internuclear potential VMorse depends on ED, the depth of the Morse potential. In general, for a given k in the range of ðm þ 1=2Þ < k 6 ðm þ 3=2Þ, there are m + 1 allowable energy levels eðkÞ n with the accompanying vibrational eigenfuncðkÞ ðkÞ ðkÞ tions denoted by w0 ; w1 ; w2 ; . . . ; wðkÞ . The allowed number of bound vibrational states and the associated energy levm els for four different depths of Morse potential are depicted in Fig. 1. It is observed that the only information the Morse potential can provide about a vibrational state is its displacement range. For example, at the ground-state energy level ðkÞ ðkÞ e0 in Fig. 1a, what V Morse can tell is the estimated vibrational range between A and B; but no detailed potential change ðkÞ
ðkÞ
between A and B at e0 can be read from V Morse . Although the use of the electronic-state potential VMorse leads to the accurate prediction of the vibrational energy levels in Eq. (2.7), its predictions about the quantum motion in each vibrational substate have severe inconsistency with those predicted by wðkÞ xÞ and P ðkÞ xÞ, as reflected in the following n ð n ð observations: • The prediction of the equilibrium bond length by the Morse potential is inconsistent with the prediction from the wavefunction wðkÞ xÞ. The action of the Morse potential ensures that once deviated from the equilibrium length n ð r0, the bound length r has a tendency to return to and remain at r0. This tendency leads to a maximum probability of finding the bond length at r ¼ r0 . Unfortunately, this prediction is inconsistent with that made from the probability density function P ðkÞ xÞ, which brings out an equilibrium bond length other than r ¼ r0 . Let us examine the n ð ðkÞ ðkÞ ðkÞ ground-state wavefunction w0 ðxÞ to raise this discrepancy. The probability density function P 0 ðxÞ ¼ w0 ðxÞ ðkÞ ðkÞ w0 ðxÞ with w0 ðxÞ given by Eq. (2.8a) ðkÞ
P 0 ðxÞ ¼
2k 1 z 2k1 e z ; Cð2kÞ
z ¼ 2kex ;
k > 1=2:
ð2:9Þ
ðkÞ ðkÞ The position xmax with the maximum value of P 0 ðxÞ is found from the condition dP 0 ðxÞ=dx ¼ 0, which yields
xmax ¼ ln
2k 1 2k > 0 ) rmax ¼ r0 þ ln > r0 : 2k 1 b 2k 1
ð2:10Þ
It can be seen that unless k ! 1, the maximum probability position rmax is distinct from the equilibrium position r0 ðkÞ ðkÞ predicted from Morse potential. The numerical distributions of P 0 ðxÞ and V Morse ðxÞ ¼ V Morse =ð h2 b2 =2lÞ ¼ k2 ð2Þ x 2 ð1 e Þ for k ¼ 2 are displayed in Fig. 2a. As predicted from Eq. (2.10), P 0 ðxÞ is stationary at xmax ¼ lnð4=3Þ 0:2877, where its maximum is achieved. The corresponding equilibrium bond length determined by ð2Þ ð2Þ P 0 ðxÞ is req ¼ rmax ¼ r0 þ ð1=bÞ lnð4=3Þ, which is different from req ¼ r0 as predicted from V Morse ðxÞ.
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
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Fig. 1. Electronic-state internuclear potentials represented by the Morse function with four different ranges of depth ðkÞ k ¼ ðED = h2 b2 =2lÞ1=2 . (a) In the range 1=2 < k < 3=2, only one bound vibrational state with energy level e0 is allowed. (b) In the ðkÞ ðkÞ range 3=2 < k < 5=2, two bound states with energy levels e0 and e1 are allowed. (c) In the range 5=2 < k < 7=2, three bound states ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ with energy levels e0 ; e1 and e2 are allowed. (d) In the range 7=2 < k < 9=2, four bound states with energy levels e0 ; e1 ; e2 and e3 are allowed. It is observed that the only information the Morse potential can provide about a vibrational state is its displacement range. ðkÞ ðkÞ For example, at the ground-state energy level e0 in (a), what V Morse can only tell is the vibrational range between A and B; the detailed ðkÞ ðkÞ change of potential between A and B at the energy level e0 cannot be read from V Morse .
ð2Þ
ð2Þ
Fig. 2. (a) The stationary positions of the probability density functions P 0 and P 1 do not coincide with the stationary position of the Morse potential at the origin. (b) The origin is a stable equilibrium position toward which the Morse potential always applies a ðkÞ restoring force f Morse even when k < 1=2. But the Morse potential cannot explain why a bound state is unable to develop when k < 1=2 ðkÞ by noting that the restoring force f Morse still exists in case of k < 1=2.
Besides the discrepancy in predicting the equilibrium bond length, a further inconsistency comes from the prediction of the number of the equilibrium bond lengths. The only possible equilibrium bond length predicted from Morse potential is req ¼ r0 for all the vibrational substates, but the probability density function P ðkÞ xÞ indicates that the number and the n ð location of the equilibrium bond length are different for different vibrational substates. To manifest this point, we conðkÞ sider the first excited vibrational state, whose probability density function w1 ðxÞ computed from Eq. (2.8b) is ðkÞ
P 1 ðxÞ ¼
2k 3 z 2k3 ð2k 2 zÞ2 ; e z Cð2k 1Þ
k > 3=2:
This function is stationary at two equilibrium positions:
ð2:11Þ
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xeq ¼ ln
4k 3
4k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 16k 15
ð2:12Þ ðkÞ
which yield two equilibrium bond lengths. The numerical of P 1 ðxÞ with k ¼ 2 is illustrated in Fig. 2a, pdistribution ffiffiffiffiffi ð2Þ showing the two equilibrium positions at xeq ¼ ln 8=ð5 17Þ 0:1314; 2:2108, where P 1 ðxÞ is stationary as local xÞ. It is hardly maximum. In general, there are n + 1 equilibrium bond lengths existing in the nth vibrational state wðkÞ n ð possible to explain the formation of such many different equilibrium bond lengths in the vibrational states wðkÞ xÞ, n ð ðkÞ n ¼ 0; 1; . . . ; ½k 1=2, by merely considering the action of the internuclear potential V Morse ðxÞ, since it is vibrationalstate-independent, and produces the only possible equilibrium point at xeq ¼ 0. ðkÞ • The Morse potential V Morse ðxÞ fails to explain the existence of nodes, where the probability P ðkÞ xÞ is zero. A node n ð xnode with P ðkÞ ð x Þ ¼ 0 is usually where the potential barrier approaches infinity; however, the Morse potential (2.1) node n is finite for any radial distance r P 0, which means that no radial distance between the two nuclei is absolutely prohibited under the action of Morse potential. Nevertheless, the probability P ðkÞ xÞ does become zero at the positions n ð ðkÞ of the roots of Ln2k2n1 ðzÞ ¼ 0, as can be seen from Eq. (2.5). Taking P 1 ðxÞ as an example, the node occurs at znode ¼ 2k 2, which amounts to xnode ¼ ln
k 1 k ) rnode ¼ r0 þ ln : k1 b k1 ð2Þ
ð2:13Þ ðkÞ
As can be seen from Fig. 2a, P 1 ðxnode Þ ¼ 0 at xnode ¼ ln 2, and notice that V Morse ðxnode Þ is a small finite value. In the nth eigenstate wðkÞ xÞ, we generally have n such nodes. If the heights of the Morse potential at all the nodes are finite, we n ð want to ask why the probability of reaching these nodes is zero and meanwhile why the probability of reaching the positions having much higher potential than xnode is not zero. Obviously, we cannot justify the existence of such nodes solely according to the action of the internuclear potential V Morse . • The action of the Morse potential is unable to clarify why bound vibrational state cannot exist in the range k 6 1=2. ðkÞ ð2=5Þ Inspecting the Morse potential V Morse with k ¼ 2=5 < 1=2 in Fig. 2b, we can see that the structure of V Morse is similar ðkÞ to those of V Morse with k > 1=2, and they all share the same properties that x ¼ 0 ðr ¼ r0 Þ is the equilibrium position ð2=5Þ ð2=5Þ ð2=5Þ ð2=5Þ with zero net force f Morse ð0Þ ¼ dV Morse ð0Þ=dx ¼ 0, and f Morse ðxÞ > 0, as x < 0, while f Morse ðxÞ < 0, as x > 0. This means that there is always a restoring force acting on the nuclei toward their equilibrium distance r ¼ r0 for the both cases of k 6 1=2 and k > 1=2. However, the Morse potential is unable to further distinguish the difference that the restoring force with k > 1=2 can form a bound state but that with k 6 1=2 cannot. ðkÞ
Above all, the vibrational-state-independent nature of the internuclear potential V Morse ðxÞ is incompatible with the vibrational-state-dependent nature of the probability density function P ðkÞ xÞ. How can a state-independent driving n ð ðkÞ force dV Morse ðxÞ=dx produce a state-dependent motion described by P ðkÞ xÞ? It is evident that the apparent internuclear n ð ðkÞ potential V Morse ðxÞ is not the sole driving potential within a diatomic molecule; it seems that there exists a state-dependent fine structure in the potential, which just produces the observed vibrational-state-dependent radial motion. In the following discussions, we will derive such a hidden fine structure from the Schro¨dinger equation and show that once the hidden fine structure is incorporated into the apparent internuclear potential, such as the Morse potential, to form a total potential, the aforementioned inconsistencies disappear automatically and the state-dependent quantum motion can be fully predicted and explained by the derived total potential.
3. The hidden potential in Schro¨dinger equation The hidden fine structure in the internuclear potential can be identified by treating the Schro¨dinger equation as an expression of energy conservation law. Firstly, we recall a classical result [20] that for a given classical Hamiltonian H c ðt; q; pÞ ¼ p2 =2m þ V ðt; qÞ; the classical Hamilton–Jacobi (H–J) equation reads oS c oS c 1 ¼ þ H c ðt; q; pÞ þ ðrS c Þ2 þ V ¼ 0; ot ot 2m p¼rS c
ð3:1Þ
ð3:2Þ
where Sc is the classical action function. By contrast, applying the transformation w ¼ expðiS=hÞ; to the Schro¨dinger equation
ð3:3Þ
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
ih
ow h2 2 r w þ V w; ¼ 2m ot
we obtain the quantum H–J equation oS 1 h 2 oS 1 2 h þ ðrSÞ2 þ V þ rS ¼ þ p þV þ rp ¼ 0: ot 2m 2mi ot 2m 2mi p¼rS If the above equation is recast into a form analogous to Eq. (3.2): oS þ H ðt; q; pÞ ¼ 0; ot p¼rS
967
ð3:4Þ
ð3:5Þ
ð3:6Þ
we find that the corresponding quantum Hamiltonian H turns out to be H ðt; q; pÞ ¼
1 2 p þ V ðt; qÞ þ Qðwðt; qÞÞ; 2m
where Qðwðt; qÞÞ is known as the quantum potential defined by h h 2 h2 2 r p r S¼ r ln wðt; qÞ: ¼ Qðwðt; qÞÞ ¼ 2mi 2mi 2m p¼rS
ð3:7Þ
ð3:8Þ
The quantum potential Q is just the aforementioned hidden potential, which, together with the externally applied potential V, completely determines a particle’s quantum motion. It is worth noting that the classical Hamiltonian Hc depends only on the externally applied potential V ðt; qÞ, whereas the quantum Hamiltonian H depends on V ðt; qÞ as well as on the internal state w where the particle lies. For a given state described by wðt; qÞ, the quantum Hamiltonian H defined in Eq. (3.7) is an explicit function of the canonical variables q and p that are regarded as independent variables. There are two roles played by the wavefunction w in the quantum Hamiltonian H. Firstly, as indicated in Eq. (3.5), it determines the canonical momentum according to pj ¼
oS o ln w ¼ ih : oqj oqj
ð3:9Þ
Secondly, it generates the quantum potential Q according to Eq. (3.8). The equations of motion for a particle moving in the quantum state w can be derived by applying the quantum Hamiltonian (3.7) to the Hamilton equations: dq oH ðwÞ 1 ¼ ¼ p; dt op m dp oHðwÞ o h2 2 r ln wðt; qÞ : ¼ ¼ V ðt; qÞ 2m dt oq oq
ð3:10aÞ ð3:10bÞ
Note that as in classical Hamilton mechanics, we have obtained oH =op and oH =oq by treating q and p as independent variables in the Hamiltonian (3.7). Each given wavefunction w represents a dynamic system described by the quantum Hamilton equations of motion (3.10), which are distinct from the classical ones in two aspects: the complex nature and the state-dependent nature. The state-dependent nature means that the quantum Hamilton equations of motion determine the quantum motion in the specific quantum state described by w. The complex nature is a consequence of the fact that the canonical variables solved from Eqs. (3.9) and (3.10) are, in general, complex variables. Complex canonical variable has a one-to-one correspondence with its associated quantum operator. For example, if we rewrite Eq. (3.9) as h ow ¼ pj w i oqj
ð3:11Þ
and compare it with the definition of the momentum operator ^ pj w ¼ pj w; we have b pj ¼
h o : i oqj
ð3:12Þ
The proof of the quantization axiom p ! ^p ¼ ihr thus becomes a one-line corollary of quantum Hamilton mechanics. Eq. (3.9) indicates that defining quantum momentum in complex domain is necessary to result in the correct momentum operator. A similar but different quantum momentum was proposed by Bohm [1] in the form of pB ¼ rS B , where SB is the phase of the wavefunction defined by w ¼ RB eiS B =h with RB and SB being real functions. If we follow the same procedures leading to Eq. (3.11) but employ the real quantum momentum pB ¼ rS B instead
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of the complex momentum p ¼ rS, we shall find that it is not possible to arrive at the correct momentum operator ^ p ¼ ihr. The other familiar example regards the operator of the angular momentum L ¼ q p. Evaluating the x component of L with p given by Eq. (3.9), we obtain h o ln w h o ln w i h o o z ¼ y z w: Lx ¼ ypz zpy ¼ y i oz i oy w oz oy Comparing the above equation to the definition, b L x w ¼ Lx w, gives b L x as o o b L x ¼ ih y z ¼ y^pz z^py : oz oy
ð3:13Þ
Following the same mechanism [21], we can identify any operator A from its related local observable Aðq; pÞ with p and q satisfying the Hamilton equations (3.10). The correctness of the quantum Hamiltonian (3.7) can be further jusb . Using Eq. (3.9) in Eq. (3.7), we have tified by showing that it leads to the correct Hamiltonian operator H h2 h2 2 1 h2 2 2 H ðt; q; pÞ ¼ ðr ln wÞ þ V r ln wðt; qÞ ¼ r wþVw ; ð3:14Þ w 2m 2m 2m which can be rewritten as h2 2 Hw ¼ r þ V w: 2m
ð3:15Þ
b w ¼ H w gives the desired operator The comparison with the definition H 2 b ¼ h r2 þ V : H 2m
ð3:16Þ
Thus the quantum Hamiltonian (3.7) is an equivalent expression of the Hamiltonian operator (3.16). Of significance in this equivalence is that the quantum Hamiltonian (3.7) and thus the Hamiltonian operator (3.16) reveal the existence of the quantum potential Q in the system’s total energy in addition to its classical component p2 =2m þ V . When the applied potential V is not an explicit function of time, the solution of the quantum H–J equation (3.6) can be expressed as Sðt; qÞ ¼ W ðqÞ Et;
ð3:17Þ
and the quantum H–J equation (3.6) is reduced to H ðt; q; pÞ ¼
1 2 oS p þ V ðqÞ þ QðwðqÞÞ ¼ ¼ E ¼ constant; 2m ot
ð3:18Þ
which is an energy conservation law for quantum systems. This energy conservation law is just an alternative expression of the time-independent Schro¨dinger equation, as can be verified by substituting p ¼ rS ¼ i hr ln w into Eq. (3.18) to obtain h2 2 r w þ ðE V Þw ¼ 0: 2m
ð3:19Þ
The equivalence between Eqs. (3.18) and (3.19) suggests that one can describe quantum motions either in terms of the probability density function w w solved from Eq. (3.19) or in terms of the canonical variables ðq; pÞ solved from the energy conservation law (3.18) and the quantum Hamilton equations of motion (3.10), which can be combined to form the quantum Newton equation: m
d2 qj oV total ¼ ¼ fj : dt2 oqj
ð3:20Þ
The total potential V Total ¼ V þ QðwÞ contains the applied potential V and the state-dependent quantum potential QðwÞ, which can be further simplified by using Eqs. (3.18) and (3.9) as V Total ¼ V þ QðwÞ ¼ E
1 2 h2 p ¼Eþ ðr ln wÞ2 : 2m 2m
ð3:21Þ
For a quantum state described by the wavefunction w, the total potential within the state w is determined from Eq. (3.21) by which the quantum equations of motion are then established according to Eq. (3.20).
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
969
The discussions so far are valid for any potential V, and we have reached the general observation that for any quantum system subjected to an external potential V, there always exists an accompanying fine-structure potential QðwÞ for each quantum state w. We can employ Eq. (3.8) to determine this fine-structure potential behind the applied potential V for any quantum system. In the hydrogen atom, V represents the Coulomb potential between proton and electron, and the fine-structure potential QðwÞ forms the shell structure observed in the hydrogen atom [22]. In an atomic nucleus, we may regard V as the Yukawa potential between two nucleons and may employ the obtained fine-structure potential to explain the shell structure observed in a nucleus. When a diatomic molecular system is concerned, V represents the electronic-state internuclear potential described by the Morse function V Morse , and the accompanying fine-structure potential QðwÞ represents the vibrational-state potential for the vibrational state w. In the next section, we will clarify that it is the total potential V Total ¼ V Morse þ QðwÞ, rather than the Morse potential VMorse alone, that is responsible for the quantum motion predicted by the probability P ðkÞ xÞ. n ð
4. Refined internuclear potential In Section 2, we have pointed out three inconsistencies between the predictions of the Morse potential VMorse and the probability P ðkÞ xÞ. Now we go on to verify that the missing potential needed to bridge the gap between VMorse and n ð P ðkÞ ð x Þ is just the quantum potential QðwÞ derived in Eq. (3.8) whose existence, as shown previously, is not by assumpn tion but is inherent in the Schro¨dinger equation. We start with the quantum Hamilton equations of motion (3.10), where, for the present case, the canonical coordinate is q ¼ x ¼ r r0 and the canonical momentum p is given by Eqs. (3.10a) and (3.9) as p¼l
dx h d ¼ ln wðkÞ n ðxÞ: dt i dx
ð4:1Þ
The quantum Hamiltonian governing the motion of x and p is provided by Eq. (3.7): H¼
p2 p2 h2 d2 ðkÞ ðkÞ ðkÞ w ðxÞ: þ V Morse ðxÞ þ QðkÞ þ V Morse ðxÞ n ðxÞ ¼ 2l 2l 2l dx2 n
ð4:2Þ
Substituting the eigenfunction wðkÞ n ðxÞ from Eq. (2.5) into the above Hamiltonian, we arrive at the energy conservation law for diatomic molecules: H¼
p2 ðkÞ ðkÞ þ V Morse ðxÞ þ QðkÞ n ðxÞ ¼ En ¼ constant; 2l
ð4:3Þ
ðkÞ with the energy level EðkÞ n already given by Eq. (2.7). Without the participation of the quantum potential Qn , the value ðkÞ 2 of the classical Hamiltonian H c ¼ p =2l þ V Morse ðxÞ is uncertain and cannot guarantee the energy conservation in the state wðkÞ n . ðkÞ Now we are in a position to remedy the aforementioned inconsistencies by replacing the role of V Morse with the ðkÞ ðkÞ refined potential V Total ¼ V Morse þ Qn , which has a dimensionless form as 2 V Total ðkÞ ðkÞ ðxÞ ¼ k2 ð1 ex Þ2 d ln wðkÞ ðxÞ V ð x Þ þ Q ¼ Morse n n dx2 h2 b2 =2l d d z ln wðkÞ z ¼ 2kex ; ¼ ðk z=2Þ2 z n ðzÞ ; dz dz
V ðkÞ xÞ n ð
ð4:4Þ
where the over-bar symbols denote dimensionless quantities. By using the wavefunctions wðkÞ n ðzÞ obtained from Eq. (2.5), the total potential for every vibrational state can thus be determined uniquely from Eq. (4.4). Especially, with the wavefunctions wðkÞ n ðzÞ listed explicitly in Eqs. (2.8a)–(2.8d), the first three total potentials can be computed as ðkÞ
V 0 ðxÞ ¼ ðk z=2Þ2 þ z=2; ðkÞ
V 1 ðxÞ ¼ ðk z=2Þ2 þ ðkÞ
V 2 ðxÞ ¼ ðk z=2Þ2 þ þ
z ¼ 2kex ;
½z2 þ 4ð1 kÞz þ 4k2 4k 2ðz 2k þ 2Þ2
;
z ¼ 2kex ; ð4:5Þ
z½z4 þ ð12 8kÞz3 þ ð24k2 68k þ 48Þz2 ½z2
2
þ ð6 4kÞz þ 4k 14k þ 12
2
z½ð32k3 þ 128k2 152k þ 48Þz þ 16k4 80k3 þ 132k2 72k ½z2 þ ð6 4kÞz þ 4k2 14k þ 122
:
970
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976 ðkÞ
The first term in V ðkÞ xÞ is the Morse potential V Morse ðxÞ, and the remaining terms are due to the vibrational-state fine n ð structure. The first inconsistency mentioned previously concerns the prediction about the equilibrium bond length. Let us examine how the total potential VTotal yields the same equilibrium bond length as predicted by the probability P ðkÞ n . ðkÞ ðkÞ The equilibrium position is where V 0 has a local minimum for which the condition dV 0 ðzÞ=dz ¼ 0 leads to zeq ¼ 2k 1 ) xeq ¼ ln
2k ; 2k 1
ð4:6Þ ðkÞ P 0 ðxÞ
which is exactly the equilibrium position predicted by as given in Eq. (2.10). From dynamic consideration, an equilibrium position must be the one on which the net applied force is zero. To verify this point, we compute the total ðkÞ force f 0 from Eq. (3.20) as d ðkÞ d ðkÞ 2k 1 ðkÞ : ð4:7Þ f 0 ðxÞ ¼ V 0 ðxÞ ¼ z V 0 ðzÞ ¼ 2k2 ex ex dx dz 2k ðkÞ According to the sign of f 0 ðxÞ, three force regions can be classified as following: ðkÞ 2k , we have f 0 ðxeq Þ ¼ 0, indicating that the net force exerting the equilibrium • Equilibrium point: At x ¼ xeq ¼ ln 2k1 point is truly zero. ðkÞ • Repulsive region: In the region x < xeq , we have f 0 ðxÞ > 0, indicating that the two nuclei are subjected to a repulsive force such that their bond length is increased toward the equilibrium position xeq . ðkÞ • Attractive region: In the region x > xeq , we have f 0 ðxÞ < 0, indicating that the two nuclei are subjected to an attractive force such that their bond length is decreased toward the equilibrium position xeq .
Examining the force action near the equilibrium point, we find that there is always a restoring force exerting the two nuclei toward their equilibrium relative position xeq . Once they enter the equilibrium position, they will remain there because the net force applied at the equilibrium position is zero. Consequently, the probability of finding x at xeq is expected to be the maximum. This point has been verified in Eqs. (2.9) and (2.10) by showing that ðkÞ 2k xeq ¼ xmax ¼ ln 2k1 is just the position maximizing P 0 ðxÞ. A general property says that the local maximum of ðkÞ ðkÞ P n ðxÞ occurs at the local minimum of V n . This property can be proved from Eq. (3.21) whose dimensionless form becomes !2 1 1 dwðkÞ xÞ ðkÞ ðkÞ n ð V n ðxÞ ¼ en þ P eðkÞ ð4:8Þ n : 2 wðkÞ d x ð x Þ n ðkÞ The minimum value of V ðkÞ xÞ is eðkÞ xÞ=dx ¼ 0; but dwðkÞ xÞ=dx ¼ 0 is just the n ð n ð n , which occurs at the position with dwn ð ðkÞ ðkÞ ðkÞ condition that P n ðxÞ ¼ wn ðxÞwn ðxÞ achieves its maximum value by noting
dP ðkÞ xÞ d dwðkÞ ðxÞ n ð ¼ wðkÞ ¼ 0; xÞwðkÞ xÞ ¼ 2wðkÞ xÞ n n ð n ð n ð dx dx dx
ð4:9Þ
since wðkÞ xÞ is a real function of x. The total potential V ðkÞ n ð n at several vibrational states specified by n for four different depths of the Morse potential is demonstrated in Fig. 3, where it can be observed that the minimum value of V ðkÞ n occurs ðkÞ precisely at the energy level eðkÞ corresponding to the vibrational state w . n n From the viewpoint of dynamic motion, a diatomic molecule must be stationary at its equilibrium position with zero velocity and zero acceleration. The Hamilton equations of motion (3.10) are helpful to clarify this point. A dimensionless expression of the Hamilton equations reads dx i dwðkÞ xÞ n ð ¼ ðkÞ ; dt wn ðxÞ dx
ð4:10aÞ
dp dV ðkÞ ðxÞ 1 dwðkÞ xÞ d2 ln wðkÞ xÞ n ð n ð ¼ n ¼ ðkÞ ; dt dx dx2 wn ðxÞ dx
ð4:10bÞ
where t ¼ ðhb2 =lÞt and p ¼ p=ðbhÞ are the dimensionless time and momentum, respectively. It is evident that the condition dwðkÞ xÞ=dx ¼ 0 exactly implies the equilibrium condition dx=dt ¼ d p=dt ¼ 0. In short, we have shown that the n ð following four positions are coincident: (a) the dynamic equilibrium position determined from Eqs. (4.10), (b) the position with minimum total potential determined from Eq. (4.8), (c) the position with maximum probability determined from Eq. (4.9), and (d) the position with zero applied force determined from Eq. (4.7).
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
971
Fig. 3. The total potential V ðkÞ n at each vibrational state specified by n for four different depths of the Morse potential. It can be occurs precisely at the energy level eðkÞ corresponding to the vibrational state wðkÞ observed that the minimum value of V ðkÞ n n n . In ðkÞ comparison with Fig. 1, the Morse potential V Morse can only provide the displacement range at enðkÞ , while the total potential ðkÞ ðkÞ ðkÞ V ðkÞ xÞ=dx at every energy level eðkÞ n ¼ V Morse þ Qðwn Þ can further offer the detailed internuclear force V n ð n .
A numerical verification of k ¼ 2 is shown in Fig. 4, where the correspondence between the maximum of P ð2Þ xÞ and n ð the minimum of V ð2Þ ð x Þ are clearly demonstrated for n ¼ 0; 1. By contrast, the equilibrium position obtained from n ðkÞ dV Morse ðxÞdx ¼ 0 is always at x ¼ 0, which is nothing to do with the correct equilibrium position given by Eq. (2.12). ðkÞ The second inconsistency between V Morse and P ðkÞ xnode with P ðkÞ xnode Þ ¼ 0. Under n regards the prediction of the node n ð ðkÞ ðkÞ the action of the Morse potential V Morse , such a node does not exist since V Morse ðxÞ is everywhere finite and no location is xÞ, node does exist in that there are positions where inaccessible. However, under the action of the total potential V ðkÞ n ð V ðkÞ xÞ approaches infinity. The relation between P ðkÞ xnode Þ and V ðkÞ xnode Þ is best illustrated via Eq. (4.8) as n ð n ð n ð !2 1 dwðkÞ xÞ n ð V ðkÞ xÞ ¼ eðkÞ ; ð4:11Þ n ð n þ ðkÞ dx 2P n ðxÞ ðkÞ ðkÞ ðkÞ ðkÞ which indicates that the position having zero probability P ðkÞ n ¼ wn wn ¼ wn wn ¼ 0 is just where the total potential ðkÞ V ðkÞ ð x Þ approaches infinity. The inaccessibility to the position with w ¼ 0 is also reflected in the equations of motion n n (4.10), which give rise to infinite velocity and acceleration at the points where wðkÞ n ¼ 0.
ð2Þ
ð2Þ
ð2Þ
Fig. 4. (a) The local minimum of V 0 coincides with the local maximum of P 0 . (b) The two local minimums of V 1 coincide with the ð2Þ ð2Þ two local maximums of P 1 , and the node position xnode at which V 1 approaches infinity is inaccessible and thus has zero probability, ð2Þ ð2Þ P 1 ¼ 0. The action of the Morse function V Morse alone cannot explain the existence of the node and cannot predict the locations ð2Þ ð2Þ maximizing P 0 and P 1 .
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C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976 ðkÞ
It has been mentioned that the Morse potential V Morse cannot explain why bound state is unable to develop when ðkÞ k 6 1=2. If we consider the action of the total potential V ðkÞ xÞ instead of V Morse , the reason becomes apparent. Accordn ð ðkÞ ing to the prediction of V 0 ðxÞ, the equilibrium position xeq for the ground state is given by Eq. (4.6), from which we can see that when k ¼ 1=2, the equilibrium position xeq and the bond length approach infinity, resulting in the dissociation of the diatomic molecules. As k is reduced to lower than 1=2, Eq. (4.6) reveals that real equilibrium position no longer exists and no bound state can be developed. A deeper understanding about the annihilation of the bound state can be gained from the force distribution (4.7), which indicates that as k 6 1=2, the internuclear force is everywhere repulsive so that no attractive force is available to form a bond state. ð1=2Þ ð1=2Þ ðxÞ, the total Fig. 5 illustrates the actual distributions of the Morse potential V Morse ðxÞ, the quantum potential Q 0 ðkÞ potential V 0 ðxÞ, and their associated internuclear forces for the two cases of k ¼ 1=2 and k ¼ 1. In case of k ¼ 1=2, ð0:5Þ ð0:5Þ ð0:5Þ ð0:5Þ although the Morse potential contributes attractive internuclear force f Morse , the total force f 0 ¼ f Morse þ f Q0 is ð1Þ everywhere repulsive, making a bound state unable to establish. When k > 1=2, the attractive force f Morse afforded ð1Þ by the Morse potential becomes large enough to overbalance the repulsive quantum force f Q0 , and allows a bound state to exist as shown in Fig. 5c and d for k ¼ 1.
5. Quantizations in fine-structure potential Besides the energy quantization, further quantization rules can be discovered from the total potential V ðkÞ xÞ and n ð R the related Hamilton equations of motion. The first noticeable quantization is concerned with the action variable p dx. Using Eq. (3.9), we can express the action variable as Z Z Z Z h 1 dwðkÞ xÞ h 1 dwðkÞ zÞ n ð n ð d x ¼ dz; ð5:1Þ p dx ¼ h p dx ¼ ðkÞ ðkÞ i d x i d z xÞ zÞ c c c wn ð c wn ð ðkÞ where c is any closed trajectory in the vibrational state wðkÞ zÞ from Eq. (2.5) further yields n . The substitution of wn ð Z Z 2k2n1ðzÞ h dðLn Þ=dz dz: ð5:2Þ p dx ¼ 2k2n1 2pi ðzÞ L c c n
To evaluate the above contour integral, we quote a useful theorem from the calculus of residue:
Fig. 5. The comparisons of the internuclear potential and force between k ¼ 1=2 and k ¼ 1. (a, b) In case of k ¼ 1=2, although the ð0:5Þ ð0:5Þ ð0:5Þ ð0:5Þ Morse potential contributes attractive internuclear force f Morse , the total force f 0 ¼ f Morse þ f Q0 is everywhere repulsive, making a ð1Þ bound state unable to establish. (c, d) In case of k ¼ 1, the attractive force afforded by f Morse becomes large enough to overbalance the ð1Þ repulsive quantum force f Q0 , and allows a bound state to exist.
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
1 2pi
Z C
F 0ðzÞ dz ¼ N P ; F ðzÞ
973
ð5:3Þ
where N and P are, respectively, the number of zeros and poles of F ðzÞ enclosed by the contour C. The application of the above residue theorem to Eq. (5.2) leads to Z p dx ¼ mh; m ¼ 0; 1; . . . ; n ð5:4Þ c
by noting that Ln2k2n1 ðzÞ is an nth-order polynomial so that its number of zeros enclosed by c is at most n, while the number of poles is always zero. The quantization rule (5.4)R reveals that in spite of the infinitely many closed trajectory c in the complex plane of x, the value of the action variable c p dx is only allowed to be the integer multiple of the Planck constant h. The other new quantization rule to be noted is the quantization of the vibration frequency. Using the fine-structure potential (4.4), we can find the force constant in each vibrational state as h i2 d2 V kn 2 xeq 2 K ðkÞ ðxeq Þ ¼ 2 eðkÞ Þ ¼ 2½2kðn þ 1=2Þ ðn þ 1=2Þ2 k2 ð1 exeq Þ2 2 ; ð5:5Þ n ¼ n k ð1 e 2 dx where xeq is the equilibrium position in the nth vibrational state. In the above evaluation of d2 V kn =d2x, we have applied the equilibrium condition dV ðkÞ xeq Þ=dx ¼ 0 and the fact that the eigenfunction wðkÞ xÞ satisfies the Schro¨dinger equation n ð n ð ðkÞ ðkÞ with the eigenvalue en . When applied to the ground vibrational state where xeq is given by Eq. (2.10) and e0 ¼ k 1=4, Eq. (5.5) yields the force constant ðkÞ
K0 ¼
h2 b2 =2l ðkÞ h2 b4 ðk 1=2Þ2 ; K0 ¼ l 1=b2
ð5:6Þ
k > 1=2;
and the related ground-state frequency becomes sffiffiffiffiffiffiffiffi ðkÞ K0 hb2 ðkÞ ðk 1=2Þ: x0 ¼ ¼ l l
ð5:7Þ
Comparing calculated from the Morse potential xMorse ¼ ffi with the above results, we see that the frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ V 00Morse =l ¼ hb2 k=l is close to the ground-state frequency x0 only when k 1=2; but for excited vibrational states with n > 0, the Morse frequency xMorse is unable to provide any information about the actual vibration frequency xðkÞ n . ðkÞ To demonstrate the self-consistency of the fine-structure inter-nuclear potential, we can further compute x0 directly from the Hamilton equation of motion (4.10a), which gives by substituting w0 ðzÞ ¼ ez=2 zk1=2 dz i ¼ zðz 2k þ 1Þ; dt 2
ð5:8Þ
whose solution can be found as zðtÞ k xðtÞ ¼ ln ¼ ln þ C 1 eiðk1=2Þt : 2k k 1=2
ð5:9Þ ðkÞ
The time response of xðtÞ indicates a dimensionless vibration period t0 equal to 2p=ðk 1=2Þ and a corresponding frequency equal to ðkÞ
x0 ¼
2p ðkÞ
T0
¼
2p ðkÞ
T 0 l=ðhb2 Þ
¼
hb2 ðk 1=2Þ; l
ð5:10Þ
which is the same as Eq. (5.7) obtained from force constant. The third method to compute the vibration frequency is via the contour integral along the trajectories determined by Eq. (5.8), leading to the same result Z Z 1 2=ð2k 1Þ 1 2 2p ðkÞ T 0 ¼ dt ¼ dz ¼ ð2piÞ ¼ 8c; ð5:11Þ i c z ð2k 1Þ i 2k 1 k 1=2 where c is any closed trajectory enclosing the equilibrium point zeq ¼ 2k 1. The contour integral is solely determined by the enclosed equilibrium point and is independent of the actual trajectory. Only one frequency is allowed in the ground vibrational state, because there is only one equilibrium point there. ðjÞ In the nth vibrational state, we have n+1 equilibrium positions xeq , j ¼ 1; 2; . . . ; n þ 1, which are the roots of ðjÞ dwðkÞ ð x Þ=d x ¼ 0, as mentioned in Section 4. The vibration about a specific equilibrium position xeq has its own force n
974
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976 ðkÞ
ðkÞ
ðkÞ
ðkÞ
ðjÞ constant K n;j determined by substituting xeq into Eq. (5.5). Therefore, n+1 force constants K n;1 ; K n;2 ; . . . ; K n;nþ1 , and n+1 related vibration frequencies can be identified in the nth vibrational state. For instance, we have two equilibrium positions for the first excited vibrational state as derived in Eq. (2.12), which, when substituted into Eq. (5.5), yields two force constants 2 h2 b4 1 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ ðkÞ K 1;1 ; K 1;2 ¼ ð5:12Þ ð16k 15Þ 16k 15 ; l 8 8
and each of the two equilibrium positions has its own vibration period as 1 2pl 1 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ ðkÞ 16k 15 : T 1;1 ; T 1;2 ¼ 2 ð16k 15Þ 8 hb 8
ð5:13Þ
ðkÞ
The determination of vibration frequency from the force constant K n;j is based on the underlying assumption that the vibrational motion takes place with respect to an equilibrium position. However, if the vibrational amplitude is so qffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ ðkÞ large that two or more equilibrium positions are enveloped within the amplitude, the relation xn;j ¼ K n;j =l becomes invalid. Such a large-amplitude vibration, which does occur in the first and other excited vibrational states, can only be detected by solving the equations of motion. As to the first excited state, the equation of motion is obtained by substiðkÞ tuting w1 into Eq. (4.10a) as dz z3 þ ð3 4kÞz2 þ ð2k 3Þð2k 2Þz ¼i ; dt 2ðz 2k þ 2Þ
z ¼ 2kex :
ð5:14Þ
The resulting trajectories in the complex plane xR xI can be classified into three sets as shown in Fig. 6a. The sets X1 and X2 contain the trajectories enclosing only one of the two equilibrium positions xð1Þ xð2Þ eq and eq listed in Eq. (2.12), and X3 contain the aforementioned large-amplitude trajectories enclosing both the two equilibrium points. The time responses of xR ðtÞ are depicted in Fig. 6b from which the vibration periods can be read. We can determine the vibration periods analytically by the residue theory without actually solving Eq. (5.14). The contour integral along a closed trajectory satisfying Eq. (5.14) gives Z Z 1 1 2ðz zk þ 2Þ ðkÞ dt ¼ dz: ð5:15Þ T1 ¼ i i c z3 þ ð3 4kÞz2 þ ð2k 3Þð2k 2Þz Note that the two equilibrium positions xð1Þ xð2Þ eq , eq listed in Eq. (2.12) are just the two poles of the above integrand. Depending on the poles enclosed by the contour c, the evaluation of the integral by the calculus of residue gives 8 2pl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðkÞ 1 > ð16k 15Þ þ 38 16k 15 ¼ T 1;1 8c 2 X1 ; > hb2 8 > < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2pl ðkÞ ðkÞ ðkÞ T 1 ¼ 2 T 1 ¼ 2pl2 18 ð16k 15Þ 38 16k 15 ð5:16Þ ¼ T 1;2 8c 2 X2 ; hb > hb > > : 2pl ðkÞ ðkÞ ðkÞ =ðk 3=2Þ ¼ T 1;3 ¼ T 1;1 þ T 1;2 8c 2 X3 ; hb2
Fig. 6. The eigen trajectories solved from Eq. (5.14) for the first excited state with k ¼ 2. (a) The trajectories in the complex plane xR xI can be classified into three sets. The sets X1 and X2 contain the trajectories enclosing only one equilibrium point, and X3 contain the large-amplitude trajectories enclosing two equilibrium points. (b) The time responses of xR ðtÞ show that the vibration periods for all the periodic trajectories are quantized into three values T 1 , T 2 , and T 3 , corresponding to the three trajectory sets X1, X2, and X3. All the trajectories in X1(X2) have the same period T 1 ðT 2 Þ, and all the large-amplitude trajectories in the set X3 have the same period T 3 ¼ T 1 þ T 2.
C.-D. Yang / Chaos, Solitons and Fractals 37 (2008) 962–976
975 ðkÞ
ðkÞ
ðkÞ
which shows that the vibration periods for all the periodic trajectories are quantized into three values T 1;1 , T 1;2 , and T 1;3 , ðkÞ ðkÞ corresponding to the three trajectory sets X1, X2, and X3 . All the trajectories in X1 (X2) have the same period T 1;1 ðT 1;2 Þ, ðkÞ ðkÞ ðkÞ and all the large-amplitude trajectories in the set X3 have the same period T 1;3 ¼ T 1;1 þ T 1;2 . Comparing Eq. (5.16) with ðkÞ ðkÞ Eq. (5.13), we can see that the force constants in Eq. (5.13) can only determine the periods T 1;1 and T 1;2 corresponding ðkÞ to the trajectories in the sets X1 and X2 enclosing only one equilibrium points, while the determination of the period T 1;3 corresponding to the large-amplitude trajectories in X3 needs the information of equation of motion (5.14). In general, for any vibrational state wðkÞ n we can determine the quantized vibration periods by evaluating the contour integral along the equation of motion (4.10a).
6. Conclusions In this paper we pointed out that an experimentally determined electronic-state internuclear potential described by ðkÞ V Morse is inadequate in describing the quantum motions in its vibrational substates and is even inconsistent with the ðkÞ probability predictions of the vibrational wavefunctions determined from V Morse . For a given electronic-state internuclear potential, we have verified that there exists an accompanying fine-structure potential V ðkÞ xÞ, which provides us n ð with all the information to determine the radial motion of a diatomic molecule in each vibrational state described by wðkÞ xÞ. We can employ V ðkÞ xÞ to derive the quantum equations of motion (4.10), to derive the internuclear forces n ð n ð ðkÞ ðkÞ f n ðxÞ ¼ V ðkÞ xÞ=dx, and to compute the force constant K n ð n for the nth vibration state. All the predictions made from ðkÞ ðxÞ, K ðkÞ , and the quantum equations of motion are consistent with the probability predictions made from V ðkÞ ð x Þ, f n n n P ðkÞ xÞ ¼ wðkÞ xÞwðkÞ xÞ. In other words, the vibrational wavefunction wðkÞ xÞ not only provides the probability distrin ð n ð n ð n ð bution for the bond length of diatomic molecules, but also provides the fine-structure internuclear potential V ðkÞ xÞ n ð via Eq. (4.4), which in turn furnishes a complete quantum dynamic model for a diatomic molecule. The proposed method of finding the hidden fine-structure potential accompanying an applied potential is equally applicable to other quantum systems. For example, we may replace the Morse potential V Morse ðrÞ with the Coulomb potential V Coulomb ðrÞ ¼ e2 =ð4pe0 rÞ between electron and proton, and by following the same procedures mentioned above we can obtain a fine-structure potential V Total ¼ V Coulomb þ QðwÞ; which correctly manifests the shell structure observed in the Hydrogen atom. We may also replace the Morse potential V Morse ðrÞ with the Yukawa potential V Yukawa ðrÞ ¼ c2 elr =r between two nucleons and reveal the fine structure potential existing in a nucleus.
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