Nonlinear quantum dynamics in diatomic molecules: Vibration, rotation and spin

Chaos, Solitons & Fractals 45 (2012) 402–415

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Nonlinear quantum dynamics in diatomic molecules: Vibration, rotation and spin Ciann-Dong Yang ⇑, Hung-Jen Weng Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan

a r t i c l e

i n f o

Article history: Received 6 April 2011 Accepted 12 January 2012 Available online 14 February 2012

a b s t r a c t For a given molecular wavefunction W, the probability density function W⁄W is not the only information that can be extracted from W. We point out in this paper that nonlinear quantum dynamics of a diatomic molecule, completely consistent with the probability prediction of quantum mechanics, does exist and can be derived from the quantum Hamilton equations of motion determined by W. It can be said that the probability density function W⁄W is an external representation of the quantum state W, while the related Hamilton dynamics is an internal representation of W, which reveals the internal mechanism underlying the externally observed random events. The proposed internal representation of W establishes a bridge between nonlinear dynamics and quantum mechanics, which allows the methods and tools already developed by the former to be applied to the latter. Based on the quantum Hamilton equations of motion derived from W, vibration, rotation and spin motions of a diatomic molecule and the interactions between them can be analyzed simultaneously. The resulting dynamic analysis of molecular motion is compared with the conventional probability analysis and the consistency between them is demonstrated. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction In recent decades, there have been fruitful applications of molecular spectra to a wide variety of fields ranging from gas kinetics to astrophysics. Among various types of molecules, diatomic molecules have always had a special attraction for physics due to the abundant data and the relative simplicity of the system as a single oscillator. The rotational modification of the vibrational wavefunctions for diatomic molecules is of special importance in astrophysics, since at the high temperatures involved the recorded band-head maximum intensities occur at high l-values. Analytical solutions to Schrödinger equation for the nuclear motion of a diatomic molecule with rotational terms included had been studied by a number of theorists, such as Kratzer [1], Dunham [2], Poschl and Teller [3] and Pekeris [4]. By ⇑ Corresponding author. E-mail addresses: [email protected] (C.-D. Yang), principlex@ yahoo.com.tw (H.-J. Weng). 0960-0779/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2012.01.006

treating solutions to Schrödinger equation as probability density functions, our understanding so far about quantum molecular motion is solely based on a probabilistic perspective. Despite the great success achieved in the molecular spectra analysis and in the probabilistic prediction of state transitions, the lack of dynamic equations of motion in quantum mechanics has long made it impossible for us to investigate quantum molecular motions in a conventional sense of nonlinear dynamics. Quantum Hamilton mechanics [5] resolved this problem by providing Hamilton equations of motion to describe quantum molecular dynamics in a deterministic way, while preserving the consistence with the probabilistic interpretation. Besides serving as a probability density function, it was recognized [6] that a solution W to Schrödinger equation also intrinsically determines a set of Hamilton equations, which are a dynamic representation of the quantum state W. The formulation of quantum Hamilton mechanics originates from the quantum Hamilton–Jacobi Equation,

C.-D. Yang, H.-J. Weng / Chaos, Solitons & Fractals 45 (2012) 402–415

@S þ Hðt; q; pÞjp¼rS @t   @S 1 ih þ p  p þ VðqÞ ¼ rp ¼ 0; @t 2m 2m p¼rS

ð1:1Þ

which is an alternative expression of Schrödinger equation via the transformation W(t, q) = exp(iS(t, q)/⁄). The quantum Hamilton–Jacobi equation is one of the nine formulations of quantum mechanics based on transformation theory [7]. The last term in Eq. (1.1) is known as the complex quantum potential defined by

Q ðWðt; qÞÞ¼

 2   h  h h r  p ¼ r2 S ¼  r2 ln Wðt; qÞ: 2mi 2mi 2m p¼rS ð1:2Þ

The canonical momentum p is related to the action function S via the law of canonical transformation

p ¼ rSðt; qÞ ¼ ihr ln Wðt; qÞ:

ð1:3Þ

pffiffiffiffiffiffiffi The appearance of the imaginary number i ¼ 1 shows that the momentum p has to be defined in a complex domain. For a given state described by W, the quantum Hamiltonian H defined in Eq. (1.1) is an explicit function of the canonical variables q and p. The Hamilton equations of motion in the quantum state W are formed by applying the above quantum Hamiltonian H to the Hamilton equations:

dq @Hðt; q; pÞ p ¼ ¼ ; dt @p m dp @Hðt; q; pÞ @ ¼ ¼ ðVðqÞ þ Q ðWðt; qÞÞÞ; dt @q @q

ð1:4aÞ ð1:4bÞ

where q and p are complex-valued vectors. The combination of Eqs. (1.3) and (1.4a) yields

dq 1 h 1 ¼ rS ¼ i rWðt; qÞ; dt m mW

q; p 2 C;

ð1:5Þ

which can be conceived of as a complex version of Bohmian mechanics [8,9]. The real-valued quantum trajectory determined by Bohmian mechanics [10–12] led to a controversy conclusion that a diatomic molecule is motionless in all its eigen-states. This difficulty could be conquered by the complex quantum trajectory q(t) determined by integrating the complex kinematical relation (1.5). Recently, the complex quantum trajectory method based on Eq. (1.5) has been developed into a potential computational tool to analyze wave-packet interference [13], wave-packet scattering [14–16], quantum vortices [17] and quantum streamline [18]. In addition, synthetic approaches were developed to generate approximate complex-valued quantum trajectories, which led to significant improvement in complex time-dependent QWB method [19] and in the calculation of tunneling probability [20]. Base on the coordinate-independent Hamilton equation b has (1.4), it was reported that every quantum operator A b W ¼ AW a complex realization A(p, q) via the relation A [5,21]. Many quantum effects have been predicted and verified from the solutions of the Hamilton equations [22–25]. The application of Eq. (1.4) to one-dimensional vibration motion in a diatomic molecule was studied in the authors’

403

previous work [26,27]. The present paper goes one step further to consider a general three-dimensional quantum motion, which describes molecular vibration, rotation and spin, simultaneously. In the next section, we derive Hamilton equations of motion for a quantum state W(r, h, /) described by spherical coordinates and then introduce Morse function to represent the internuclear potential in a diatomic molecule. The related Schrödinger equation is solved to find the eigenfunctions Wnv lml ðr; h; /Þ whose three-dimensional quantum dynamics are represented by spherical Hamilton equations. The resulting radial quantum dynamics are then solved in Sections 3 and 4 to analyze the interactions between molecular vibration and rotation motions in ground state and excited states, respectively. Finally, the azimuth quantum dynamics are solved in Section 5 to show that spin is the remnant angular motion when orbital angular momentum is zero; in other words, spin is the ‘‘zero dynamics’’ of nonlinear quantum dynamics. 2. Three-dimensional quantum dynamics of a diatomic molecule Referring to Fig. 1, the equivalent single-particle model of a diatomic molecule with reduced mass m = m1m2/ (m1 + m2) is described by spherical coordinates (r, h, /), where r denotes the bond length, and (h, /) denotes the orientation of the molecule. By expanding the inner product p  p and the divergence r  p in spherical coordinates, quantum Hamiltonian H in Eq. (1.1) can be expressed by

" !# 1 h  2 h @ 2 ln W 2 p þ p þ H¼ 2m r i r r i @r2 " ! 1 h  h @ 2 ln W 2 p þ ph cot h þ þ 2mr 2 h i i @h2 !# 2 1 2 @ ln W p2/  h þ 2 @/2 sin h þ Vðr; h; /Þ ¼

1 2 L2 Pr þ þ Vðr; h; /Þ: 2m 2mr2

ð2:1Þ

Fig. 1. The equivalent single-particle model of a diatomic molecule with reduced mass m = m1m2/(m1 + m2), whose rotation and vibration motions are described by spherical coordinates (r, h, /).

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In comparison with their classical counterparts, the momenta P2r ; L2 and L2z defined in quantum Hamiltonian (2.1) contain additional quantum correction terms:

P2r ¼ p2r  2ih 2

L ¼

p2h þ

2 pr 2 @ ln W ;  h r @r 2

 i hph cot h  h L2z 2

sin h

2

ð2:2aÞ

2

@ ln W @h2 2

; L2z ¼ p2/  h

@ 2 ln W @/2

;

ð2:2bÞ

where the terms containing Plank constant stem from quantum correction. The dynamic representation of a state W in spherical coordinates can be derived from Eq. (1.4a) with H given by Eq. (2.1),

 2 2 d ln U 2 d ln U 2  h ¼ m2l h ¼ constant; ð2:5aÞ d/ d/2  2 d ln H 2 d ln H 2  h cot h L2 ¼ h dh dh

L2z ¼ h

2

2

2

 h

d ln H

L2z

2

¼ lðl þ 1Þh ¼ constant; 2 dh2 sin h ! 2 2  2 2 h d ln R h 2 d ln R d ln R  þ H¼ 2 dr 2m 2m r dr dr þ

þ

L2 þ VðrÞ ¼ E ¼ constant: 2mr2

ð2:5bÞ

ð2:5cÞ

@H p h cot h h  @ ln W h cot h ¼ h þ ¼ ; þ h_ ¼ @ph mr 2 i 2mr2 imr2 @h i 2mr 2

ð2:3bÞ

Before applying quantum Hamilton mechanics to a molecular system, we have to specify the representative potential V in Eq. (2.1). There is no loss of generality by adopting Morse function to represent the internuclear potential in a diatomic molecule, since potentials for various molecules coalesce into a single curve, which can be represented by Morse potential with a properly scaled internuclear distance [30]. The standard Morse potential is a 3-parameter function expressed by

p/ @H h @ ln W /_ ¼ ¼ ¼ : @p/ mr 2 sin2 h imr2 sin2 h @/

ð2:3cÞ

V Morse ðrÞ ¼ ED ½1  ebðrr0 Þ 2 :

r_ ¼

@H pr h 1 h @ ln W h 1 ¼ ; ¼ þ þ @pr m i mr im @r i mr

ð2:3aÞ

The total potential VTotal = Q + V defined in Eq. (1.4b) is the sum of the intrinsic quantum potential Q and the applied potential V: 2

V Total ¼

h  ð4 þ cot2 hÞ 8mr 2

! 2 h  @ 2 ln W 1 @ 2 ln W 1 @ 2 ln W þ 2 þ  2 2 r 2m @r 2 @h2 r 2 sin h @/ þ Vðr; h; /Þ:

ð2:4Þ

In case of a time-independent central-force field Vðr; h; /Þ = V(r), Schrödinger equation has a separable solution in the form of W(r, h, /, t) = R(r)H(h)U(/)eiEt/⁄. The wavefunction components R(r), H(h) and U(/) were found to be solutions of the following three conservation laws [28,29],

ð2:6Þ

As illustrated in Fig. 2, 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 molecular spectroscopic data. Because Morse potential VMorse(r) is a central-force field, the corresponding Schrödinger equation can be decoupled into three ordinary differential equations as shown in Eq. (2.5). The solutions to Eqs. (2.5a) and (2.5b) are given readily as Hlml ðhÞ ¼ m Pl l ðcos hÞ and Uml ð/Þ ¼ eiml / , while Eq. (2.5c) becomes

    1 d 2m lðl þ 1Þ 2 dR R ¼ 0: r þ ðE  V ðrÞÞ  Morse 2 r 2 dr dr r2 h 

ð2:7Þ

By rewriting the radial wavefunction as R(r) = u(r)/r and changing the independent variable r to y ¼ ebðrr0 Þ , Eq. (2.7) can be recast into the following form:

Fig. 2. Morse function is more accurate than the quadratic function in expressing the internuclear potential within a diatomic molecule in higher excited vibration states.

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    1 d du 2m E  ED 2ED K r20 þ 2 2 þ  uðyÞ ¼ 0; y  E D y dy y2 r2 dy y2 y b h 2

K¼

h lðl þ 1Þ : 2mr 20

ð2:8Þ

This equation has no analytical solution, but an approximate approach is to expand r20 =r2 with respect to y  1 and retain up to the second-order terms [4]. With this approximation, Eq. (2.8) becomes

#   " 1 d du E  ED  C 0 2ED  C 1 y þ þ  ðED þ C 2 Þ uðyÞ ¼ 0; y dy dy y2 y ð2:9Þ 2 2

  3 3 C0 ¼ 1 þ 2 ; 2 b b b     lðl þ 1Þ 4 6 lðl þ 1Þ 1 3  2 ; C2 ¼  þ 2 : C1 ¼ 2 2 b b b b b b lðl þ 1Þ

ð2:10Þ

In order to put Eq. (2.9) into a standard differential equation, we introduce three parameters:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ED þ C 2 ;

2ED  C 1 k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ED þ C 2

a2 ¼ 4ðE  ED  C 0 Þ ð2:11Þ

and express a solution u(y) by the following form:

uðyÞ ¼ e

z=2 a=2

z

FðzÞ;

z ¼ 2gy ¼ 2ge

bðrr 0 Þ

:

ð2:12Þ

In terms of F(z), Eq. (2.9) is reduced to the hypergeometric equation: 2

z

d F 2

dz

þ ða þ 1  zÞ

nmax

#  " 2 k1 k  C 1 =2 1 ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ; 2 k2 þ C 2 2 

ð2:17Þ

2 2

where E ¼ E=ðb  h =2mÞ; ED ¼ ED =ðb  h =2mÞ, and

g¼

When the molecule vibrates beyond a critical quantum number nv, it begins to dissociate. The condition of dissociation can be determined from Eq. (2.15), which indicates that to ensure Rnv ;l ðrÞ ! 0 as z ? 0 (r ? 1), the value of a has to satisfy a P 0. Once a becomes negative, the wavefunction Rnv ;l ðrÞ does not converge to zero as r ? 1, and the bond length r may become infinite to yield dissociation. With a given by Eq. (2.14), we rewrite the condition a P 0 as nv 6 (k  1)/2. For a given bound energy ED , k2 of Morse potential, the maximum allowable quantum number nv is found as

dF 1 þ ðk  a  1ÞF ¼ 0: dz 2

ð2:13Þ

where [] denotes Gauss symbol. In a pure vibration motion with l = 0, the maximum vibration quantum number is given by nmax = [k  1/2] from Eq. (2.17). As the molecule starts to rotate with l – 0, the value of nmax decreases. In general, to ensure the existence of a bound state with rotation quantum number l and vibration quantum number nv, the bond energy of Morse potential ED ¼ k2 has to satisfy the condition

2nv þ 1 1 þ k> 4 4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 8lðl þ 1Þ 4 6 2 ; ð2nv þ 1Þ þ  2 b b2 b

nv ¼ 0; 1; 2; . . .

ð2:18Þ

The violation of condition (2.18) may lead to dissociation. It can be seen from Eq. (2.18) that large bond energy is required to hold a diatomic molecule rotating with large angular momentum l. Up to this stage, we have constructed the eigenfunction wnv lml ðr; h; /Þ associated with the eigen-energy Env ;l as

wnv lml ðr; h; /Þ ¼ Anv lml Rnv l ðrÞHlml ðhÞUml ð/Þ m

To ensure the existence of a bound solution to the above hypergeometric equation, the following condition must be imposed on the constants k and a:

ðk  a  1Þ=2 ¼ nv ;

nv ¼ 0; 1; 2; . . . :

ð2:14Þ

Under this condition, a solution to Eq. (2.13) can be expressed by the nth-order hypergeometric polynomial F(nv, a + 1; z). Then the radial wavefunction turns out to be

Rnv ;l ðrÞ ¼ uðzÞ=r ¼ r1 ez=2 za=2 Fðnv ; a þ 1; zÞ:

ð2:15Þ

The bound condition (2.14) also leads to the energy quantization of a diatomic molecule. Inserting the definition of a from Eq. (2.11) into Eq. (2.14) yields a2 ¼ ðk  2nv  1Þ2 ¼ 4ðE  ED  C 0 Þ from which we can solve for the rotation-dependent energy levels Env ;l as

Env ;l ¼ ED þ C 0 

ðED  C 1 =2Þ2 2ED  C 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnv þ 1=2Þ ED þ C 2 ED þ C 2

 ðnv þ 1=2Þ2 ;

ð2:16Þ

where we note that the three constants C0, C1 and C2 are functions of the angular quantum number l as defined in Eq. (2.10).

¼ Anv lml ez za=2 r1 Fðnv ; aþ1; zÞPl l ðcos hÞeiml / ; ð2:19Þ where Anv lml is a normalization factor. The threedimensional quantum dynamics for a diatomic molecule in the quantum state Wnv lml ðr; h; /Þ ¼ wnv lml ðr; h; /ÞeiEt=h is described by the equations of motion (2.3) with eigenfunction wnv lml ðr; h; /Þ given by Eq. (2.19)

dr 1 @ ln Wnv lml 1 1 d lnðr Rnv l ðrÞÞ; ¼ þ ¼ ds i ir i dr @r dh 1 @ ln Wnv lml 1 cot h ¼ þ ds ir2 i 2r 2 @h   1 d ln Hlml ðhÞ cot h ; ¼ 2 þ ir 2 dh @ ln Wnv lml d/ 1 ml ¼ ; ¼ ds ir 2 sin2 h @/ r 2 sin2 h

ð2:20aÞ

ð2:20bÞ ð2:20cÞ

where r ¼ br is the dimensionless bond length and s = t(hb2/m) is the dimensionless time. In a quantum state specified by the three quantum numbers nv, l and ml, the vibration of molecular bond length r(t) and molecular orientations h(t) and /(t) can be expressed as functions of time by solving Eq. (2.20). The force driving the molecule to vibrate and rotate is produced by the total potential VTotal = VMorse(r) +

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Q(w(r, h, /)), which is state-dependent and different state wnv lm is subjected to different total potential. Substitution of wnv lm into Eq. (2.4), we obtain the total potential in a dimensionless form as

V Total ¼

V Morse þ Q 2

 =2m b2 h

tion. The molecular bond length rðsÞ as a function of time can be solved analytically from Eq. (3.3) as

r ðsÞ ¼ b  ln

¼ V Morse þ Q

¼ req þ lnð1 þ Aeias=2 Þ:

  1 cot2 h @ 2 ln Rnm l ðr Þ  ¼ k2 ð1  erþb Þ2 þ 2 1 þ r 4 @r 2 2 2 @ ln Uml ð/Þ 1 @ ln Hlml ðhÞ 1  2  : ð2:21Þ r r 2 sin2 h @h2 @/2 Standard quantum mechanics makes use of the wavefunction Wnv lml ðr; h; /Þ to predict probability of finding the molecule’s appearance at a position (r, h, /). A position with a large value of W⁄W implies that there is a relatively high possibility for the molecule to reach there. But the probability interpretation cannot further explain why some place has high accessibility but some is hardly accessible. A more fundamental factor determining the molecule’s location is the total potential V Total . We will see in the next section that the position with maximum W⁄W is just the lowermost point of V Total , which presents the least resistance to quantum motion; the node with W⁄W = 0 is otherwise the position at which V Total approaches to infinity. 3. Rotation-dependent vibration in ground state The three-dimensional quantum dynamics derived in Eq. (2.20) are helpful to provide a dynamic description for the coupled vibration and rotation motions of a molecule. We consider vibration–rotation interaction in the ground state in this section. In the ground vibration state with nv = 0, the radial wavefunction Rnv ;l ðrÞ is given by

R0;l ðrÞ ¼ uðzÞ=r ¼ r 1 ez=2 za=2 ;

z 1 ¼ b  lnða=2gÞ  2g 1 þ Aeias=2

z ¼ 2gebðrr0 Þ :

ð3:1Þ

The dependence of R0,l(r) on angular quantum number l is reflected in the relation of a and g to l by noting

2k2  C

1 ffi1 a ¼ k  2nv  1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

k þ C2 4 2k2  lðl þ 1Þð4b  6Þ=b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1; 4 k2 þ lðl þ 1Þðb þ 3Þ=b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ ED þ C 2 ¼ k2 þ lðl þ 1Þðb þ 3Þ=b4 :

ð3:2aÞ ð3:2bÞ

Substituting R0,l(r) into Eq. (2.20a) yields the equation of motion for the rotation-dependent vibration in the ground state

  dz z2 d uðzÞ i ¼ zðz  aÞ: ln r  ¼ ds r 2 i dz

ð3:3Þ

The above radial dynamics has a stable equilibrium point at zeq = a, i.e., req ¼ b  lnðzeq =2gÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 4 4 2 k2 þ lðl þ 1Þðb þ 3Þ=b k lðlþ1Þð2b  3Þ=b 1 A:  ¼ b  ln @ 2 4 4 2 2 k þ lðl þ 1Þðb þ 3Þ=b k þ lðl þ 1Þðb þ 3Þ=b ð3:4Þ

The other equilibrium point of the radial dynamics (3.3) is at zeq = 0, i.e., r eq ¼ 1, corresponding to an unbound mo-

ð3:5Þ

The time-dependent term in r ðsÞ denotes the deviation of the vibration from the equilibrium point req . The trajectories of r on the complex plane are depicted in Fig. 3a for three angular quantum numbers l = 0, 3, 6 with molecular constants k = 2 and b = 3. As can be verified from the expression (3.4), the equilibrium bond length r eq increases monotonically with angular quantum number l. The time response of rðsÞ in Eq. (3.5) shows that the radial vibration is a periodic motion with a period given by

T¼

2p m 4p )T¼ 2 a=2 hb a 0

4

11

4pm B 2k2  lðl þ 1Þð4b  6Þ=b C ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1A : 4 hb2 k2 þ lðl þ 1Þðb þ 3Þ=b

ð3:6Þ

It is worth noting that this period of oscillation is independent of the actual trajectories and is quantized with respective to angular quantum number l. This trajectoryindependent property can be proved by the residue theorem

Z

I

dz 4p ¼ zðz  aÞ a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p 9 þ lðl þ 1Þ=16 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ 18  lðl þ 1Þ=8  9 þ lðl þ 1Þ=16

T¼

2 dt ¼ i

c

ð3:7Þ

where the contour c is an arbitrary closed trajectory solved from Eq. (3.3). The contour integral depends only on the pole enclosed by the contour. The residue evaluated at the stable equilibrium point zeq = a is 1/a, which then leads to the result of Eq. (3.7) with molecular constants given by k = 2 and b = 3. The time responses of the real part of rðsÞ are shown in Fig. 3b for three angular quantum numbers l = 0, 3, 6. It is observed that the numerically computed periods are identical to the theoretic predictions from Eq. (3.7). As angular quantum number l increases, both the molecular bond length and the period of vibration increase correspondingly. However, there is an upper bound on l, beyond which the period T becomes negative and the molecule tends to dissociate. To ensure a positive period T P 0 in Eq. (3.7), we come up with the constraint l 6 10. The equilibrium bond length r eq for l lower or equal to 10 is listed in Table 1. From Fig. 3c, we see that the peak of the radial probability density function coincides with the equilibrium position req for each angular quantum number l. This coincidence has its theoretic origin. We recall the definition of the radial probability density function

Pnv ;l ðr Þ ¼ 4pr 2 Rnv ;l ðrÞRnv ;l ðr Þ ¼ 4pðrRnv ;l ðrÞÞ2 ;

ð3:8Þ

by noting that Rnv ;l ðr Þ given by Eq. (2.15) is a real function of r . On the other hand, according to the dynamic Eq. (2.20a), the equilibrium position r eq satisfies the condition

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Fig. 3. The ground-state quantum trajectories for three angular quantum numbers l = 0, 3, 6 with b = 2, k = 3. The equilibrium positions in part (a), the peak positions of the probability density P nv ;l in part (c) and the lowermost points of the total potential V Total in part (d) are coincident. All these positions shift to right with increasing angular quantum number l, because the increasing centrifugal force separates the two atoms further. Part (b) shows that the period of oscillation is quantized with respect to the angular quantum number and independent of trajectories.

Table 1 Equilibrium bond lengths for the first three vibration states nv = 0, 1, 2. l/req/nv

0

1

2

3

4

5

6

7

8

9

10

11

0

ð1Þ r eq

2.18

2.21

2.28

2.38

2.51

2.69

2.91

3.21

3.62

4.30

6.78

⁄⁄⁄

1

r eq

2

ð1Þ

1.79

1.82

1.88

1.98

2.10

2.27

2.49

2.80

3.29

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

r eq

ð2Þ

3.30

3.36

3.48

3.67

3.96

4.38

5.10

7.20

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

ð1Þ r eq ð2Þ r eq ð3Þ r eq

1.69

1.73

1.79

1.89

2.04

2.24

2.53

3.08

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

2.85

2.91

3.02

3.22

3.53

4.07

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

5.04

5.21

5.62

6.56

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

⁄⁄⁄

dr 1 d ¼ lnðr Rnv l ðrÞÞ ¼ 0: ds i dr

ð3:9Þ

It appears that the equilibrium condition (3.9) is just the condition requiring the radial probability density function P nv ;l in Eq. (3.8) to have an extreme value. It is worth noting that the coincidence of the peak position of the probability density function with the equilibrium point is a quantum result different from the prediction of classical mechanics, wherein the turning

points of the dynamics, rather than the equilibrium points, correspond to peak probability. The internal mechanism underlying the observed probability distribution of w⁄w is governed by the total potential V Total defined in Eq. (2.21). In the ground state, the total potential is reduced to the following form

V Total ðr; hÞ ¼ k2 ð1  erþb Þ2 þ gerþb þ

cot2 h 1 @ ln Hlml ðhÞ  2 ; r 4r2 @h2 ð3:10Þ

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The variation of V Total ðr ; hÞ at heq = p/2 with respect to the bond length r is plotted in Fig. 3d. The comparison of Fig. 3a and c with Fig. 3d indicates that the position with maximum probability is actually the position where the total potential V Total ðzÞ achieves its minimum value. However, this quantum result is also different from its classical counterpart for which maximum probability is attained when the potential achieves its maximum value in the case of harmonic oscillation. All the properties obtained earlier from the dynamics Eq. (3.3) can be re-derived from the action of the radial total force f rTotal ðrÞ. As an illustration, we consider the case with l = 0 for which Hlml ðhÞ ¼ 1 and the total potential in Eq. (3.10) has a simple expression,

V Total ðr; heq Þ ¼ k2 ð1  erþb Þ2 þ gerþb ;

ð3:11Þ

where we note a = 2k  1 and g = k from Eq. (3.2) for the case of l = 0. The radial total force f rTotal ðr Þ now can be determined from V Total ðrÞ as

f r ðr Þ ¼  dV Total Total dr

  g ¼ 2k2 eðrbÞ 1  2  eðrbÞ : 2k

2k  1 a  ¼ b  ln ¼ r eq : 2k 2g

ð3:13Þ

This position is exactly the equilibrium position r eq already obtained in Eq. (3.4) by using the equilibrium condition dr =ds ¼ 0. The evaluation of f rTotal ðrÞ in the vicinity of req gives f rTotal ðr Þ < 0 if r > req , and f rTotal ðrÞ > 0 if r < r eq . The sign of f rTotal ðr Þ in the neighborhood of req indicates that the two atoms attract each other when their bond length is longer than r eq and repel each other when their bond length is shorter than r eq . Consequently, there is always a restoring force to make r return to its equilibrium position req . In conjunction with the restoring force f rTotal ðrÞ, a force constant K for the diatomic molecule can be defined. This force constant is generated by the total potential V Total ðrÞ, which is a combination of Morse potential VMorse and the quantum potential Q. A quantum force constant is defined as the second-order derivative of V Total ðrÞ evaluated at the equilibrium position r eq : 2

d V Total K¼ ðr eq Þ ¼ 2ðk  1=2Þ2 ) K dr 2 ¼

l

ðk  1=2Þ2 :

ð3:16Þ

which is monotonically decreasing over the entire range of r. To describe this dissociation process quantitatively, we have to express the bond length r as a function of time. Substituting k = 1/2 and a = 0 into Eq. (3.3), we have

dz i ¼ z2 ; ds 2

z ¼ erþb :

ð3:17Þ

Solving the above equation for rðsÞ by separating real and imaginary parts yields:

xI ðsÞ ¼ ð2n  1=2Þp;

s  0: ð3:18Þ

It is evident that irrespective of the initial separation r 0 , the distance r R between the two atoms increases monotonically and eventually leads to dissociation. In summary, we have discussed the rotation-dependent vibrational motion from four aspects: the dynamic equation of motion, the probability density function, the total potential and the total force. All the four aspects of consideration yield consistent result. 4. Rotation-dependent vibration in excited states In excited states, the quantum dynamics (2.20) has multiple equilibrium points, and the corresponding probability density function and total potential possess multiple-shell structure. The radial wavefunction in the first excited state with nv = 1 is given by

R1;l ðzÞ ¼ uðzÞ=r ¼ ez=2 za=2 r 1 ðz  a  1Þ

ð4:1Þ

and the radial equation of motion is obtained from Eq. (2.20a) as

dz i z3  ð2a þ 3Þz2 þ aða þ 1Þz ¼ : ds 2 za1

ð3:14Þ

Knowing the force constant, we now can determine the period of vibration by the usual definition

rffiffiffiffiffi m 4p m ¼ 2 T ¼ 2p : K h  b ð2k  1Þ

1 ð1 þ e2rþ2b Þ: 4

ð4:2Þ

There are two stable equilibrium points in the first excited state, which can be solved from the condition dz/ds = 0 as

2

h b4

V Total ðr; heq Þ ¼

r R ðsÞ¼b þ lnðs=2Þ; ð3:12Þ

The position free from radial force is found from the condition f r ðr Þ ¼ 0, which leads to

r ¼ b  ln

ular dissociation occurs at k ¼ ðED Þ1=2 ¼ 1=2. This critical value is confirmed by condition (2.18), which gives k > 1/ 2 to guarantee the existence of a bound state in the ground state nv = l = 0. The occurrence of dissociation can also be detected by the variation of total potential V Total ðrÞ. When k > 1=2; V Total ðrÞ has a shell structure with its lowermost point at the equilibrium bond length r eq , as shown in Fig. 3d. In case of k = 1/2, the shell structure of V Total ðrÞ disappears as can be seen from Eq. (3.11):

ð3:15Þ

This result is the same as Eq. (3.6) evaluated at l = 0. A diatomic molecule begins to dissociate, when its vibration motion becomes unstable and its period of vibration approaches to infinity. Eq. (3.15) indicates that molec-

ð2Þ zð1Þ eq ; zeq ¼

2a þ 3  2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8a þ 9 : 2

ð4:3Þ

By evaluating the constant a at nv = 1 with k = 2 and b = 3, we have

18  lðl þ 1Þ=8

a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3: 9 þ lðl þ 1Þ=16

ð4:4Þ

With this value of a, the two equilibrium bond lengths appear to be l-dependent

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Fig. 4. The reciprocal relation between the probability density function and the total potential in the first and second excited states. The nodes of the probability density function correspond to the locations at which the total potential approaches to infinity. On the other hand, peak positions of the probability density function correspond to the lowermost positions of the total potential, which are easiest to be accessed.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a þ 3  8a þ 9 r ð1;2Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ b  lnðz =2 g Þ ¼ b  ln eq eq 4 9 þ lðl þ 1Þ=16 ð1;2Þ

ð4:5Þ

The variation of r eq with respect to angular quantum number l is listed in Table 1. The maximum allowable value of l in the first excited state is smaller than that in the ground state. As can be seen in Table 1, ð2Þ the outer equilibrium bond length req disappears first as l greater than 7, and the inner equilibrium bond ð1Þ length req also disappears when l further increases beyond 8. The probability density function P 1;l ðrÞ of the first excited state is demonstrated in Fig. 4a for several values of l. As expected, there are two peaks in the curves of P1;l ðr Þ, which exactly coincide with the two equilibrium points r ð1;2Þ rÞ and eq . One of the major difference between P 1;l ð P 0;l ðr Þ is the existence of a node in P 1;l ðrÞ. To explain the occurrence of such a position with zero probability, we need the information regarding the distribution of the total potential in the first excited state:

Vðr ; hÞ ¼ ðk  z=2Þ2 þ þ

cot2 h 1 @ 2 ln Hlml ðhÞ  2 r 4r 2 @h2

z2 þ ð2a  2Þz þ a2 þ 4a þ 3 2ðz  a  1Þ2

;

ð4:6Þ

which is plotted in Fig. 4b. It is evident that the node just locates at the position where the total potential Vðr ; hÞ approaches to infinity. The appearance of an infinite potential barrier at the node accounts for the absolute inaccessibility of this point. This infinite potential barrier separates Vðr ; hÞ into two shells. The lowermost point in the inner shell corð1Þ responds to the inner equilibrium point req , while the lowermost point in the outer shell corresponds to the outer ð2Þ equilibrium point req . For the purpose of comparison, Morse potential VMorse is also shown in Fig. 4b. The comparison between VMorse and P 1;l ðr Þ reveals that without the participation of the quantum potential Q in the total potential VTotal = VMorse + Q, Morse potential alone cannot produce the multiple-shell structure observed in the probability density function P1;l ðrÞ as shown in Fig. 4a.

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(a) Evolution of total potential in the ground state

VTotal

l =6

l =7

l =8

l =9

l =10

l =5 l =4 l =3 l =2

l =1

VMorse

l =0

b =2

Re(r ) (b) Evolution of total potential in the first excited state

VTotal

inner shell

outer shell l =7

l =6

l =5 l =4 l =3 l =2 l =1 l =0

VMorse b =2

Re(r ) Fig. 5. Evolution of the total potentials due to change of molecular angular momentum l in the ground and the first excited vibration states. The molecular bonding strength gets weaker and the equilibrium bond length gets longer in a quantum state with larger angular momentum.

The analogy between total potential VTotal and probability density function Pnv ;l ðr Þ can be found in all quantum states. As a further verification of this point, we consider the radial wavefunction in the second excited state

R2;l ðzÞ ¼ ez=2 za=2 r1 ½z2  ð4 þ 2aÞz þ ða2 þ 3a þ 2Þ;

ð4:7Þ

from which the radial dynamics can be derived as dz 1 z4 þ ð8 þ 3aÞz3 þ ð3a2  11a  10Þz2 þ aða2 þ 3a þ 2Þz ¼ : ds 2i z2  ð4 þ 2aÞz þ ða2 þ 3a þ 2Þ ð4:8Þ

In the above nonlinear dynamics, three stable equilibrium ð1Þ ð2Þ ð3Þ points zeq ; zeq ; zeq can be solved from the following thirdorder polynomial equation

z3 þ ð8 þ 3aÞz2 þ ð 3a2 11a  10Þzþaða2 þ 3aþ 2Þ ¼ 0; ð4:9Þ with their numerical values listed in Table 1 for several values of the angular quantum number l. To ensure the existence of all the three equilibrium points, the maximum

value of l has to be lower than or equal to 3. As l > 3, the dissociation process begins from the outermost equilibð3Þ rium point zeq . The probability density function P 2;l ðrÞ and the total potential VTotal in the second excited state are shown, respectively, in Fig. 4c and d for l 6 3. The three peak positions of P 2;l ðr Þ and the three lowermost positions of VTotal ð1Þ ð2Þ ð3Þ coincide at the three equilibrium points zeq ; zeq ; zeq determined from Eq. (4.9). On the other hand, the two nodes in P2;l ðrÞ corresponds to the two locations with infinite potential barrier which divides the total potential VTotal into three shells as illustrated in Fig. 4d. To demonstrate how a large angular momentum destabilizes vibration motion and eventually leads to molecular dissociation, Fig. 5 shows the evolution of the total potentials by changing the molecular angular momentum in the ground and the first excited vibration states. A close inspection on the depth and location of the total potential VTotal indicates that the molecular bonding strength gets weaker and the equilibrium bond length gets longer in a quantum state with larger angular momentum. As l

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increases beyond 10 in the ground state or beyond 7 in the first excited state, the lowermost point of VTotal, i.e., the equilibrium bond length, approaches to infinity and gives a signature of molecular dissociation. The depth difference between the inner-shell potential and outer-shell potential in the first excited state in Fig. 5b helps to explain why a molecule vibrating in the outer-shell is loosely bound and easier to dissociate than that vibrating in the inner shell. 5. Orbital and spin angular dynamics So far our discussions on molecular quantum dynamics are restricted to the radial vibration motion governed by Eq. (2.20a). To take angular dynamics into account, all the three equations in Eq. (2.20) have to be considered. Of significance is that the molecular quantum dynamics governed by Eq. (2.20) not only describes orbital angular dynamics but also spin angular dynamics. Firstly, we wish to show how spin motion emerges naturally from Eq. (2.20) in the ground state where orbital angular momentum is zero. Setting nv = l = ml = 0 and substituting R00(z) = ez/2za/2/r, H00(h) = U0(/) = 1 into Eq. (2.20) yields the 3D quantum dynamics in the ground state:

dr i i ¼  ðz  aÞ ¼  ð2gerþb  aÞ; ds 2 2 dh i cot h ¼ 2 ; r ds 2

d/ ml ¼ ¼ 0; ds r2 sin2 h

ð5:1Þ

where a and g are given by Eq. (3.2) with l = 0. Since angular velocity vanishes in the / direction, angular motion in the h direction is the only possible source of spin. The integration of the h dynamics with h = hR + ihI gives the trajectories as shown in Fig. 6a. According to the behavior of the h dynamics, three spin regions can be defined, which correspond to three spin modes:  Spinless mode: within the region X0 , {(hR, hI)j jsinhhIj < jsinhRj}, hR(s) exhibits stable oscillation around the equilibrium points ±p/2. The sign of dhR/ds in this mode changes alternatively and the mean angular velocity is zero as illustrated in the middle part of Fig. 6b.  Spin-down mode: within the region X , {(hR, hI)j sinhhI P jsinhRj}, hR(s) is monotonically decreasing and the sign of dhR/ds is strictly negative, having a negative mean angular velocity.  Spin-up mode: within the region X+ , {(hR, hI)jsinhhI 6 jsinhRj}, hR(s) is monotonically increasing and the sign of dhR/ds is strictly positive, having a positive mean angular velocity. Although orbital angular momentum is completely depressed by the condition l = ml = 0, non-zero angular motion does exist in the spin-up and spin-down regions. To relate this remnant angular motion to spin, the next step is to identify the magnitude of this remnant angular momentum. In the regions of X and X+, we note that cot(h) = cot(hR + ihi) is a periodic function of hR and its mean value over a period of hR can be computed as hcothi = htanhi = i, h 2 X; hcothi = i, h 2 X+, and hcothi =

0, h 2 X0. Taking mean value of both sides of the h dynamics in Eq. (5.1), we obtain the expected result:



8 > h=2; 8hðsÞ 2 X  < dh dh 2 2  mr ¼ h r ¼ 0; 8hðsÞ 2 X0 : dt l¼ml ¼0 ds l¼ml ¼0 > : h=2; 8hðsÞ 2 Xþ ð5:2Þ

In the ground state where orbital angular motion vanishes, the remnant angular motion in the h direction emerges as the spin dynamics. By the language of nonlinear dynamics, we could say that spin is just the ‘‘zero dynamics’’ of orbital motion. Fig. 6b gives a numerical verification of Eq. (5.2) in a dimensionless form. Adopting molecular constants k = 2 and b = 3 in the computation, we find the average value of the remnant angular velocity from Eq. (5.2) as

hdh=dsi ¼  ¼

1 1 1 1 ¼ 2 r 2eq 2 ðb  lnða=2gÞÞ2 1=2 ð2  lnð5=6ÞÞ2

0:105;

ð5:3Þ

which is confirmed in the spin-up and spin-down responses in Fig. 6b. The quantum motion described by Eq. (5.1) occurs on a vertical plane of / = constant. By choosing this plane as the y-z plane, it turns out that spin is a h-rotation motion around the x axis, as compared to the orbital motion that is a /-rotation motion around the z axis. Spinless trajectories on the y-z plane are shown in Fig. 6c, where we can see that the spinless motion oscillates around the equilibrium points h = ±p/2. The direction of the spinless rotation changes alternatively and results in zero mean angular velocity. Shown in Fig. 6d are spin trajectories, whose rotation direction is fixed, being counterclockwise for spin-up motion and clockwise for spin-down motion. The distinction between spinless and spin motions is caused by the special structure of the total potential in the ground state,

V Total ðr ; hÞ ¼ k2 ð1  erþb Þ2 þ gerþb þ

cot2 h ; 4r 2

ð5:4Þ

whose distributions in the spinless region X0 and spin region X± are illustrated, respectively, in Fig. 6e and f. Total potential in the spinless region has two isolated layers around the h direction. A trajectory across the two layers is prohibited due to the infinite potential barrier present between them. On the contrary, the total potential in the spin region has one complete layer around the h direction, which allows the trajectories to encircle the central peak of the potential clockwise or counterclockwise, giving rise to spin-down or spin-up motion. In the ground state nv = l = ml = 0, we have shown that the molecular angular motion is solely contributed by spin dynamics. Once l – 0, spin dynamics and orbital dynamics are both present in the angular motion. To expound the interaction between the two angular dynamics, we consider the excited rotation state with nv = 0,l = 1 and ml = 0, whose wavefunction is given by R01(r) = r1ez/2za/2, H10(h) = cosh and U0(/) = 1. The related equations of motion are derived from Eq. (2.20) as

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(a) θ R (τ ) vs. θ I (τ ) in three spin modes θI

(b) dθ R / dτ vs. τ in three spin modes dθ R dτ

Ω

Ω0

spin-up : Ω +

spinless : Ω 0 θR

τ

spin-down : Ω

Ω+

(d) Trajectories in spin modes

(c) Trajectories in spinless modes z

θ

spin-down : Ω -

z

θ

y

y

z spin-up : Ω +

(f) Total potential in spin modes

(e) Total potential in spinless mode

VzT otal

VT otal

y

y

z

z

Fig. 6. The comparisons of spin and spinless modes in the ground state from three aspects. Part (a) classifies the three spin modes on the complex h plane. Part (b) compares the time responses of dhR(s)/ds for the three modes. Part (c) and (d) compare the vibration–rotation motion on the real y-z plane (/ = constant) for the spinless and spin modes. Part (e) and (f) compare the shell structure for the spinless and spin modes.

dr i ¼  ð2gerþb  aÞ; ds 2   dh i cot h d/ ml ; ¼  2  tan h þ ¼ ¼ 0; r ds 2 ds r 2 sin2 h

ð5:5Þ

where a and g are given by Eq. (3.2) with l = 1. The h(t) trajectories on the complex hR  hI plane are illustrated in Fig. 7a. Like the situation discussed in the ground state,

 three modes can be identified: spinless mode X0 X00 , spin-up mode (X+) and spin-down mode (X). Especially, there are two regions corresponding to the spinless mode. The X0 region contains closed h(t) trajectories that enclose only one equilibrium point of the h(t) dynamics at

pffiffiffiffiffiffiffiffi heq ¼ kp  cos1 2=3, while closed trajectories in the X00 region enclose two equilibrium points, as shown in Fig. 7a. Evaluating the mean values of tanh and coth over one period of oscillation, we find the average angular momentum in the three modes as following



8 > 3h=2; 8h 2 X  < dh dh mr2 ¼ h r 2 ¼ 0; 8h 2 X0 ; X00 : dt l¼ml ¼0 ds l¼ml ¼0 > : 3h=2; 8h 2 Xþ ð5:6Þ

Because the azimuth / dynamics around z axis is prohibited by the condition ml = 0, the only way to display the

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(b) Total potential in spinless mode Ω 0

(a) θ R (τ ) vs. θ I (τ ) Ω-

θI Ω 0¢

Ω0

Ω0

cθ1

Ω 0¢

Ω 0¢ Ω0

Ω0

cθ1

cθ1

Ω+ θR

(c) Total potential in spinless mode Ω0

(d) Total potential in spin modes Ω ± spin-up : Ω

spin-down : Ω

Fig. 7. Spin and spinless motions in the excited state with nv = 0, l = 1 and ml = 0. Part (a) classifies four regions on the complex h plane: X0 ; X00 are spinless modes; X+, X are spin modes. Total potentials for the four modes are shown in part (b), (c) and (d). The corresponding spin and spinless trajectories are plotted over the surfaces of the total potentials to reveal the influence of the shell structure on the molecular vibration and rotation.

orbital angular motion specified by l = 1 is through h _ dynamics. In comparison with hmr 2 hi l¼ml ¼0 in Eq. (5.2), 2_ the h angular momentum hmr hil¼1;ml ¼0 in Eq. (5.6) contains an additional component ⁄, which, as expected, is contributed from the orbital angular motion l = 1. To see how the trajectories move up and down under the influence of the total potential V Total , in Fig. 7b–d, the trajectories solved from Eq. (5.5) are plotted over the surface of the total potential

V Total ¼ V Morse þ Q ¼ k2 ð1  erþ2 Þ2 þ gerþb þ

1 ðcot2 h þ 4 sec2 hÞ: 4r 2 ð5:7Þ

Total potential in the spinless X0 mode has a quadraticshell structure with each shell containing pffiffiffiffiffiffiffiffione of the four 1 equilibrium points at h ¼  cos 2=3 and heq ¼ p eq pffiffiffiffiffiffiffiffi cos1 2=3. The trajectories solved from Eq. (5.5) with initial conditions belonging to X0 fall on one of the four isolated shells and oscillate therein around their associated

equilibrium point. Comparatively, total potential in the spinless X00 mode has a double-shell structure with each shell containing two equilibrium points, as shown in Fig. 7c. The h dynamics in X0 and X00 regions are all periodic with zero mean angular velocity. On the other hand, the h dynamics in X± regions are monotonically increasing or decreasing, because the total potential in X± regions has one complete layer around the h direction, as shown in Fig. 7d, which allows the trajectories to encircle the central potential peak, clockwise or counterclockwise, and produces the observed spin-down and spin-up motions. Angular momentum in other excited states can be analyzed in a similar way to yield the following general results:

8 > < ðl þ 1=2Þh; h 2 X _ ¼ 0; hmr 2 hi h 2 X0 ; > : ðl þ 1=2Þh; h 2 Xþ 8 h 2 X > < 0; 2 hmr 2 /_ sin hi ¼ ml h  ; h 2 X0 : > : 0; h 2 Xþ

ð5:8Þ

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In general, the angular momentum Ph ¼ mr 2 h_ derived from Eq. (2.3b) contains both spin and orbit motions:

Ph ¼ mr2 h_ ¼

 1 @ W h cot h h ¼ Lh þ S; þ i W @h i 2

ð5:9Þ

where Lh is the h-component orbital angular momentum, and S = (i h=2) coth is the complex representation of the (local) spin angular momentum. It can be seen that spin _ being independent of S = (i h=2) coth is intrinsic to mr2 h, the wavefunction W. Mean value of the local spin can be calculated by

hSi ¼

1 2p

I

Sdh ¼

ch

 h 4pi

I

cot hdh:

ð5:10Þ

ch

Noting that the residue of coth at each pole h = np, n 2 Z, is equal to 1, we have from the residue theory

hSi ¼

h h ð2pinh Þ ¼ ns ; 4pi 2

ns ¼ 0; 1; 2; 3; . . . ;

ð5:11Þ

where ns is the number of poles of coth enclosed by the contour ch. It is worth noting that upon arriving at the result (5.11), we do not specify the type of particles, neither the type of the applied potential. The spin quantization rule (5.11) says that the mean value of the spin angular momentum is only allowed to be an integer multiple of h  =2. Eq. (5.11) provides us with a geometrical method to identify spin of a given particle by inspecting its h trajectory ch and counting the number of points np, n 2 Z within it. 6. Conclusions This paper reveals the internal dynamics embedded in a molecular quantum state. There are external and internal representations for a quantum state W. External representation is commonly adopted in standard quantum mechanics by exploiting probability density function W⁄W to measure accessibility of each spatial point. In this representation, physical quantities are real-valued and have no memory about their past motions. On the other hand, in quantum Hamilton mechanics, physical quantities are complex-valued and have internal representation that provides time histories for physical quantities. The internal dynamics of a molecule is governed by deterministic Hamilton equations of motion, which controls the internal mechanism underlying the externally observed random events. The use of quantum Hamilton mechanics allows us to apply all the analytical methods developed in nonlinear dynamics to investigate molecular quantum dynamics. We conclude with a brief summary of what has been achieved here by applying quantum Hamilton mechanics to describe the coupled rotational and vibration quantum motion in diatomic molecules. 1. For a given molecular wavefunction Wnv lml ðr; h; /Þ, we show that there exists an accompanying total potential VTotal, which completely determines the vibration and rotation quantum motion of a diatomic molecule. It is revealed that a node of W⁄W just locates at the position where the total potential approaches to infinity. The appearance of an infinite potential barrier at the node

accounts for the absolute inaccessibility of this point. On the other hand, the position with maximum W⁄W is identified as the position where the total potential VTotal achieves its minimum value. 2. We discuss the molecular quantum motion from four aspects: the dynamic equations of motion, the probability density function W⁄W, the total potential VTotal and the total force fTotal = @VTotal/@r. All the four aspects of consideration are found to yield consistent results 3. The probability density function W⁄W is not the only information that can be extracted from a molecular wavefunction W. The Hamilton equations of motion determined from W give the period T(l) and force constant K(l) of the bond-length vibration as explicit functions of the orbital angular quantum number l, which otherwise are not manifested in the probability density function W⁄W. 4. One of the additional information that can be extracted from W is the molecular spin dynamics. It is a common belief that a wavefunction W solved from Schrödinger equation can only describe spinless motion. However, our study shows that Hamilton equations of motion determined from W reveal the coexistence of spin-up, spin-down and spinless motions in W.

References [1] Kratzer A. Die ültraroten Rotationsspektren der Halogenwasserstoffe. Zeit f Physik 1920;3:289–307. [2] Dunham JL. The energy levels of a rotating vibrator. Phys Rev 1932;41:721–31. [3] Pöschl G, Teller E. Bemerkungen zur Quantenmechanik des anharmonischen oszillators. Zeits f Phys 1933;83:143–51. [4] Pekeris CL. The rotation-vibration coupling in diatomic molecules. Phys Rev 1934;45:98–103. [5] Yang CD. Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom. Ann Phys 2006;321:2876–926. [6] Yang CD. Complex mechanics. Progress in nonlinear science, vol. 1. Hong Kong: Asian Academic Publisher; 2010. [7] Styer DF et al. Nine formulations of quantum mechanics. Am J Phys 2002;70:288–97. [8] Yang CD. A new hydrodynamic formulation of complex-valued quantum mechanics. Chaos Solitons Fract 2009;42:453–68. [9] John MV. Modified De Broglie-Bohm approach to quantum mechanics. Found Phys Lett 2002;15:329–43. [10] Bohm D. A suggested interpretation of the quantum theory in terms of hidden variables I, II. Phys Rev 1952;85:166–93. [11] Bohm D, Hiley BJ. The undivided universe. New York: Routledge; 1993. [12] Holland PR. The quantum theory of motion. Cambridge: Cambridge University Press; 1993. [13] Chou CC, Sanz AS, Miret-Artes S, Wyatt RE. Hydrodynamic view of wave-packet interference: quantum caves. Phys Rev Lett 2009;102. 250401.1–4. [14] Wyatt RE, Rowland BA. Computational investigation of wave packet scattering in the complex plane: propagation on a grid. J Chem Theor Comput 2009;5:443–51. [15] Chou CC, Wyatt RE. Quantum trajectories in complex space: onedimensional stationary scattering. J Chem Phys 2008;128. 154106.1–10. [16] David JK, Wyatt RE. Barrier scattering with complex-valued quantum trajectories: taxonomy and analysis of isochrones. J Chem Phys 2008;128. 094102.1–9. [17] Chou CC, Wyatt RE. Quantum vortices within the complex quantum Hamilton–Jacobi formalism. J Chem Phys 2008;128. 234106.1–10.

C.-D. Yang, H.-J. Weng / Chaos, Solitons & Fractals 45 (2012) 402–415 [18] Chou CC, Wyatt RE. Quantum streamlines within the complex quantum Hamilton– Jacobi formalism. J Chem Phys 2008;129. 124113.1–12. [19] Goldfarb Y, Schiff J, Tannor DJ. Complex trajectory method in timedependent WKB. J Chem Phys 2008;128. 164114.1–21. [20] Goldfarb Y, Degani I, Tannor DJ. Semiclassical approximation with zero velocity trajectories. J Chem Phys 2007;338:106–12. ^ ¼ i [21] Yang CD. The origin and proof of quantization axiom p ! p hr in complex spacetime. Chaos Solitons Fract 2007;32:274–83. [22] Yang CD. Wave-particle duality in complex space. Ann Phys 2005;319:444–70. [23] Yang CD. Stability and quantization of complex-valued nonlinear quantum systems. Chaos Solitons Fract 2009;42:711–23. [24] Yang CD. Complex tunneling dynamics. Chaos Solitons Fract 2007;32:312–45.

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[25] Yang CD, Chang TY. Quantum index theory: relations between quantum phase and quantum number. Internat J Nonlinear Sci Numer Simul 2009;10:907–25. [26] Yang CD. Complex dynamics in diatomic molecules Part I: Fine structure of internuclear potential. Chaos Solitons Fract 2008;37:962–76. [27] Yang CD. Complex dynamics in diatomic molecules. Part II: Quantum trajectories. Chaos Solitons Fract 2008;38:16–35. [28] Yang CD. Quantum dynamics of hydrogen atom in complex space. Ann Phys 2005;319:399–443. [29] Yang CD. Solving quantum trajectories in Coulomb potential by quantum Hamilton–Jacobi Theory. Int J Quantum Chem 2006;106:1620–39. [30] Jhung KS, Kim IH, Hahn KB, Oh KH. Scaling properties of diatomic potentials. Phys Rev A 1989;40:7409–12.