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Volume 5 Issue 5 - August 15, 2008  |
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Nonlinear Dynamics Governing Quantum Transition
Behavior Ciann-Dong Yang*, Shiang-Yi Han, and Fei-Bin Hsiao Department of Aeronautics and Astronautics,
National Cheng Kung University *Email: cdyang@mail.ncku.edu.tw International Journal of Nonlinear Science and Numerical Simulation: 8 (3), 411-416, 2007。SCI Category: Mathematics Applied, Rank 1/150 |
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 AbstractIn quantum mechanics, the only known quantitative index describing a state transition process is the occupation probability. In this paper, we propose three new quantitative indices, namely, complex transition trajectory, quantum potential, and total quantum energy, to give an elaborate description of state-transition behavior, which otherwise cannot be manifested in terms of occupation probability. The new quantitative indices and the nonlinear dynamics governing quantum transition behavior are derived from the quantum Hamilton mechanics that provides a unified description of the entire transition process from an initial state to a final state via an entangled state. The proposed nonlinear analysis of transition behavior will be applied to the transition in a hydrogen atom from the 1s state to the 2s state. Introduction of complex mechanics In conventional quantum mechanics (QM), a particle’s position and momentum in a quantum system can only be obtained in terms of an expectation value through a quantum operator. To reflect this situation, one can imagine that there is a bee in a house with no window. Suppose that the bee has a special power allowing it to pass through walls to go outside. Of course, we cannot see the bee while it stays outside the house. It can only be seen after passing through the wall and coming back into the house again. Hence, what we observe is that the bee appears all of a sudden and disappears later if it passes in and out of the house. In such a condition, we have no idea about when it will come back to the house for the reason that we cannot see anything outside the house, and what we can do is to estimate the probability of being stung by the bee according to the position we are inside the house. This is the probability interpretation proposed by QM to describe a quantum system. Let us replace all the walls of the house by glass, and then there will be no doubt that we can see everything outside the house as we stay inside. Now, we still can observe where the bee is and even how it moves after it is passing through the glass wall to go outside. There is no problem for us to predict its flight path, position, velocity and heading. In other words, we can be told when and where the bee comes back into the house in a deterministic way, without estimating the stinging probability. In fact, the actual world we live in is a space like the one inside the house, which is a real space, a classical world, while there is an imaginary space for the area outside the house. Only small particles like electrons and atoms can go through the wall to enter the imaginary space. We can say that the quantum Hamilton mechanics (QHM) plays the role of the glass wall, extending the space dimension from real to complex. Through this dimension extension, every skill, method and equation we have used in classical mechanics (CM) can be used equally in the quantum world. As a result, one can obtain the whole dynamic information of a quantum particle, such as its trajectory, momentum, action function, potential and total energy, without using operators and probabilities. With the help of the glass wall provided by QHM, it is the first time that the quantum world can be realized in a deterministic way, and be described from a causal aspect. In the last few years, the first author of this paper has developed QHM and used it to explain and describe many mysterious quantum phenomena, such as the formation of matter wave1, shell structure distribution in a hydrogen atom2, tunneling effect and multi-path integration3,4, electron spin dynamics5,11,12, electron’s scattering trajectory6, the origin of quantum operators7,8, quantum motion in complex space9,10, quantum chaos13 and so on. The essence of complex mechanics can be grasped via the schrödinger equation:  (1)
Due to the presence of the imaginary number, solutions of the Schrödinger equation are complex functions. However, scientists have always regarded the imaginary sign in (1) as a mathematical tool, which contains no physical meaning. Accordingly, they adopted the probability interpretation to get a real expectation value through ，ψ*ψ. On the other hand, QHM obtained the quantum Hamilton-Jacobi (H-J) equation similar to the classical one by rewriting Eq.(1) via the transformation ψ=exp(iS/ħ) :  (2)
where in the brackets one sees the quantum Hamiltonian H,  (3)
with the quantum potential  defined
as (4)
The quantum Hamiltonian becomes the classical one if we neglect the quantum potential in Eq.(3). The quantum Hamiltonian is a complex function due to the participation of the quantum potential . In CM,
the Hamiltonian Hc
depends on the external potential V(q) only; but in
QHM, the state ψ where the particle stays also plays a role
in the quantum Hamiltonian H(ψ). For a given
wavefunction ψ(q), one can obtain the momentum
p=∇S from Eq.(2): (5)
As we expect, it is a complex momentum. According to the quantum Hamiltonian H in Eq.(3), one can derive the quantum Hamilton equations of motion as  (6a)  (6b)
where the position q and the momentum p are two independent complex variables. It is totally the same manner of expressing and solving equations of motion as was done in CM, except that p and q are defined in complex space. If we combine Eq.(6a) with (6b), then we obtain the quantum Newton’s second law:  (7)
Therefore, we can solve quantum mechanical problems by the methods developed in classical mechanics. Analysis and Discussions In this research, we focus on a particle’s transitional behavior from one state to another, and propose a classical interpretation for it through QHM. Unlike QM, which can only provide probabilistic description to quantum transition, QHM gives more specific and precise dynamic indices to analyze the transition behavior. Regarding the phenomenon of quantum state transition, one can imagine that there are two houses built on a hillside, one located at a higher altitude than the other. And there is again a bee which plays the particle’s role, which can fly out of the house by passing through the wall. It flies toward the higher house from the lower one, for both houses represent two eigenstates. When the bee is flying between the two houses, we say it is in an entangled state. As far as we know, if the two houses are constructed by QM, then they are opaque. In such circumstances, one is totally blind about how and where the bee flies outside and when it will arrive at the other house. On the other hand, the houses are transparent if they are constructed by QHM, via which the bee’s flight path can then be observed and its arrival time can be predicted, no matter which house the observer stays in. In the following, we exploit electron transition in the hydrogen atom as a demonstration of QHM. Consider a hydrogen electron transitioning from a ground state (1s) to the first excited state (2s). The wavefunctions used to describe this transition are given by  , t
≤ 0 , 0 <
t < π/(2ω) , t
≥ π/(2ω)
Fig.1 Four different approaches to describing the
transition from the 1s state to the 2s state of the
hydrogen atom. Part (a) describes the transition by occupation
probability; part (b) describes the transition in terms of the
quantum trajectory connecting the 1s eigen-trajectory via the
transient trajectory to the 2s eigen- trajectory; part (c)
describes the transition by the evolution of the total potential,
and part (d) describes the transition according to the time response
of the total energy. where ψ100=e-r/a0 and ψ200=(2-r/a0)e-r/a0 represent the wavefunctions for 1s and 2s states, respectively. According to Eq.(6a), Eq.(3) and Eq.(4), the equations of motion, the total potential and the total energy can be used as quantitative indices to describe the transition process. Fig.1a depicts the time histories of the probability densities for the electron in the two states, wherein we have normalized the time to be  . The blue
line denotes the probability density  with  for the
electron in the 1s state, which is decreasing from unit in the
transition process, and becomes 0 as the electron reaches the
2s state. The red line denotes the probability density  in the
2s state, which increases from 0 to 1 during the transition
process. Fig.1b depicts the electron’s trajectory obtained by
integrating the equations of motion (6a) in the spherical
coordinates: ,  ,  (8)
where ρ=r/a0 is the normalized radial distance with a0 being the Bohr radius. It is obvious that the transition trajectory in the time interval 0 < t < π/(2ω) connects the eigen-trajectories smoothly, giving a continuous measure for the transition process. The other index describing the transition process is the total potential  as shown in Fig.1c, where we see that the total potential transfers from the 1s state, which contains only one shell, to the 2s state, which has two shells. The total energy of the electron is given by  , which
transfers from the value 
in the 1s state to the value 
in the 2s state, as shown in Fig.1d. It is worthy to notice
that the total energy of the electron is of complex value during the
transition process and only its real part in shown in Fig.1d. A
related issue that a complex Hamiltonian may have real energy
spectrum has been studied widely in the
literature14,15,16,17.Conclusions By extending the space dimension from real to complex, we can describe the quantum world by classical mechanics. In this research, we show that the entire quantum motions before, during, and after the transition process can be described in a unified way so that quantum trajectory smoothly connecting an initial state to a final state via an entangled state can be solved from the same set of nonlinear Hamilton equations of motion. We point out that apart from the occupation probability afforded by a given transitional wavefunction Ψ(t,q), exact nonlinear dynamics governing the transition process can also be extracted from Ψ(t,q), which provides us with the detailed force interaction within the entangled state, with the internal mechanism underlying the state transition process, and with several new quantitative indices measuring the progression of the transition process, such as complex transition trajectory, Hamilton quantum potential, and total energy. References
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=iħ∇ in complex spacetime”, Chaos, Solitons and
Fractals, (32), 274-283, 2007.
