





Nonlinear Dynamics Governing Quantum Transition
Behavior CiannDong Yang^{*}, ShiangYi Han, and
FeiBin Hsiao


  

Abstract In 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 statetransition 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 wave^{1}, shell structure distribution in a hydrogen
atom^{2}, tunneling effect and multipath
integration^{3,4}, electron spin dynamics^{5,11,12},
electron’s scattering trajectory^{6}, the origin of quantum
operators^{7,8}, quantum motion in complex
space^{9,10}, quantum chaos^{13} 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 HamiltonJacobi (HJ) 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/(2 ω)
, t
≥ π/(2 ω)
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 eigentrajectory 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=(2r/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 eigentrajectories 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
literature^{14,15,16,17}.
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
 C. D. Yang, “Waveparticle duality in complex space ”, Ann of
Phys., (319), 444470, 2005.
 C. D. Yang, “Quantum dynamics of hydrogen atom in complex
space”, Ann of Phys., (319), 399443, 2005.
 C. D. Yang, “Complex tunneling dynamics”, Chaos, Solitons
& Fractals, (32), 312345, 2007.
 C. D. Yang, C. H. Wei, “Parameterization of all path integral
trajectories”, Chaos, Solitons & Fractals, (33),
118134, 2007.
 C. D. Yang, “On modeling and visualizing singleelectron spin
motion”, Chaos, Solitons & Fractals, (30), 4150, 2006.
 C. D. Yang, “Solving quantum trajectories in Coulomb potential
by quantum HamiltonJacobi theory”, Int. J. of Quantum Chemistry,
(106), 16201639, 2005.
 C. D. Yang, “Quantum Hamilton mechanics: Hamilton equations of
quantum motion, origin of quantum operators, and proof of
quantization axiom”, Ann of Phys., (321), 28762926, 2006.
 C. D. Yang, “The origin and proof of quantization axiom
p→=iħ∇ in complex spacetime”, Chaos, Solitons and
Fractals, (32), 274283, 2007.
 C. D. Yang, C. D., “Modeling Quantum Harmonic Oscillator in
Complex Domain”, Chaos, Solitons, & Fractals, (30), 342
– 362, 2006.
 C. D. Yang, “Quantum Motion in Complex Space”, Chaos, Solitons
& Fractals, (33), 1073 1092, 2007.
 C. D. Yang, “Complex spin and antispin dynamics: A
generalization of de BroglieBohm theory to complex space”, Chaos,
Solitons & Fractals, (38), in press, 2008.
 C.D. Yang, “Spin: The Nonlinear Zero Dynamics of Orbital
Motion”, Chaos, Solitons and Fractals, (37), 1158, 2008.
 C.D. Yang, and Wei, C. H., “Strong Chaos in 1D Quantum
System”, Chaos, Solitons & Fractals, (37), 988, 2008.
 N. Hatano, “NonHermitian delocalization and eigenfunctions”,
Phys. Rev. B (58), 8384 – 8390, 1998.
 C. M. Bender, J. H. Chen, D. W. Darg, and K. A. Milton,
“Classical Trajectories for Complex Hamiltonians”, J. Phys. A:
Math. Gen (39) 42194238, 2006.
 S. S. Ranjani, A. K. Kapoor, and P. K. Panigrahi, “Quantum
HamiltonJacobi analysis of PT symmetric Hamiltonians”, Int. J.
Mod. Phys. A (20), No. 17, 4067  4077, 2005.
 A. Mostafazadeh, “Real Description of Classical Hamiltonian
Dynamics Generated by a Complex Potential”, Phys. Lett. A
(357), 177180, 2006.


  






