Wave-Particle Duality and Quantum-Classical Analogy

Abstract

Mathematical analogy between the systems of weakly coupled oscillators and multi-level quantum system allows considering them in the framework of a unified approach. In particular, an asymptotic interpretation of wave-particle duality provides an efficient analytical tool for solving both linear and nonlinear non-stationary dynamical problems of classical and quantum mechanics.

Appendices

Appendix A

In accordance with the procedure of the multi-scale expansions we suppose (see, e.g., [25])

$$ \begin{aligned} & \psi_{j} (\tau_{0} ,\tau_{1} , \ldots ) = \psi_{j,0} (\tau_{0} ,\tau_{1} ,) + \varepsilon \psi_{j,1} ( \tau_{0} ,\tau_{1} , \ldots ) + O(\varepsilon^{2} ),\;j = 1,2. \\ & \tilde{\tau } = \tau_{0} ;\;\tau_{1} = \varepsilon \tau_{0} , \ldots \\ & \frac{d}{{d\tilde{\tau }}} = \frac{\partial }{{\partial \tau_{0} }} + \varepsilon \frac{\partial }{{\partial \tau_{1} }} + O(\varepsilon^{2} ) \\ \end{aligned} $$

$$ \begin{array}{*{20}c} {O(1):\frac{{\partial \psi_{j,0} }}{{\partial \tau_{0} }} - i\psi_{j,0} = 0 \to \psi_{j,0} = f_{j,0} (\tau_{1} , \ldots )e^{{i\tau_{0} }} ,\,j = 1,2} \\ \begin{aligned} O(\varepsilon ):\frac{{\partial \psi_{1,1} }}{{\partial \tau_{0} }} - i\psi_{j,1} + \frac{{\partial f_{1,0} }}{{\partial \tau_{1} }}e^{{i\tau_{0} }} - i\beta [(f_{1,0} e^{{i\tau_{0} }} - f_{{_{1,0} }}^{*} e^{{ - i\tau_{0} }} ) - \\ (f_{2,0} e^{{i\tau_{0} }} - f_{2,0}^{*} e^{{ - i\tau_{0} }} )] = 0; \\ \frac{{\partial \psi_{2,1} }}{{\partial \tau_{0} }} - i(\psi_{2,1} + \gamma \,f_{2,0} e^{{i\tau_{0} }} - \gamma \,f_{2,0}^{*} e^{{ - i\tau_{0} }} ) + \frac{{\partial f_{2,0} }}{{\partial \tau_{1} }}e^{{i\tau_{0} }} \\ - i\beta [(f_{2,0} e^{{i\tau_{0} }} - f_{2,0}^{*} e^{{ - i\tau_{0} }} ) - (f_{1,0} e^{{i\tau_{0} }} - f_{1,0}^{*} e^{{ - i\tau_{0} }} )] = 0; \\ \end{aligned} \\ \end{array} $$

(29)

The condition of the absence of the secular terms leads to the equations of the principal asymptotic approximation:

$$ \left\{ {\begin{array}{*{20}c} { - i\frac{{df_{1,0} }}{{d\tau_{1} }} + \beta (f_{1,0} - f_{2,0} ) = 0} \\ { - i\frac{{df_{2,0} }}{{d\tau_{1} }} + \beta [(1 - \gamma /\beta )f_{2,0} - f_{1,0} ] = 0} \\ \end{array} } \right. $$

(30)

Exclusion of the secular terms is equivalent to averaging over the fast time.

Appendix B

Non-smooth variables were used in connection with the elaboration of the method of non-smooth transformation for analysis of vibro-impact or close to them systems [31, 46, 47, 62]. Far going extension of this method was presented in [46]. It turned out unexpectedly [21, 22] that functions $\tau (\tau_{1} ),\;e(\tau_{1} )$ describe adequately the intensive energy exchange in the systems, very different from the vibro-impact ones by both their physical content and original mathematical formulation. They describe the beats by the most simple and obvious manner due to the choice of the appropriate non-smooth variables $\tau (\tau_{1} ),\;e(\tau_{1} )$.

The form) of the solution corresponding to LPT leads naturally to introducing the saw-tooth time. After such transformation the solution is described by the smooth functions of the non-smooth time (see Appendix D).

Appendix C

The canonical transformation of the variables for the periodic FPU chain has a view [50]:

$$ \begin{array}{*{20}l} {Q_{j} = \sum\limits_{k = 0}^{N - 1} {\sigma_{j,k} } \zeta_{k} } \hfill \\ {\sigma_{j,k} = \left\{ {\begin{array}{*{20}c} {\frac{1}{\sqrt N },} & {k = 0} \\ {\sqrt {\frac{2}{N}} \sin \left( {\frac{2\pi kj}{N} + \gamma } \right),} & {k = 1, \ldots \left[ {\frac{N - 1}{2}} \right]} \\ {\frac{{( - 1)^{j} }}{\sqrt N },} & {k = \frac{N}{2}} \\ {\sqrt {\frac{2}{N}} \cos \left( {\frac{2\pi kj}{N} - \gamma } \right),} & {k = \frac{N}{2} + 1, \ldots ,N - 1} \\ \end{array} } \right.} \hfill \\ {\gamma = \frac{\pi }{4}} \hfill \\ {\sigma_{j,k} = \frac{1}{\sqrt N }\left[ {\sin \left( {\frac{2\pi kj}{N}} \right) + \cos \left( {\frac{2\pi kj}{N}} \right)} \right]} \hfill \\ \end{array} $$

(31)

The quadratic and quartic constituents of the Hamilton function in the normal coordinates of the linearized system are written as follows

$$ H_{2} = \sum\limits_{k = 1}^{N - 1} {\frac{1}{2}(\eta_{k}^{2} + \omega_{k}^{2} \xi_{k}^{2} )} $$

$$ H_{4} = \frac{\beta }{8N}\sum\limits_{k,l,m,n = 1}^{N - 1} {\omega_{k} \omega_{l} \omega_{m} \omega_{n} C_{k,l,m,n} \xi_{k} \xi_{l} \xi_{m} \xi_{n} } $$

$$ C_{k,l,m,n} = - \Delta_{k + l + m + n} + \Delta_{k + l - m - n} + \Delta_{k - l + m - n} + \Delta_{k - l - m + n} $$

(32)

$$ \Delta_{r} = \left\{ {\begin{array}{*{20}l} {( - 1)^{r} ,\;if} \hfill & {r = mN,\;m \in {\rm Z}} \hfill \\ 0 \hfill & {otherwise.} \hfill \\ \end{array} } \right. $$

Appendix D

The advantages of the techniques based on the use of the non-smooth variables are evident while dealing with the nonlinear beats (see Appendix B). They can not be presented as linear combination of NNM because the superposition principle is not valid in this case. It was shown earlier [21,22,23,24, 26, 30, 50], that an efficient temporal description of LPT in nonlinear chains is attained in the terms of the non-smooth functions of slow time $\tau (\tau_{2} ),\;e(\tau_{2} )$. They are plotted in Fig. 2 where $\tau (\tau_{1} )$ has to be changed to $\tau (\tau_{2} )$ and $e(\tau_{1} )$ to $e(\tau_{2} )$.Then the dependent variables can be presented as [47]

$$ \theta = X_{1} (\tau ) + Y_{1} (\tau )e\left( {\frac{{\tau_{2} }}{a}} \right),\quad \Delta = X_{2} (\tau ) + Y_{2} (\tau )e\left( {\frac{{\tau_{2} }}{a}} \right) $$

(33)

After substitution into the equations of motion of the effective particles one obtains the equations with respect to smooth functions of non-smooth variables $X_{i} (\tau ),\;Y_{i} (\tau );\;i = 1,2$.

The possibility of similar substitutions is based on the statement that every periodic process, independently on the class of its smoothness, is expressed by the unique manner as an element of the algebra of hyperbolic numbers through the variables $\tau$ и e [62]:

$$ x(\tau_{2} ) = x(\tau ,e) = X(\tau ) + e\,Y(\tau ),\;\,e\left( {\tau_{2} } \right) = \frac{d\tau }{{d\tau_{2} }}. $$

(34)

where $X(\tau ) = \frac{1}{2}\left[ {x(\tau ) + x(2 - \tau )} \right],\quad \,Y(\tau ) = \frac{1}{2}\left[ {x(\tau ) - x(2 - \tau )} \right]$ so, that $x\,(\tau ,e) \equiv x(\tau ) + e\,Y\left( {\tau \,(t),e\,(t)} \right)$.

At that, the pair $\{ 1,e\}$, where $e^{2} = 1$ is a basis, and the algebraic operations as well as differentiation or integration over time preserve the structure of hyperbolic number. This property provides applicability and convenience of the corresponding transformations while solving the differential equations [47].

Interestingly that the hyperbolic numbers which are frequently used for a simplest illustration of the Clifford algebra, were known from the middle of XIX century as abstract mathematical objects without any connection with vibration processes. On the other side, the elliptic complex numbers with the basis {1, i} (i² = −1) and corresponding trigonometric functions turned out, in essence, the main tool for the description of such processes.

The analytical presentation of the solution for the FPU chain in the terms of power series over slow time (the periodicity of the process is taken into account by introducing the independent variable τ) can be written as follows:

$$ X_{i} = \sum\limits_{l = 0}^{\infty } {X_{j,l} } \tau^{l} ,\;Y_{i} = \sum\limits_{l = 0}^{\infty } {Y_{j,l} } \tau^{l} , $$

(35)

where $j = 1,2$, and the plot corresponding to expressions (35) is presented in Fig. 10.

Fig. 10

Time behavior of the symmetric FPU chain corresponding to LPT in coordinates $\theta ,\Delta$

Close results can be obtained for the asymmetric FPU potential [58]. The extension to the case of the chain interacting with an elastic foundation is also possible. Then a minimum frequency in the spectrum of the linearized system differs from zero [28]. The coherence domains in this case are formed, contrary to the FPU chain, in the low part of the spectrum (closely to the frequency gap) if one deals with a soft nonlinearity (similarly to the Frenkel-Kontorova model). With such change all the results presented above are valid. They can be applied to the analysis of the nonlinear dynamics of crystalline oligomers in the vicinity of the optic branch of the dispersion curve. This allows one to clarify the most efficient mechanisms of the energy exchange and the transition to energy localization in such systems. Let us note that the localization of the vibrations in polyatomic molecules and their role in the relaxation processes are considered in monograph [45] (see also citations there).

Appendix E

We consider the two-level approximation corresponding to the Gross-Pitaevsky equation (GP) [2, 52] as a starting point. The GP equation describes a series of the quantum processes, e.g., the Bose-Einstein condensation (BEC), in the framework of self-consistent field approach that is a source of nonlinearity

$$ i\hbar \frac{{\partial\Psi \left( {r,t} \right)}}{\partial t} = - \frac{{\hbar^{2} }}{2m}\nabla^{2}\Psi \left( {r,t} \right) + \left[ {V_{trap} \left( r \right) + g_{0} \left| {\Psi \left( {r,t} \right)} \right|^{2} } \right]\Psi \left( {r,t} \right). $$

(36)

Then the time evolution of the wave function of GP Eq. (36) can be presented as a superposition of the wave functions corresponding to two natural basic states between which a tunneling is possible:

$$ \Psi \left( {r,t} \right) = a_{1} \left( t \right)\Phi _{1} \left( r \right) + a_{2} \left( t \right)\Phi _{2} \left( r \right) $$

(37)

The functions $\Phi _{1,2} \left( r \right)$ in (37), depending on the space coordinate, are expressed through symmetric $\Phi _{ + } \left( r \right)$ и and anti-symmetric $\Phi _{ - } \left( r \right)$ stationary GP states as follows [52]:

$$ \begin{aligned} &\Phi _{1} \left( r \right) = \frac{{\Phi _{ + } +\Phi _{ - } }}{2} \\ &\Phi _{2} \left( r \right) = \frac{{\Phi _{ + } -\Phi _{ - } }}{2} \\ \end{aligned} $$

(38)

We take into account that $\int {\left| {\Phi _{1,2} \left( r \right)} \right|}^{2} dr = 1$ and $\int {\Phi _{1} \left( r \right)\Phi _{2} \left( r \right)} dr = 0$.

Then, with taking into account (38), we obtain the discrete equations:

$$ \begin{aligned} i\hbar \frac{{\partial a_{1} }}{\partial t} = (E_{1}^{0} + U_{1} \left| {a_{1} } \right|^{2} )a_{1} - \kappa a_{2} , \hfill \\ i\hbar \frac{{\partial a_{2} }}{\partial t} = (E_{2}^{0} + U_{1} \left| {a_{2} } \right|^{2} )a_{2} - \kappa a_{1} \hfill \\ \end{aligned} $$

(39)

This system is similar to that for two weakly interacting classical nonlinear oscillators with cubic nonlinearity, in slow time. Its analysis is performed in [52], and in terms of LPTs in [22, 27]. The detailed development of the LPT concept in application to nonlinear problems is presented in [23, 24, 26, 30].

If one of the parameters of the quantum system (39) depends linearly on the time, the similar procedure leads to the equations which describe a particular case of the nonlinear LZT:

$$ \begin{aligned} & i\dot{a}_{ + } = \varepsilon ta_{ + } +\Omega a_{ - } + \gamma \left| {a_{ + } } \right|^{2} a_{ + } , \\ & i\dot{a}_{ - } =\Omega a_{ + } + \gamma \left| {a_{ - } } \right|^{2} a_{ - } \\ \end{aligned} $$

(40)

The limiting magnitudes of the tunneling probabilities (when $t \to \infty$) for system (40) were calculated in [19], and full description of the process is presented in [13]. Here the mathematical analogy is clearly seen with the classical system of weakly coupled nonlinear oscillators the linear stiffness of one of which changes in time. It is possible to note that derivation of both quantum and corresponding classical equations includes the averaging procedure. In quantum case such averaging is a stage of the self-consistence field procedure. In the classical system it is that over the fast time in the method of multiple scale expansions.