Abstract Studying possible laws, rules, and mechanisms of time-evolution of quantum wavefunctions leads to deeper understanding about the essential nature of the Schrödinger dynamics and interpretation on what the quantum wavefunctions are. As such, we attempt to clarify the mechanical and geometrical processes of deformation and bifurcation of a Gaussian wavepacket of the Maslov type from the viewpoint of length-scale hierarchy in the wavepacket size relative to the range of relevant potential functions. Following the well-known semiclassical view that (1) Newtonian mechanics gives a phase space geometry, which is to be projected onto configuration space to determine the basic amplitude of a wavefunction (the primitive semiclassical mechanics), our study proceeds as follows. (2) The quantum diffusion arising from the quantum kinematics makes the Gaussian exponent complex-valued, which consequently broadens the Gaussian amplitude and brings about a specific quantum phase. (3) The wavepacket is naturally led to bifurcation (branching), when the packet size gets comparable with or larger than the potential range. (4) Coupling between the bifurcation and quantum diffusion induces the Huygens-principle like wave dynamics. (5) All these four processes are collectively put into a path integral form. We discuss some theoretical consequences from the above analyses, such as (i) a contrast between the δ-function-like divergence of a wavefunctions at focal points and the mesoscopic finite-speed shrink of a Gaussian packet without instantaneous collapse, (ii) the mechanism of release of the zero-point energy to external dynamics and that of tunneling, (iii) relation between the resultant stochastic quantum paths and wave dynamics, and so on.
Abstract Studying possible laws, rules, and mechanisms of time-evolution of quantum wavefunctions leads to deeper understanding about the essential nature of the Schrödinger dynamics and interpretation on what the quantum wavefunctions are. As such, we attempt to clarify the mechanical and geometrical processes of deformation and bifurcation of a Gaussian wavepacket of the Maslov type from the viewpoint of length-scale hierarchy in the wavepacket size relative to the range of relevant potential functions. Following the well-known semiclassical view that (1) Newtonian mechanics gives a phase space geometry, which is to be projected onto configuration space to determine the basic amplitude of a wavefunction (the primitive semiclassical mechanics), our study proceeds as follows. (2) The quantum diffusion arising from the quantum kinematics makes the Gaussian exponent complex-valued, which consequently broadens the Gaussian amplitude and brings about a specific quantum phase. (3) The wavepacket is naturally led to bifurcation (branching), when the packet size gets comparable with or larger than the potential range. (4) Coupling between the bifurcation and quantum diffusion induces the Huygens-principle like wave dynamics. (5) All these four processes are collectively put into a path integral form. We discuss some theoretical consequences from the above analyses, such as (i) a contrast between the δ-function-like divergence of a wavefunctions at focal points and the mesoscopic finite-speed shrink of a Gaussian packet without instantaneous collapse, (ii) the mechanism of release of the zero-point energy to external dynamics and that of tunneling, (iii) relation between the resultant stochastic quantum paths and wave dynamics, and so on.
We analyze the Schrödinger dynamics and the Schrödinger function (or the so-called wavefunction) in the following four aspects. (1) The Schrödinger equation is reconstructed from scratch in the real field only, without referring to Newtonian mechanics nor optics. Only the very simple conditions such as the space-time translational symmetry and the conservation of flux and energy are imposed on the factorization of the density distribution in configuration space, giving rise to a two-dimensional real vector. On returning to the original Schrödinger equation, the imaginary number arises naturally. (2) Like the Langevin equation in a Brownian dynamics, we formulate a single-event path dynamics in quantum mechanics, contrasting with the Schrödinger distribution function. The path thus attained is referred to as one-world path, which represents, for instance, a path of a singly launched electron in the double-slit experiment that leaves a spot at the measurement board, while many of accumulated spots give rise to the fringe pattern. We start from the Feynman-Kac formula to draw a relation between a stochastic dynamics and the parabolic differential equations, to one of which the Schrödinger equation is transformed. (3) To highlight the roles of the flux and energy conservation in the Schrödinger dynamics, we build that the quantum Maupertuis-Hamilton principle, which reveals the symplectic structure in the parameter space. (4) We track how the inherent quantum nature like the Huygens-principle-like properties is built. We show that classical trajectory components in semiclassics are demanded to branch into many coherent pieces beyond the semiclassical regime and dissolve into the deep dynamics of genuine full quantum dynamics.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.
A detailed framework for a new formalism of phase-space path integral is presented, the outline of the theory of which has been reported elsewhere [K. Takatsuka, Phys. Rev. Lett. 61, 503 (1988)]. This path integral is described in terms of a phase-space distribution function, which was proposed earlier by the present author under the name of the dynamical characteristic function (DCF). The DCF is characterized by two independent wave functions and with two phase spaces. In the present paper, we extend the DCF by utilizing the Feynman kernel in place of the two independent wave functions. This is called the identity DCF. The dynamics of the DCF for general wave packets is reduced to that of the identity DCF. Some characteristics of the identity DCF are presented. For example, the quantum-mechanical time-evolution operator is represented in terms of the identity DCF and of a complete set of quantum q (coordinate) and p (momentum) operators. A semiclassical theory for the identity DCF is also developed, its primary aim being the study of heavy-particle dynamics such as molecular collisions.It is shown that the propagation of the identity DCF is represented by classical trajectories, each of which is associated with an amplitude factor and with the action integral as the quantum phase factor. This amplitude factor is free from singularity, unlike the primitive semiclassical expression of the Feynman kernel, which is well known to diverge at caustics. The physical significance of the identity DCF is analyzed in the context of chaotic dynamics. In particular, it is shown that its amplitude factor is an explicit function of the eigenvalues of a Jacobian matrix [\ensuremath{\partial}(qf,pf)/\ensuremath{\partial}(qi,pi)], where (qi,p) and (qf,pf) are the initial and final points, respectively, of a classical trajectory. These eigenvalues are well known in the study of the classical chaos of Hamilton systems and have been used to construct a measure of the linear (and local) stability of an orbit, the Liapounov characteristic exponents, and so on. Further, if the identity DCF is applied to a periodic orbit, the amplitude factor becomes equivalent to the square root of the determinant of a matrix, which is the so-called monodromy matrix minus the unit matrix. This matrix is one of the standard quantities used in bifurcation theory to investigate the stability of fixed points. In fact, the amplitude factor is closely related to the so-called residue, which was proposed by Greene [J. Math. Phys. 20, 1183 (1979)] in order to predict the occurrence of global chaos in classical systems. It is emphasized that all these classical quantities arise naturally in our representation of quantum mechanics.
We study a quantum theory in terms of action decomposed function (ADF), a class of quantum wave function, towards many-dimensional applications to quantum dynamics of heavy particles as in chemical reactions. The equation of motion for the complex-valued amplitude of ADF represents a coupling between the internal diffusive motion of a wave packet and dynamics of its group velocity in a hierarchical manner ascending from classical to purely quantum mechanics via semiclassical dynamics. We attempt to solve this equation of motion dividing it into two stages: a semiclassical level and beyond. In this paper, as the first stage, we develop a semiclassical approximation in the Lagrange picture of classical phase flow. In the Euler picture (as in the standard WKB picture), continuous integration of the stability matrix along the paths is required. By adopting the Lagrange picture, on the other hand, we represent the semiclassical amplitude in terms of what we call deviation determinant, which can be evaluated readily in many-dimensional systems. Numerical tests show that ADF reproduces quantum wave packets at each space-time point along classical paths very well. However, the ADF in this stage is not free of the semiclassical singularity. In other words, the wave functions diverge at turning points or caustics, depending on the initial conditions chosen. This divergence is known to take place at points where classical paths smoothly distributed in phase space have ``focuses'' in configuration space (or momentum space) and reflects an intrinsic relationship between quantum and classical mechanics. Therefore, it is by studying the mechanism of removing the singularity that the essential feature of quantum mechanics will be clarified. This aspect will be discussed in a companion paper [K. Takatsuka and S. Takahashi, Phys. Rev. A 89, 012109 (2014)] as the second stage of many-body quantum theory.
I compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. I concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. I find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which I show to be the result of a simple beating phenomenon between paths in the semiclassical propagator. When the interaction of the classical manifold with the hyperbolic fixed point leads to escape from the remnant separatrix, representing the onset of classical chaos, the interference associated with the tendril leads to a long-lived accumulation of quantum amplitude on top of the cosine barrier. This effect also has a semiclassical interpretation, meaning that it arises from the interference between classical paths, but one must expand to second order in Planck's constant to describe the behavior correctly. Both the nodal structure and this pinning of amplitude on the barrier are dynamic mechanisms for the quantum inhibition of mixing. I then couple the system to a bath of harmonic oscillators in order to study the effect of the environment on these mechanisms. Although an oscillator bath environment brings dissipation as well as noise to the problem, the noise effect dominates for the high temperature, weak coupling regime I study. When there is sufficient noise to render the interfering classical paths indistinguishable, I find that the quantum interference gets erased. This dephasing occurs at very early times, long before there is appreciable dissipation of energy to the environment. Consequently, one can argue that the presence of an environment, even if its effect would be negligible in a nonchaotic setting, allows for the possibility of quantum mixing.
The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0) is expressed in a didactic form. The expression found is a product of pure real factors and pure phase factors. This makes it very easy to derive the expression for the probability density from the wavefunction. The physical meaning of each of the factors is analysed. Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von jedem Faktor wird analysiert. While writing a paper [1] about the time evolution of different wavepackets I wanted to find a didactic expression for the ψ(x,t) wavefunction of a Gaussian initial state. The particular expression that was finally constructed is different from those found in quantum mechanics texts [2 ,3]. Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c) show the time development of the real part of factor 1 , factor 2 and of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d) shows the time development of the probability density ρ(x,t). The x, y scale is the same for all time instants. Atomic units are used. a = 2.5 Bohr = 0.13 nm, nm. The atomic time unit is 2.41 × 10 -17 s. See the text for details. Figure 2. t-dependent parts of the wavefunction. (a) shows the time dependent prefactor of factor 2 as function of time. (b) and (c) show the time dependence of the terms of factor 3 . Their real (full curve) and imaginary (broken curves) parts are plotted against time. The thin dashed horizontal lines in (b) show the asymptotes for t= ∞ . Atomic units are used. See the text for details. Initial state Our initial state is a simple Gaussian wavepacket of the form This wavepacket is a product of three factors: A normalization factor that makes the norm of the wavefunction unity. A plane wave factor that accounts for the non-zero momentum p 0 of the wavepacket. A bell-shaped localizing function with half width at half maximum . Time evolution The time development of the initial ψ 0 (x) state is given by [ 2 ]: This Fourier integral can be calculated easily with Gaussian integrals and leads to a wavefunction like Transformation of into didactic form Now we want to transform this into something more informative. First note that the centre of the wavepacket is moving with the group velocity . Hence it is worth writing instead of x 0 into the first term of the numerator in the exponential. Working this out gives the following result It is getting clearer already! Now let us get rid of the complex denominators! Utilizing this we get finally where arg z is the phase of the complex number z, i.e. . Our ψ(x,t) has three main factors ((7), (8) and (9)). The first factor (7) is a product of two pure real coefficients and a plane wave. This plane wave part of factor 1 and the entire second (8) and third (9) factors are pure phase factors , i.e. their magnitude is one. Hence it is very easy to calculate the probability density ; one has only to calculate the square of the pure real coefficients of factor 1 which gives: The three terms of ψ(x,t) are as follows. Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having the same form as ψ 0 (x) but the centre of gravity of the Gaussian is moving with speed and its width is increased to . The maximum value of the Gaussian is decreasing as its width increases making the area under ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is shown in figure 1(a). Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in x. One can see from figure 1(b) that this factor oscillates faster for larger |x| values. This accounts for the fact that the higher momentum components of the initial Gaussian ψ 0 (x) move with higher velocities. The function which describes the time-dependent prefactor of the phase is . This function (cf figure 2(a)) is not monotonic in time. Its value is zero for t = 0 and t= ∞ and has a maximum at . Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor. This phase factor is a product of two terms. The first term is a monotonic function of time while the second one is oscillating. The phase of the first term is zero for t = 0 (a(t) is pure real) and -π/4 for t= ∞ (a(t) is pure imaginary). The second term is where and it accounts for the time development of the plane wave component in factor 1 . These two phase factors are plotted in figures 2(b) and 2(c) against time. Acknowledgment This work was partially supported by the Hungarian OTKA grant No F 014236. References [1] Márk G I Influence of the wavepacket shape to its time development to be published [2] Cohen-Tannoudji C, Diu B and Laloë F 1977 Quantum Mechanics (New York: Wiley) [3] Merzbacher E 1970 Quantum Mechanics 2nd edn (New York: Wiley)