The corresponding effects of features in the potential on classical, semiclassical, and quantum mechanics are probed using the technique of functional sensitivity analysis. It is shown that the classical and quantum functional sensitivities are equivalent in the classical (small ℏ) and harmonic limits. Classical and quantum mechanics are known to react in qualitatively similar ways provided that features on the potential are smooth on the length scale of oscillations in the quantum wave function. By using functional sensitivity analysis, we are able to show in detail how the classical and quantum dynamics differ in the way that they sense the potential. Two examples are given, the first of which is the harmonic oscillator. This problem is well understood by other means but is useful to examine because it illustrates the detailed information about the interaction of the potential and the dynamics which can be provided by functional sensitivity analysis, simplifying the analysis of more complex systems. The second example is the collinear H+H2 reaction. In that case there are a number of detailed and striking differences between the ways that classical and quantum mechanics react to features on the potential. For features which are broad compared to oscillations in the wave function, the two react in qualitatively the same way. The sensitivities are oscillatory, however, and there are phasing differences between the classical and quantum sensitivity functions. This means that using classical mechanics plus experimental data in an inversion scheme intended to find the ‘‘true’’ potential will necessarily introduce sizeable errors.
The corresponding effects of features in the potential on classical, semiclassical, and quantum mechanics are probed using the technique of functional sensitivity analysis. It is shown that the classical and quantum functional sensitivities are equivalent in the classical (small ℏ) and harmonic limits. Classical and quantum mechanics are known to react in qualitatively similar ways provided that features on the potential are smooth on the length scale of oscillations in the quantum wave function. By using functional sensitivity analysis, we are able to show in detail how the classical and quantum dynamics differ in the way that they sense the potential. Two examples are given, the first of which is the harmonic oscillator. This problem is well understood by other means but is useful to examine because it illustrates the detailed information about the interaction of the potential and the dynamics which can be provided by functional sensitivity analysis, simplifying the analysis of more complex systems. The second example is the collinear H+H2 reaction. In that case there are a number of detailed and striking differences between the ways that classical and quantum mechanics react to features on the potential. For features which are broad compared to oscillations in the wave function, the two react in qualitatively the same way. The sensitivities are oscillatory, however, and there are phasing differences between the classical and quantum sensitivity functions. This means that using classical mechanics plus experimental data in an inversion scheme intended to find the ‘‘true’’ potential will necessarily introduce sizeable errors.
The paper considers the sensitivity of quantum dynamics systems with respect to parameters or features in the underlying Hamiltonians. The analysis allows for the calculation of the gradient of the wave function with respect to any well defined system parameters in the Hamiltonian. Differential sensitivity equations are derived from the Schrödinger equation and their solution is expressed in terms of a Green’s function. For scattering problems the necessary asymptotic Green’s function can be exclusively expressed in terms of the available solution to the original Schrödinger equation. Also considered are problems from time-dependent quantum mechanics. In this latter situation sensitivity theory allows for the calculation of gradients of the time evolution operator with respect to system parameters. The scattering formalism is illustrated with a study of the anisotropy coefficient sensitivity in the collision of an atom and a ridgid rotor. The time-dependent theory was applied to a collision-free molecular beam translating through a region of space containing electric and magnetic fields. This case examined the sensitivity of the molecular transition probabilities to field parameters. These brief examples serve to illustrate the potential usefulness of sensitivity analysis in quantum dynamics.
There are significant differences in the structures of course presentations of classical mechanics and quantum mechanics, but also essential links that connect the two approaches. A large part of the differences in approach do not involve quantization, but rather the traditional use of instantaneous values to describe macroscopic objects and position probability densities to describe microscopic objects. A pedagogic reformulation of the classical problem to bridge this gap is presented here. The method is numerical in nature, but for illustrative purposes it is herein applied to two specific cases for which analytic solutions also exist, the Kepler–Coulomb potential and the isotropic harmonic oscillator. Using this approach, classical systems can be examined in the presence of perturbations in the same way as is used in quantum mechanics courses. Semiclassical quantization can also be introduced to extend the connection to a quantum-mechanical treatment.
There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy. Shakespeare , Hamlet, Act 1 Scene 5 The challenge of semiclassical mechanics: Reassessing the views Semiclassical mechanics significantly broadens our understanding of the relationship between classical and quantum mechanics. It reveals that there is much more to the relation between these theories than simply showing that classical behavior can be recovered from quantum systems in the limit of large actions ( ħ → 0) or large quantum numbers ( n → ∞). Although these limits are an important piece of the puzzle, the usual reductionist account of the quantum–classical relation in terms of these limits alone does not tell the whole story. More specifically, research in semiclassical mechanics has uncovered a number of surprising theoretical and experimental correspondences . Indeed the program of semiclassical research can be broadly understood as the theoretical and experimental task of uncovering these correspondences and extending them in new ways. On the theoretical side, semiclassical methods such as EBK quantization, Gutzwiller's trace formula, and closed orbit theory show how classical structures can be used to construct quantum quantities. These semiclassical methods have proven to be not just useful calculational tools, but also important investigative tools, in that they are often able to provide physical insight into the structure of the quantum dynamics, and hence lead to new discoveries.
A major difficulty in comparing quantum and classical behavior resides in the struc- tural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator equations only formally identical to the classical equations of motion. By taking the expectation of these equations, the well-known Ehrenfest theorem provides identities which, however, are not a closed system of equations which allows to evaluate the time evolution of the system. The formalism of the effective action seems to offer a possibility of comparing quantum and classical evolutions in a system- atic and logically consistent way by naturally providing approximation schemes for the expectations of the coordinates which at the zeroth order coincide with the classical evolution [1]. The effective action formalism leads to equations of motion which differ from the classical equations by the addition of terms nonlocal in the time variable. This means that for these equations an initial value problem is not meaningful and they have to be interpreted in an appropriate way. Here we analyze situations in which the nonlocal terms can be reasonably approximated by local ones, so that the quantum corrections do not modify the locality of classical equations. In the simplest approximation, the effective Lagrangian differs from the corresponding classical one by a renormalization of both the potential- and the kinetic-energy terms. We shall not discuss the causal formalism used, for example, in refs. [2-4], as in the approximation considered this would lead to the same local equations. The present contribution describes the beginning of a systematic study of semiclassical evolutions using the effective action formalism. In the first part, after introducing the formalism of the effective action and its expansion in powers of hbar (loop expansion) in the context of quantum mechanics, we concentrate on the structure of the first-order corrections in li. These corrections are evaluated to the second order in the derivative expansion [5], by two different methods. The first is based on a Euclidean approach [6], the second one on an adiabatic approximation in evaluating functional determinants. In the second part of the article we put the formalism at work, choosing as our case study a two-dimensional (2D) anharmonic oscillator of the kind considered in molecular physics. The results of the simulations show that by increasing lithe effective dynamics tends to regularize the classical motion and becomes qualitatively very similar to the quantum evolution provided the energy is sufficiently small. The evaluation of the effective dynamics in more general cases will be presented in a forthcoming paper.
By using the formulation of quantum mechanics with classical pictures, the motion of a quantum is analysed for linear harmonic oscillator. On this occasion it is postulated that the quantum mechanical momentum of the quantum is approximately equal to the corresponding classical mechanical momentum in the region of validity of the coordinates. According to this analysis it is shown that the quantum mechanical motion well corresponds to the classical motion, and that the wave equation determines the associated wave along the whole orbit of the quantum in the configuration space and the uniqueness of the wave throughout the orbit plays an important role in quantum mechanics. In addition to the above, an attempt is made to make clear the well-known fact that classical mechanics with the quantization gives approximately the same results as quantum mechanics.
A major goal of the semiclassical theory is to understand and treat a quantum system in the regime of large quantum numbers using information from its classical limit. In this framework, quantum phenomena are addressed through a path integral perspective, i.e. the weighted interference of phases accumulated along classical trajectories. While this formalism has been successful in explaining interference phenomena, for instance, in mesoscopic single-particle quantum systems, the recent extension of the semiclassical theory allows now also the treatment of interacting quantum many-body systems. This thesis aims to contribute to these ongoing efforts to apply semiclassical techniques to quantum many-body systems. It is focused on bosonic quantum many-body systems, where the thermodynamic limit of a large number of particles represents an alternative version of a semiclassical limit. Within the derivation of the corresponding semiclassical theory, mean-field equations, i.e. effective nonlinear wave equations describing the matter wave, re-emerge as Hamilton's equations of motion of an abstract Hamilton function, defining the above thermodynamic limit as the classical limit. Within this thesis, two different topics are studied. The first one deals with the so-called “Out-of-Time-Order Correlator” (OTOC), the expectation value of the squared commutator of two local operators at different times. The OTOC provides a direct probe for the presence of chaos in the classical limit (if any) of a quantum many-body system. This is based on that for short times the OTOC directly relates to the stability of classical solutions upon changes in their initial conditions. One of the defining properties of chaos is that this stability displays an exponential growth, with the rate called the classical Lyapunov exponent. Immediately, this implies an exponential growth of the OTOC for short times, from which the Lyapunov exponent can be extracted. For later times, one observes a saturation of the OTOC as a consequence of unitary quantum time evolution. The Ehrenfest time, the time scale which marks the onset of quantum interference, separates these different dynamical behaviors. In this thesis a thorough understanding of the concept of chaos in the classical limit of quantum many-body systems is provided, and with that the underlying interference mechanisms involved in the early exponential growth and the later saturation of OTOCs are identified. It is found that the pre-Ehrenfest time exponential growth is given through the interference of multiple contributions, all of them essentially following only a single solution of mean-field equations. Conversely the post-Ehrenfest time behavior stems from the contributions of fundamentally different mean-field solutions, which display correlated dynamics only for a limited amount of time, of the order of Ehrenfest time. The second topic covered in this thesis deals with the coherent transport of cold bosonic atoms through an Aharonov-Bohm ring structure. This setup consists of two semi-infinite wave guides attached on the opposite sides of a wave guide ring which is pierced by a synthetic gauge field. Within the ring, the atoms are further subject to both a weak disorder potential and particle-particle interaction. In the non-interacting case, the disorder-averaged transmission probability as a function of the encircled flux displays the well-known Al'tshuler-Aronov-Spivak (AAS) oscillations, i.e. oscillations with a frequency twice as large as Aharonov-Bohm oscillations. This thesis provides further insight into the influence of a weak particle-particle interaction on AAS oscillations, for which numerical results predict an interaction-based inversion of peaks. Here, using semiclassical techniques to solve the corresponding mean-field problem, this inversion is successfully reproduced, and a first indicator of the relevant mechanism leading to the inversion is found through contributions of self-averaging scattering path constellations.