Classical mechanics is empirically successful because the probabilistic mean values of quantum mechanical observables follow the classical equations of motion to a good approximation (Messiah 1970, 215). We examine this claim for the one-dimensional motion of a particle in a box, and extend the idea by deriving a special case of the ideal gas law in terms of the mean value of a generalized force used to define “pressure.” The examples illustrate the importance of probabilistic averaging as a method of abstracting away from the messy details of microphenomena, not only in physics, but in other sciences as well.
Classical mechanics is empirically successful because the probabilistic mean values of quantum mechanical observables follow the classical equations of motion to a good approximation (Messiah 1970, 215). We examine this claim for the one-dimensional motion of a particle in a box, and extend the idea by deriving a special case of the ideal gas law in terms of the mean value of a generalized force used to define “pressure.” The examples illustrate the importance of probabilistic averaging as a method of abstracting away from the messy details of microphenomena, not only in physics, but in other sciences as well.
Textbooks in quantum mechanics frequently claim that quantum mechanics explains the success of classical mechanics because the mean values [of quantum mechanical observables] follow the classical equations of motion to a good approximation, while the dimensions of the wave packet be small with respect to the characteristic dimensions of the problem. The equations in question are Ehrenfest's famous equations. We examine this case for the one-dimensional motion of a particle in a box, and extend the idea deriving a special case of the ideal gas law in terms of the mean value of a generalized force, which has been used in statistical mechanics to define pressure. The example may be an important test case for recent philosophical theories about the relationship between micro-theories and macro-theories in science.
Every vertex of the magic cube of physics discussed so far introduced a major element in the development of physics and the whole of science. Mechanics introduced the basic concept of equations of motion, or time-evolution of a physical system given certain initial condition and the associated idea of determinism, even with its generalization in the form of chaos, the dependence of the outcomes of the temporal evolution on the initial conditions. These concepts taken for many particles, or elements of a physical system, led to the probabilistic description of many particle systems, in the realm of statistical physics and thermodynamics, whereby the averages over time of various observable quantities were related to the ensemble averages, that involve classical, so-called Kolmogorov, probability measures, at least for certain “equilibrium” situations. The second law of thermodynamics introduced the concept of entropy and its probabilistic increase, the first truly emergent, probabilistic law in physics (temperature and pressure, being other emergent quantities, apart from entropy, the measure of disorder of a statistical system). In the context of Newtonian gravity, one gets, at least in retrospect the geometric description of gravity in non-relativistic, Galilean spacetime. Galilean relativity, based on the asymmetric mixing of space and time, with the absolute, universal time, true for all observers, gets generalized in special theory of relativity to observer dependent space and observer dependent time, viewed as “slices” of a new concept of spacetime, with the new concept of causal structure, that distinguish timelike, null and spacelike events. This in turn is made into a full dynamical geometric structure of spacetime with a relativistic description of gravity stemming from curvature of spacetime, and a dynamical causal structure, tied to gravity. In our next story, the story of quantum theory, we will encounter fundamental, intrinsic probabilities, that can interfere, unlike the classical probabilities of statistical thermodynamics, and which are consistent with equations of motion of individual elements, and not, as in statistical thermodynamics, with temporal averages over the time evolution of many elements. This is the big new lesson of quantum theory, that transcends the lessons of classical spacetime physics.
The following equation pair shows the close connection between quantum and classical mechanics. F = m × a = ×( ) ( )average F m average a The first equation is the basis of classical mechanics. It summarizes Newton’s laws, where F is the force, m is the mass, and a is the acceleration. The second equation is the quantum-mechanical generalization. It looks almost the same except for the word average. Averaging is necessary because particles in quantum mechanics are associated with waves, and one cannot precisely determine the position or motion of a wave. Only the averages are known.
While ultimately they are described by quantum mechanics, macroscopic mechanical systems are nevertheless observed to follow the trajectories predicted by classical mechanics. Hence, in the regime defining macroscopic physics, the trajectories of the correct classical motion must emerge from quantum mechanics, a process referred to as the quantum to classical transition. Extending previous work [Bhattacharya, Habib, and Jacobs, Phys. Rev. Lett. 85, 4852 (2000)], here we elucidate this transition in some detail, showing that once the measurement processes that affect all macroscopic systems are taken into account, quantum mechanics indeed predicts the emergence of classical motion. We derive inequalities that describe the parameter regime in which classical motion is obtained, and provide numerical examples. We also demonstrate two further important properties of the classical limit: first, that multiple observers all agree on the motion of an object, and second, that classical statistical inference may be used to correctly track the classical motion.
The motion of macroscopic objects such as soccer balls, vehicles and planets can be well described using classical mechanics. However, classical mechanics is inadequate to describe the movements or the structure of microscopic objects such as atoms, molecules or nuclei. To properly describe the properties of microscopic objects, the theory of quantum mechanics was developed. While quantum mechanics can be applied to macroscopic objects, it turns out that for objects of large dimensions the quantum mechanical laws evolve naturally into classical mechanics. In general, solving a problem in quantum mechanics is far more difficult than solving it in classical mechanics. Therefore, the laws of classical mechanics are employed when they can be applied safely.
There seems to be much interest in the literature (1) in associating quantum mechanics (in particular the Schrodinger equation) with a classical joint phase space probability F(x,p,t) such as appears in classical Liouville theory etc. Here, we consider that P(x)dx = dx/L for a particle at rest and analyze the extra physics which appears when one considers a single particle bouncing back and forth in a box, which is what a person in a frame moving with -v would see (albeit with a Lorentz contracted L.) We suggest that the constraint L is important as it is linked with physical forces which appear when the particle moves. In other words, a particle at rest is linked with a static probability, while for motion, one must have a probability which is linked to more than x and t, but to force considerations in x and t, a dynamical probability. (Experiment bears this out as we discuss.) In particular, we argue that in the case of a moving particle, two physical concepts related to force appear, namely pressure which is an average over time and impulse hit which is only linked with space. Thus, two different particles may have the same p, but different v and pressure values. We thus suggest a physical grouping of (p,x) and (pressure, t). We then note that one needs to somehow preserve the notion of P(x)=1/L, i.e motion does not introduce a bias to x, nor to time. In a rest frame, there is no motion and so there is neither p nor pressure, but P(x)=1/L pertains. If one tries to define a probability with two groupings, one for pressure, t and the other for p,x, then if one multiples a p1 case by a p2 case, the p,x grouping should yield p1+p2, x. Furthermore, a p,x grouping should imply mixed terms, e.g. px etc, otherwise one does not really have an association between the two variables. This suggests a pf(x) form linked to the p-x grouping. Given the notion of special relativity, it seems that one may replace pressure with the notion of energy and create a Lorentz invariant probability. This would then lead to the free particle quantum wavefunction (probability) exp(-iEt+ipx), but we argue that the roots are in classical physics and a desire to physically describe the physical information present for a particle bouncing back and forth in a box.