In physics, there are deterministic theories such as classical mechanics and statistical theories such as classical statistical mechanics and quantum mechanics. Many interpretations are applied to statistical theories and there is sometimes debate as to the application of such theories in various physical scenarios. In a previous note, we considered cycling due to exp(iEt) in quantum mechanics and tried to argue that such cycling may also appear in statistical mechanics. We made use of the example of the ground state of the quantum oscillator which has a quantum mechanical density of the same form as the classical statistical mechanical result. In this note, we wish to consider the idea of cycling in statistical mechanics using some simple examples. In addition, we wish to also examine the notion of losing information to pass from a deterministic theory to a statistical one. Finally, we wish to consider some possible aspects of the classical statistical density in a potential. It seems that even in classical statistical mechanics, the picture presented by the ‘averaging’ may actually be quite different from the physical scenario which may contain deterministic behavior which is lost in the calculation of statistical results.
In physics, there are deterministic theories such as classical mechanics and statistical theories such as classical statistical mechanics and quantum mechanics. Many interpretations are applied to statistical theories and there is sometimes debate as to the application of such theories in various physical scenarios. In a previous note, we considered cycling due to exp(iEt) in quantum mechanics and tried to argue that such cycling may also appear in statistical mechanics. We made use of the example of the ground state of the quantum oscillator which has a quantum mechanical density of the same form as the classical statistical mechanical result. In this note, we wish to consider the idea of cycling in statistical mechanics using some simple examples. In addition, we wish to also examine the notion of losing information to pass from a deterministic theory to a statistical one. Finally, we wish to consider some possible aspects of the classical statistical density in a potential. It seems that even in classical statistical mechanics, the picture presented by the ‘averaging’ may actually be quite different from the physical scenario which may contain deterministic behavior which is lost in the calculation of statistical results.
Both quantum mechanics and classical statistical mechanics predict the same momentum and spatial density forms forms for the solution of an oscillator. In the case of quantum mechanics, the ground state is considered. One may ask whether this is a coincidence, and if not, why two seemingly different theories should lead to the same result for the oscillator case. We argue in this note that it is the form of the conservation equation of the oscillator which leads to x and p having the same probability distribution.Furthermore, given that both quantum mechanics and classical statistical mechanics are of a statistical nature, one may inquire into the actual nature of the statistics being used, the meaning behind averages being used. For example, the classical deterministic solution of an oscillator can be written in statistical terms by removing time dependence and using density = dx/v. This does not mean, however, that the actual physical motion is no longer deterministic. In addition, we try to argue that quantum mechanics may be a more detailed statistical model including dynamics which satisfies the more static statistical result of spatial density for classical statistical mechanics.
Statistical mechanics seems to be based on the idea of distributing a fixed amount of energy among a number of particles. (Usually this number is very large.) Alternatively, one may think of a single particle changing its state (e.g. momentum) during many time intervals. In both cases, one takes an average of the energy, but this means one needs weights for each single particle state. These weights seem to be based on the idea that each ‘vector’ of states has the same probability of occurrence. Thus, if one has N particles and an energy eN, one vector involves one particle having eN and the rest 0. Many vectors, however, may be created with N particles having energy near e. Thus, the weight for a single particle to have energy e is much higher than for it to have energy E. The weight is ultimately taken to be the Boltzmann factor. In quantum mechanics, it seems a very different assumption is made. Unlike classical physics where one follows a particle in space and time, knowing its acceleration, position and velocity at each x, in quantum mechanics, collisions with the potential seem to be granular, and the main assumption seems to be that a quantum particle may be in any momentum plane wave state at a given time. Like statistical mechanics, there is a set of weights, but one no longer uses the scenario of distributing energy among different states to determine these weights. Instead, the weights are obtained by noting that a quantum average at a point x is taken using fp exp(ipx)/ (Sum over fp exp(ipx)). The weights must ensure that: -1/2m (Sum over p p2/2m fp exp(ipx)) / (Sum over fp exp(ipx)) = E-V(x) at each x point. Thus, the plane wave behavior of quantum particles becomes a dominant factor. For the case of V(x)=0, the plane wave nature of the possible single particle momentum states leads to an overall (Sum over fp exp(ipx)) which has the appearance to two plane waves. This is very different from the classical statistical mechanical case. In this note, we attempt to examine the difference of these two statistical theories, and understand why in one case, the ground state of an oscillator, the two become the same.
Quantum mechanics is often compared with classical mechanics because both may deal with a single particle subject to a potential. Classical mechanics is said to be deterministic describing x(t) and its derivatives i.e showing how a particle moves and accelerates while bound state quantum mechanics produces a real wavefunction W(x) such that W(x)W(x) yields a spatial density profile. The comparison is carried further with the correspondence principle which states that at high energy levels quantum mechanics should mimic classical mechanics. Classical mechanics does not simply describe x(t) and its derivatives i.e. deterministic motion. It also includes two body scattering, force and torque balances for a stationary object etc. In the case of the classical statistical mechanics of an equilibrium ideal gas, the Maxwell-Boltzmann (MB) distribution in the absence of a potential is C1 exp(- pp/(2mT) ). If a potential V(x) is added, one may find P(x) two different ways using Newtonian ideas. The first is to consider the gas as a collective entity like a tiny macroscopic classical object at x. Then one may argue that equilibrium means the system is stationary and apply Newtonian statics i.e. a balance of pressures and -dV/dx to obtain exp(-V(x)/T) as is often done in textbooks. Alternatively, one may consider a single particle and apply Newtonian mechanics differently, namely use a conservation of energy equation: p1p1/2m + V(x1) = p2p2/2m + V(x2). Again one obtains exp(-V(x)/T). The spatial part of probability may be based on Newtonian mechanics indicating that it is not stochastic. The stochastic function is C1 exp(-pp/2mT). Interestingly, however, the two Newtonian approaches are different with respect to a single particle. A collective nonmoving set of particles acts as a stationary object, yet within it each particle seems to be “bouncing” off of an imaginary wall creating pressure. Thus one has an impulse free particle picture yet obtains the correct result exp(-V(x)/T). Alternatively one may think of a single particle accelerating which it does if it is not colliding with other particles. This is a very different picture of the single particle than of one bouncing back and forth hitting an imaginary wall. In fact, there is usually a wall holding the gas together and pressure and bouncing particles are the real picture there, but within the gas one would normally think of a single particle accelerating. Nevertheless there are two scenarios which yield the same result. The idea of pressure balance is more in keeping with an overall statistical picture of the gas. In other words, one may actually ignore the idea of following a single particle as it accelerates. (That is not to say that single particle acceleration does not exist - it does.) Quantum mechanics we argue is a statistical theory. In the case of a single bound particle one has a momentum distribution W(x)=wavefunction = Sum over p a(p)exp(ipx). One may apply conservation of average energy at each x to obtain the time-independent Schrodinger equation. In such a picture, like with the statistical mechanical gas, we argue that one may imagine impulse hits with the potential instead of considering acceleration. The quantity vrms(x) from KEclassical= .5m vrms(x)vrms(x) = -1/2m d/dx dW/dx / W does display acceleration, but this is a mathematical average. As energy levels become higher the correspondence principle is said to apply, but a classical particle exhibits acceleration. The quantum bound state does not, but like in the case of the statistical mechanical gas it may be that two scenarios lead to the same result. In other words, the quantum bound W(x) still has crests and troughs, but with low resolution one may only see an envelope roughly touching W(x)W(x) which matches a classical C2/v(x) where v(x) is classical velocity. Given that quantum mechanics is a statistical theory it may not be necessary to have it explicitly display acceleration.
We consider the possibility that both classical statistical mechanical systems as well as quantum mechanical time independent systems evolve according to the Hamiltonian H. This suggests that given the energy of the system, the Hamiltonian operating on a density or wavefunction will bring back the density or wavefunction with a weight factor of E, energy. For the quantum mechanical case, E represents a frequency. We argue that it is a frequency for both the quantum and statistical mechanical case, related to how quickly the system interacts under the Hamiltonian. We also examine an aspect of interference in the quantum wavefunction which may be related to a 2x2 model of a particle which cycles back and forth as it moves with an average velocity v in a certain direction. We argue that the plane waves stirred up in a potential (or in the Fourier series of the wavefunction) are associated with periodic time motion (due to zitterbewegung). We argue that as these plane waves interfere, they create a new constant average energy throughout a system with a potential which acts as the period of periodic time behaviour, namely exp(iEt)
Classical mechanics was developed in the 1600s and is considered a complete theory in that it matches observables in the macroscopic world. In the 1920s, however, it was found that different mechanical rules seem to apply to the electron in an atom. In this note we try to argue that classical mechanics is a subset of quantum mechanics i.e. it is based on partial information or the dropping of information. In (1) we noted that a classical spatial density Cdx/v(x,) where v(x) is velocit, is sometimes presented in the literature (2). This expression is proportional to dt associated with a fixed length dx so the density is a relative one. If one considers two bound classical systems (a particle in a box) with different constant speeds (in the interior) then dt is different for each, but Cdx/v(x) does not distinguish. Thus we argue that information has been lost. The higher v particle not only has a higher impulse, but an increased frequency and we argued in (1) that quantum mechanics takes note of this extra frequency by increasing the number of crests associated with higher constant velocities. In this note, we point out that quantum mechanics itself is linked to the dropping of information. If one considers free particle wavefunctions W(x) for different momenta (i.e velocities if mo is the same), one has exp(i p1 x), exp(i p2 x) etc. These clearly have different information. They describe direction, but more importantly they have an associated wavelength proportional to 1/p. P(x)=W*(x)W(x)=constant and so this information, which is physical, is lost . P(x) is related to the question: What is the probability for a particle to be at x? If one considers cos(px) as a nonlocal spatial distribution which shows motion i.e. negative and positive parts of cos(px), then sin(px) is a spatially shifted distribution which helps to indicate direction, but also shows that all x points are the same. One has a peak of cos(px) at x=0 and a lower value at x=delta, but due to the motion, the peak may later be at delta just like a classical wave. This is what W*(x)W(x)=constant captures. It shows that even though cos(px) maps to x in a certain way with sin(px) in an extra dimension, overall the free particle has no preference for one x point versus another. There are two points we wish to make. First, classical mechanics seems to be based on P(x) i.e. it drops information about a wavelength. Space is broken into tiny dx units and the particle has a constant velocity in each even if it is accelerating. This is the Newtonian picture and leads to dp/dt = Force or pp/2m + V(x) = E. This information, however, is contained completely in quantum mechanics because for a bound state with V(x) one has: -1/2m d/dx dW/dx / W + V(x) = En. -1/2m d/dx dW/dx / W is exactly KE(x) classical. Quantum mechanics, however, displays more information than the classical picture. For example, only certain En are allowed and W*(x)W(x) leads to a spatial density which does not appear in classical mechanics. This leads to a second point. exp(ipx) contains information about a wavelength which is lost if one only considers P(x) for a free particle. If there is an interaction (two electrons shifted in space, a V(x) or a single or double slits etc) exp(ipx) acts like a probability and adds in OR or interaction situations. In such a case the information regarding the wavelength becomes physically crucial. In other words for an interaction this information does not drop out as it does in the classical scenario of pp/2m + V(x) = E. Thus we argue that classical mechanics is a subset of quantum mechanics. Wavelengths are tiny in the macroscopic world and so are simply not observed, allowing the subset scenario of pp/2m + V(x)=E (nonrelativistic) to be an excellent approximation. Indeed for every energy level of a quantum bound state this relationship also holds. It is the spatial density which is altered.
In this note, we try to develop bound state quantum mechanics directly from statistical arguments. The key idea, we argue, is that a statistical treatment of a system considers “free particles” together with a stochastic equilibrium driving mechanism. If an acceleration driving potential exists (as in classical statistical mechanics), it alters the results, but does not change the main scenario of free particles interacting through a stochastic mechanism at a point because one does not follow particles in time, noting how they accelerate as they travel. (For example, a Maxwell-Boltzmann gas distribution is based on two body elastic scattering whether there is a potential or not. The potential affects the density, but not the relative momentum probabilities at any x.) For the quantum case, the potential V(x) is the stochastic equilibrium driver, but on average kinetic energy plus potential must equal total energy. Again, one does not follow the quantum particle in time trying to see if or how it accelerates.