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.
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.
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.
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 classical statistical mechanics, C exp[ -( mv*v/2 + V(x))/T) represents density i.e. the probability for a particle to have a velocity v and be at x. If applies this probability to Shannon’s entropy formula, one obtains C/T ( mv*v/2 + V(x)) exp[ -( mv*v/2 + V(x))/T) + exp[ -( mv*v/2 + V(x))/T) ln(C). Thus, it seems energy times density + density times ln(C) is the entropy density, a well known result in classical statistical mechanics, although sometimes not explicitly stated. The first term integrated over x is essentially the kinetic energy, while the second integrated over velocity yields the potential energy term. Thus, the first term partial density seems to be a purely momentum dependent kinetic energy piece times probability and the second a purely x dependent potential energy piece. (There is still an integral over velocity for the first term and one over x for the second to obtain the full entropy.) In quantum mechanics, we argue that a(p1)exp(ip1 x) a(p2) exp(ip2 x) serves as the probability where W(x)=Sum over p a(p) exp(ipx) = wavefunction. This leads to an expression for entropy density with two terms, one of which may be immediately integrated over momentum and the other over space. If one wishes to make a link to the classical statistical mechanical case, one may ask what requirement is necessary for the momentum related piece to represent average kinetic energy and the spatial part, average potential energy. The requirement, we find, yields the wavefunction of the ground state harmonic oscillator. For other cases, the partial density is not equivalent to average energy. Thus, we try to establish a link between entropy density and energy density, but require the kinetic energy related term to be strictly a function of momentum and the potential related term strictly one of x. In quantum mechanics, one may form averages at a point x. For example, KEave(x)= [Sum over p p*p/2m a(p) exp(ipx)]/W(x). If one examines a p/-p pair with a(p)=a(-p) (for example), cos(px) terms arise which are positive at some x and negative at others. We argued in previous notes that this represents stochastic flow in the opposite direction from that expected by -dV/dx. In other words, -dV/dx is an average which consists of stochastic force addends which may give momentum hits to the right or left, but which add to yield -dV/dx. As a result, one may have negative contributions to the kinetic energy (and density as well). In this note, we argue one may also have negative contributions to the entropy density due to this phenomenon.
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.
In a previous note (1), we argued that sound wave equations (with certain boundary conditions) derived from the Boltzmann transport equation conservation relations match the form of a quantum mechanical solution of a particle in a box with an infinite potential at the walls. Furthermore, we argued that in both the sound and quantum pictures there exists a dispersion velocity u= d/d/x ln(density) (classical) and u= -i d/dx ln(W) where W is the wavefunction in quantum mechanics. Both represent forward backward motion at different x points. Thus, there seems to be a link between classical sound and quantum mechanics at least in terms of some density type solutions and a backwards forwards motion. It is also known, however, that for the ground state of a ground oscillator, the quantum solution is of the same form as the static Maxwell-Boltzmann (MB) solution C exp(- mvv/2mT - .5kxx/T). In this case, quantum mechanics still has a dispersion velocity with forward backward motion, but statistical mechanics has a zero u(x,t). The Boltzmann transport conservation relations yield the MB solution on the basis of a balance of pressure and force. This leads to the question: Why do the two approaches yield the same solution in this one case, even though the underlying physics seems to be quite different? We argue that even though u(x,t)=0 for the MB static solution, there is still dispersive flow which is forward at the left endpoint and backwards at the right. Thus, in both the case of sound and the MB solution with a potential there may be dispersion which has links to quantum mechanics.
Quantum mechanical spatial probability densities may be obtained by solving the time independent Schrodinger equation. Statistical mechanical spatial densities may be obtained by maximizing Shannon´s entropy d(r)ln[d(r)] (where d(r) is the spatial density) subject to appropriate constraints. A classical spatial probability density may be obtained from 1/v(x) where v(x) is the velocity at position x obtained from kinetic energy + potential energy (x) =total energy. Recently, there have been attempts to link these various spatial probability densities. For example for the ground state wavefunction of a harmonic oscillator, the Schrodinger equation can be obtained from varying Shannon´s entropy with constraints because V(x) W*(x) W(x) (where W(x) is the wavefunction) can be written in the form of Shannon´s entropy times a constant [3]. Furthermore, Casal et al have maximized Tsalis´ entropy to obtain good approximations to the ground state wavefunctions of oscillators and Coulomb potentials. In [1], the classical density 1/v(x) has been calculated using energy levels from quantum mechanics, namely E=hbar w (n+.5). This density has in turn been used to calculate entropy and compared with results from wavefunctions of the Schrodinger equation with good results even for low n values even though for n=0, the wavefunction W(x)=1/C exp(-ax2) and 1/v(x) are not at all similar. The purpose of this note, is to attempt to examine the relationship between these different spatial densities.