In Part I of this note we argued that the quantum bound state expectation value < -1/(2m) d/dx d/dx> = Integral dx WW (-1/2m d/dx dW/dx /W) in the high energy limit is equivalent to the first part of the classical action: Integral dt .5mvv As a result, we identified WW=spatial density (bound state) with 1/v(x) in the high energy limit. 1/v(x) is used in the literature (1) as probability classical(x) when calculating Shannon’s entropy classically (as a bound for high energy quantum entropy). In this note, we wish to examine further some ideas associated with this link to the classical action.
In Part I of this note we argued that the quantum bound state expectation value < -1/(2m) d/dx d/dx> = Integral dx WW (-1/2m d/dx dW/dx /W) in the high energy limit is equivalent to the first part of the classical action: Integral dt .5mvv As a result, we identified WW=spatial density (bound state) with 1/v(x) in the high energy limit. 1/v(x) is used in the literature (1) as probability classical(x) when calculating Shannon’s entropy classically (as a bound for high energy quantum entropy). In this note, we wish to examine further some ideas associated with this link to the classical action.
Linking quantum mechanics to classical mechanics seems to have been an early goal in quantum theory as represented by the correspondence principle. More recently there have been attempts to link the entropy of both a single bound quantum state and a thermal statistical system of quantum states to classical entropies (1),(2). For the former, the entropy is proportional to Integral dx density(x) ln[density(x)] where density(x)= delta t = 1 / v(x) with v(x) being velocity. Thus, density, which is a probability in Shannon’s entropy equation, is a measure of the change in the value of delta t(x) in this classical entropy. Bound state quantum mechanics is said to be time-independent, but by the correspondence principle (1) the probability of the envelope of the wavefunction i.e. Envelope(W(x))* Envelope(W(x)) = delta t(x) so quantum probability (albeit linked to the envelope) is related to classical time intervals which adjust with x to yield the proper ration v(x)=delta x / delta t(x) with delta x being constant. In this note we argue that the expectation value: Integral W(x) d/dx dW/dx = Integral WW KEclassical(x) for a bound state is, in the high energy limit Integral dt KEclassical(x) i.e. a portion of the action, just as in the classical case (i.e. the correspondence principle). This leads to the association of the higher energy limit of the envelope of W times itself with delta t(x).
In (1), the spatial quantum Shannon’s entropy: - Integral dx Wn(x)Wn(x) ln[ Wn(x)Wn(x)] ((A)) for a bound state is considered, where Wn(x) is the nth energy eigenstate. In the limit of n very high, quantum entropy is approximated by: - Integral C/v(x) ln{ C/v(x)} dx ((B)) where C/v(x) is the normalized classical entropy with .5mv(x)v(x) + V(x) = En. In such a case, W(x)W(x) still has peaks and zero points, but if resolution in the classical world is low, the integral is approximated by the classical C/v(x) which is said to pass through the peaks of W(x)W(x). In this note, we consider a bound classical particle moving first to the right and then to the left. In such a case, it seems entropy is zero. The argument in (1) is that the time a particle spends in a fixed dx is dx/v(x) and that this should represent an unnormalized probability. We argue this may represent an unnormalized probability for an ensemble of classical systems with the same energy, but different starting points x,t. If one wishes to know the entropy of a single system, we argue it is 0 and not ((B)). The goal of this note is to find a quantum expression for spatial entropy which tends to 0 in the high n limit. Our approach is to introduce a cut off-length dx which is very small with respect to the system length and is also small in the classical world. Next, W(x)W(x) which has humps and zero values is examined. If the base of any hump is within dx, entropy density is set to 0 within this dx, otherwise it is given by ((A)). As humps become narrower, the entropy becomes smaller as more and more dx units are replaced with 0. In this way, we try to study the change in quantum bound entropy as the system becomes more and more classical with higher En values. Thus one has S(dx, n) which may be studied for various dx and n values.
A classical bound system may be said to have a spatial density proportional to the amount of time dt a particle spends in a constant dx region i.e. C/v(x) (1) We stress that in this scenario there is a well defined and constant dx so that “dt” may be linked to velocity v(x). We also note that there is no spatial representation of action-reaction here even though this exists in Newton’s theory. In a classical Maxwell-Boltzmann gas there are many particles which each act dynamically according to .5mvv+V(x) = E (where v and E are particular to the particle when it is not colliding). At the same time there is a spatial density which “represents” macroscopic static chunks of matter which are balanced through pressure differences (P=spatial density(x) RT) and -dV/dx density where V(x) is the potential. From the pressure equation, T is proportional to energy which in turn is proportional to RMS speed squared. RMS speed may then be linked to a dt if a fixed dx is understood. In the classical statistical case, however, spatial density is directly linked to the number of particles in a region. Time enters in the pressure expression through a term proportional to the average kinetic energy i.e. pv where p is like an impulse and v a flux. What happens in a single particle quantum bound state? In a sense it is like a classical bound state as the equation: prms(x)prms(x)/2m + V(x) = En holds for any energy level. Can one then take vrms(x), choose a fixed dx and argue that spatial density should be equal to dt or C/vrms(x)? We argue that one cannot due to the lack of defined “dx” quantity. Why is dx problematic to define? We argue that it is because quantum mechanics is based on action-reaction x-probaiblity distributions linked to impulse. For example, exp(ipx) for a free particle captures only p or impulse (not velocity) and cos(px) and sin(px) are its action-reaction regions.These lead ultimately to a density is given by the well-known W*(x)W(x) where W(x)=Sum over p a(p)exp(ipx), although kinetic energy with pp/2m = .5pv is used to find W(x) as well. . W*(x)W(x) has crests and troughs similar to cos(px) in exp(ipx), except that these now have varying heights and widths. We try to argue that even though each x point maps to a different KEave(x), the hump physically represents action-reaction in the system (just as the humps in cos(px) do except for a single p) while KE(x) represents an average of p’s. This action-reaction is linked with impulses. In general, if one takes d/dx of the time-independent Schrodinger equation one finds an average impulse term, -dV/dx and KE(x) dW/dx / W where dW/dx / W is the average impulse. At peaks, dW/dx =0 and so an average force through time*impulse term balances -dV/dx=Newtonian force. These peaks are also associated with W*(x at peak)W(x at peak) which is the spatial density. We interpret this as representing the average time the single particle spends near x at peak. This value, however, follows from ideas of impulse and not from speed as there is no well defined dx even though there is a well defined vrms(x). Thus the ratio of W*(x)W(x) at two peaks is not the inverse of the ratio of vrms(x) except apparently when the energy level n becomes very high. We investigate these ideas.
Classical statistical mechanics is often derived using counting methods applied to all possible configurations a system may take subject to a total volume and a total energy constraint. Given that there are numerous configurations, there should also be an average of these configurations described by f(e) for the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions. In previous notes we have argued that one may apply a dynamical constraint to the average configuration. In particular, we argued that there should be a time reversal balance for two body elastic collisions. This immediately yields the various distributions with only consideration given to the average state and no counting techniques used. The counting of configurations, however, emphasizes that statistical mechanical systems contain “fluctuations” and that a specific dynamic constraint (i.e. time reversal two body scattering) applies only to the average. Quantum bound states, we argue, are very similar. They represent a statistical system which has many fluctuations (i.e. a particle may be knocked out with an momentum p with probability a*(p)a(p) where W(x)=wavefunction = Sum over p a(p)exp(ipx)). There is, however, also a dynamical constraint which applies to the average, namely KE(x)+V(x)=E which is a classical mechanical constraint. Here KE(x)= [-1/2m d/dx dW/dx /W] ((1)). If fluctuations disappear in the high energy scenario, one may assume there is an almost fixed momentum p at each x. ((1)) still applies and there is still a wavefunction which is often in the form of crests and troughs. If an almost unique p exists at each x, one expects W(x) to vary in height at peak values. This may be verified from actual W(x) solutions. Given that the constraint on the statistical average is KE(x)+V(x)=E, one would expect that as p becomes an almost unique value at each x, that the quantum expectation Integral dx WW KE(x) with KE(x) given by ((1)) should also approach a classical value in other words Integral dx WW —> integral dt. One does not need to impose the correspondence principle to arrive at this result, one only needs to consider the case in which fluctuations in a quantum system disappear allowing the average p at each x to consist of almost a single p value.
In part I we argued that classical mechanics divides a one dimensional line into equally spaced dx regions. These are small, but not zero because classical spatial density is C1 dt = C1 dx/ v(x) where v(x) is velocity which is assumed to be constant within dx to first order. In terms of probability, if a particle is within a certain dx, it is not in the adjacent ones i.e a probability distribution reaches zero at the endpoints of dx. We argued that for a constant velocity classical density is P(x)=constant and so one must keep two spatial probability distributions (in two dimensions or separated by i) such that the modulus is 1. This makes the two ideas compatible. This idea seems to apply to quantum mechanics with the stipulation that dx is linked to 1/p. Thus one has exp(ipx) as a dynamical probability as discussed in Part I. In this note we wish to consider this in more detail together with the Schrodinger equation. In particular we focus on the high energy limit. We argue that exp(ipx) should hold for a small dx region classically which may include several wavelengths proportional to 1/p. Thus there is still the idea of the distribution and its shift as two dimensions, but we argue that the time-independent Schrodinger equation may be an approximation as it considers exp(ipx) and exp(-ipx) when creating Wn(x) and it is not clear that such interference holds in a very high n limit.
In a previous note (1) we argued that both the relativistic A= -mo t sqrt(1-vv) and nonrelativistic A=t .5mvv classical free particle actions may be written with v=x/t and x and t varied independently to yield: dA/dx partial = p and dA/dt partial = -E. Thus A is equivalent to -Et + px for a free particle with x and t being treated as independent. We argue in this note that such an assumption is equivalent to a statistical picture for a free particle even at the classical level because x and t are treated as independent. Thus a constant p may occur at any x because no time is linked to x. Given that all space points are the same for constant p one expects a constant probability at all x points. Knowing p exactly means not knowing anything about which x one has. Secondly, one may create an ensemble of free particle p classical actions for a situation containing a potential V(x). Given that x and t are independent, there cannot be any notion of single p accelerating, although a vrms from an ensemble may. At a given x (x known exactly) one does not know which p is present. Thus the ideas of uncertainty of p and x follow immediately from this picture. If one considers a single particle and an ensemble of free particles in the x-t independent scenario, then each free particle action p is linked to A=-Et+px where E=pp/2m. This ensemble would arguably be stirred up by a potential V(x). If one wishes to consider an equilibrium scenario for which there is an overall En for the bound system one would expect an overall Ent for the system and for -pp/2m t pieces to disappear. Thus we postulate that pp/2m +{p part} G(p’s k’s) = En for all p where {p part}G(p’s, k’s) is an unknown function and V(x)=Sum over k Vk exp(ikx). Next one may consider the spatial situation for which V(x) is the classical potential for a single particle. The kinetic energy follows from an ensemble of free particles with the overall -Et removed. From classical considerations one would expect to have a KE(x). Thus one expects: KE(x) = {Sum over p a(p) f(px) } / {Sum over a(p)f(px) } with f(px), an orthonormal basis set, because the action contains px. (It is orthonormal because one wishes the identity of p to exist if not at a point, then over all of x.) We try to argue in this note that f(px)=exp(ipx), the momentum eigenstate which follows from A=-Et+px. We find G(p’s, k’s) which implies entanglement or mixing of a given p state with other p and k states from the potential V(x)=Sum over k Vk exp(ikx). Thus one expects an entropy linked to this momentum entanglement. Given the form exp(ipx), finding a set of ap’s which satisfy pp/2m + G(p’s, k’s)=En leads to a spatial distribution W(x)=Sum over p a(p)exp(ipx) so there should be an entropy contribution from this spatial part as well. In classical statistical mechanics one has both momentum and spatial contributions to entropy. We argue for S=Sp+Sx, but note that even though px has x, p symmetry as does the Heisenberg uncertainty principle var x var p >= hbar/2, Sp and Sx may not be symmetric in terms of important parameters such as n, energy level. In other words, Sp the entanglement entropy with V(x) may be the driver entropy sensitive to energy level changes while Sx the resulting spatial density may not be as sensitive. Furthermore the overall entropy Sp+Sx gives information about uncertainties in x and p as is already known in the literature where bound state Sp+Sx bounds are given.