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).
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 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 the literature (e.g. (1)), the expression - density(x) ln(density(x)) is used as Shannon’s spatial entropy for a quantum system (bound or not). For the bound case, the correspondence principle is said to hold for high energy levels (2). In that case, quantum density = W(x)W(x) (W(x)=real wavefunction) is said to be equivalent to the classical density C/v(x) where v(x) the velocity is given by .5mv(x)v(x) + V(x) = E. Classical density yields a curve which passes through peak regions of quantum density humps. In the classical case, particle motion is completely determinate. C/v(x) as a density (probability) is proportional to dt, the amount of time a particle spends passing through a fixed length dx. If the particle moves quickly, it spends little time, and hence there is less time for a measurement (i.e one may miss the particle due to its speed). We argue in this note that such a scenario does not seem to be in keeping with the ideas of Shannon’s entropy where one has probabilities for certain mutually exclusive events at each ti. For example at each ti, one may toss a coin. If it is not “heads”, this means it is tails. For the classical particle in the correspondence principle, each dx region has its own probability proportional to dt = dx/v(x). Thus, each dx is a region of its own in terms of probability. Thus, if Ci measurements of the presence of the particle are made within dx of xi out of N trials, the probability that a measurement is made in dx at xi is Ci/N and the probability that it is not is 1-Ci/N. Thus, unlike Shannon’s picture, various arrangements do not have the same weight or probability. We investigate this scenario in more detail and argue that it does not follow the approach of Shannon’s entropy. We also try to connect with the quantum situation.
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.
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 (1), Shannon’s spatial entropy is computed using - [W*W] ln[W*W], where W(x) is the wavefunction and a classical envelope approximation: W*(x)W(x)=C/v(x), where v(x) is classical velocity, is used (according to the idea of the correspondence principle). A momentum entropy: - Integral dp a(p)a(p) ln[a(p)a(p)] (one dimension) is also calculated. Recently in (2), we argued that one should form Shannon’s entropy using the probability = a(p1)exp(ip1 x) a(p2)exp(ip2 x). This leads to the following if one sums the first term over p1,p2 and integrates the second over x: Entropy= Integral dx -2W x d/dx W - Integral dp a(p)a(-p) ln[a(p)a(-p)] ((1)) one dimension In (1), the spatial entropy is found to behave as ln(n) for high n whereas for ((2)), spatial entropy is always 1. Thus, there is a very large difference between the two. In (1), a momentum entropy -a(p)a(p)ln[a(p)a(p)] is added to spatial entropy and in ((1)) a very similar momentum piece appears. For the case of the oscillator, W(x) and a(p) are both Gaussians and so the overall result for entropy from (1) and from this note both behave as ln(n) when n is large, but the factor of ln(n) differs. Ours is half as large for the case of a quantum oscillator. For our case, all of the entropy comes from the momentum term in the large n limit. In this note, we consider these ideas in more detail.
The classical Fisher information, F, in one dimension is given in the literature as: Integral dx d(x) [ d/dx ln( d(x) ) ]2 ((1)) where d(x) is the spatial probability density. For quantum mechanics, (1) uses d(x)=W*(x)W(x) for quantum mechanics, where W(x) is the complex wavefunction. The Cramer Rao inequality is used to show how 1/ F is related to the variance of d(x). For a bound state wavefunction, which is real, one may show Fisher’s information leads to Integral dx v(x)v(x) d(x) which is proportional to the average kinetic energy. Using the correspondence principle, d(x), for high energy levels, should approach C/v(x) where v(x) is the classical velocity. On the other hand, using the definition of F directly with d(x)=C/v(x) leads to a different result. This seems unusual. We try to investigate in this note to see that quantum features exist even in the high energy limit. The Fisher information calculated with W(x)W(x)=C/v(x) seems to capture quantum information, while that with d(x)=C/v(x), information about the envelope function.