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 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.
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), 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.
Shannon’s entropy is calculated using probabilities P(i) i.e. S= - Sum over i P(i) ln(P(i)). A probability is related to an uncertainty associated with event i, but this event may not be a physical event, but rather a measurement. In other words, measurements in a system may be made according to one distribution or another. This distribution may be converted into something which looks like a probability even though the system is completely deterministic. The entropy then calculated is based on the measurement distribution and not the physical system. As a first example, consider the completely deterministic motion of a bound classical system between two turning points. Person A may send values of x (one dimensional space) to person B, based on a certain distribution instead of sending the values based on time i.e. x(t1),x(t2) etc. One distribution which may be used is the uniform one i.e. P(x)= dx/L. Another distribution, used in the literature (1), is based on the time spent in each constant dx i.e. P(x) = Cdx/v(x) where v(x) is velocity. These two distributions lead to different Shannon’s entropies, but the underlying system is completely deterministic which begs the question: Why does one associate an entropy with it? The entropy seems to be associated with the sending of information. We try to apply these ideas to a quantum bound state. In the literature (2), one sometimes sees a Shannon’s entropy based on W(x)W(x) = P(x), where W(x) is the single particle bound wavefunction. This is a distribution, but is it based on uncertainty in the system? W(x)W(x) often consists of a set of crests of different heights and widths. We argue that the inherent uncertainty in x is due to the width of a crest. In other words each crest may be normalized to 1 and a Shannon’s entropy for each crest calculated. One must then weight the different crests. If one calculates the classical time associated with the base length of crest, this could be used as a weight which in the high energy level limit is associated with P(x)=Cdx/v(x) from (1), but it seems there is no reason to base the weight on time. The uncertainty in x is due to a superposition of exp(ipx)s which are not linked to time. Alternatively, one may use an unnormalized weight of the length of the base as uncertainty is only in x. One may then add a Shannon’s momentum entropy using a(p)a(p)=P(p) where W(x)=Sum over p a(p)exp(ipx).
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.
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.