Classically there seems to be the sense of an isotropy in space and time and so statistically this becomes linked with degeneracies, say in a classical gas in space with no potential. Quantum mechanics of a free particle, however, suggests that different momenta, which represent physically different impulses in space, are not associated locally with isotropy in space or time. There is a wavelength in space and period in time (hbar/p, hbar/E where E=pp/2m nonrelativistically) and probabilities apply to these regions. We have argued that in general one might associate an entropy in both space and time due to the probabilities within a wavelength/period although generally in the literature this does not seem to be done. We have also argued previously that the quantum free particle wavelength and period follow from degeneracy associated with the Lorentz invariant: A = -Et+px. Here we point out that there is an added degeneracy, namely the direction of p in space. Thus the probabilistic exp(ipx) associated with a constant momentum can be further associated with various probability distributions in space if the free particle interacts with an n-slit system. In such a case, the particle has a fixed p and direction approaching the slit, but locally there is no spatial isotropy. This local probabilistic feature converts into a probability distribution for different p vector directions creating a unique spatial distribution on a screen when many identical particles, each with the same p, interact with the n-slit system. Each particle, however, begins with a specific p vector and after the interaction has the same magnitude of p, only a different direction. The local anisotropy associated with the wavelength is now converted into a more global spatial distribution due to extra uncertainties associated with impulse hits with the n-slit system. As a result, what one sees as entropy due to an interaction with a n-slit system is associated with degeneracy and the bias to believing that a free particle behaves in a deterministic manner at length scales of the order of a wavelength.
Classically there seems to be the sense of an isotropy in space and time and so statistically this becomes linked with degeneracies, say in a classical gas in space with no potential. Quantum mechanics of a free particle, however, suggests that different momenta, which represent physically different impulses in space, are not associated locally with isotropy in space or time. There is a wavelength in space and period in time (hbar/p, hbar/E where E=pp/2m nonrelativistically) and probabilities apply to these regions. We have argued that in general one might associate an entropy in both space and time due to the probabilities within a wavelength/period although generally in the literature this does not seem to be done. We have also argued previously that the quantum free particle wavelength and period follow from degeneracy associated with the Lorentz invariant: A = -Et+px. Here we point out that there is an added degeneracy, namely the direction of p in space. Thus the probabilistic exp(ipx) associated with a constant momentum can be further associated with various probability distributions in space if the free particle interacts with an n-slit system. In such a case, the particle has a fixed p and direction approaching the slit, but locally there is no spatial isotropy. This local probabilistic feature converts into a probability distribution for different p vector directions creating a unique spatial distribution on a screen when many identical particles, each with the same p, interact with the n-slit system. Each particle, however, begins with a specific p vector and after the interaction has the same magnitude of p, only a different direction. The local anisotropy associated with the wavelength is now converted into a more global spatial distribution due to extra uncertainties associated with impulse hits with the n-slit system. As a result, what one sees as entropy due to an interaction with a n-slit system is associated with degeneracy and the bias to believing that a free particle behaves in a deterministic manner at length scales of the order of a wavelength.
Classical mechanics samples space as being uniform i.e. each x point having the same weight. Thus the number of x states in a length L is proportional to L or in 3-space to volume V. The ln of this value then appears in classical entropy calculations. In the case of a particle moving in space the same idea holds, but this means the particle spends equal amounts of time in each dx region. Thus we argue this result of a constant probability in space is based on time sampling. Time sampling is the sampling done for an equilibrium gas. The system does not change on average in time, so it is as if one has a snapshot. Thus one has a constant probability in a length L in time sampling. One may argue that this constant is information so ln(probability(x))=constant1 or P(x)=constant 2. The relativistic/nonrelativistic action may be written as A = -Et+px if one uses v=x/t. This Lorentz invariant form allows one to clearly separate t and x with E and p as constants and may also be considered to be information. In such a case, if one ignores time, then px is the spatial contribution to A and this does represent a constant probability in space. Rather there is scaling of space by p. If x=constant/p, then each p has its own unique length, called a wavelength, in which space sampling is the same i.e. creates the same distribution form, but with 1/p as the wavelength. This we argue is space sampling of a spatial distribution and has physical consequences (i.e. two slit interference, measuring small regions etc). One may note that the wavelength=hbar/p depends on the magnitude of p. In actuality, p has a direction as this is space sampling. If px is considered to be information, then ln(probability(x))= px or probability x = exp(ipx). To obtain time sampling, one needs to freeze out motion i.e. exp(ipx)exp(-ipx) (AND situation). Thus the manner in which one samples an underlying distribution may give rise to a different distribution. Arguing that a particle spends the same amount of time in each dx region and so the spatial probability is constant completely ignores the form px from the classical action. exp(ipx) is associated with a wavelength, thus x samples are within different lengths for different p’s. This suggests an associated entropy which is not additive because one loses information when adding p1+p2. Probability exp(ip1x)exp(ip2x) = exp(i (p1+p2)x), however, is conserved as is the overall impulse p1+p2 which depends only on the net momentum, whereas p1 entropy depends on a specific value.
Quantum mechanics is based on two different probability schemes, namely W(x) =wavefunction and W*(x)W*(x)=P(x), with one being linked to the other. Why should there be two probability schemes? In (0) we argued that there are two spatial behaviour schemes because an impulse p (momentum) is automatically associated with two spatial distributions cos(px) and sin(px) in exp(ipx), yet a particle with constant p moves as x=p/m t. Thus if one is only concerned with motion, each x point has the same weight, and this is physically measurable as a photon or electron travels a certain distance in a certain time in free space. On the other hand, an impulse interaction is associated with exp(ipx) is a completely different spatial picture, and this too is measurable as in a 2-slit experiment. Thus it seems the two schemes are measured differently. We argue that this is an important issue because if there are two x-probability schemes there should be different entropies associated with each. Ttraditionally (1), one sees two quantum entropies: - W*(x)W(x) ln {W*(x)W(x)} and - a*(p)a(p) ln{ a*(p)a(p) } where W(x) = Sum over p a(p)exp(ipx). These are associated with the x=p/m t scheme or x-dependence based on motion or average motion. W*(x)W(x) represents spatial density which should be measured presumably by some sort of scattering, but we ask whether some information is lost in this approach? In particular, we note that a wavefunction may be obtained for a bound state. In the high energy limit, the envelope over the density W*(x)W(x) is said (2) to match C/v(x) from classical physics where .5mv(x)v(x) + V(x) = E. C/v(x), however, is proportional to delta t (with delta x being constant) and has nothing to do with the strength of an impulse hit. In the classical case, there is a single v(x) and so a single p(x) in each delta x thus no impulse entropy within a delta x i.e. at a point x. In quantum mechanics, however, we argue that a constant p automatically has an x probability distribution exp(ipx) associated with it and there might be an entropy linked to this.This entropy is associated with a probability to receive a p impulse at x and is so not linked to interference while the W*(x)W(x) is. In (3) we introduced the impulse or wavelength entropy (for a free particle |cos(px)|/C ln{| cos(px)|/C},, but here we wish to investigate its properties more closely.
In general, x and t are considered free variables taking on all real values (with t>0). A particle’s velocity may be calculated from x(t). For the Lorentz invariant A= -Et+px one may consider E for a free particle to follow from Integral F dx and p from Integral Fdt. If one wishes to put these on an equal footing, then Fdx requires a dt and Fdt a dx. This means t is replaced with dt and x with dx in -Et+px. What does it mean to have t and x intervals associated with a specific E,p and continuous x(t) motion? There does not seem to be either an x or t interval. There are, however, dx and dt intervals associated with the creation of E and p from p=0.In other words, these dt, dx intervals are physically real properties just like p and E and are manifested experimentally. If one imposes time reversal symmetry, not only should these intervals represent a kind of probability, but they should be symmetric. Thus, x=0 and t=0 are starting points such that A=0. A is a Lorentz invariant and so the end point should also have A=0 for time reversal to apply. This time reversal leads to two symmetric distributions, one in time with hbar/E as a period and one in space with hbar/p as a wavelength.This, however, does not show the direction of motion, say in the spatial distribution. Thus the symmetric x-probability distribution should ultimately be linked to both p and -p i.e should have a vector nature, although this has to be formally proved. A full x-probability distribution linked with p should show the direction of motion. One may write F(px) for the even symmetric function described above and G(px) and G(-px) for skewed full probability distributions such that G(px) + G(-px) = 2F(px). This implies G(px) = F(xp) + B(px) where B is an odd function. We discussed the idea of G(px) being the sum of an odd and even function in previous notes, but did not try to give a reason why F(px) should be an even function with probability endpoints of 0. In this note, we argue that this is due to time reversal invariance. A free Newtonian particle follows x= vt and this should hold on average for a quantum particle which travels through space measured by an external ruler and clock. Above, however, we discussed probability distributions due to p. In fact their sharpness d/dx is proportional to p. How does one eliminate x dependence? As we argued in a previous note (2), one may take the AND condition G(px)G(-px) which should remove all motion.This should be a constant. It is not just motion which is removed, but overall impulse which is the cause of the x-distributions. The square nature of G(px)G(-px) explains why quantum mechanics deals with square root probabilities. These are the probabilities which are linked to impulse, hence to interactions, but the spatial distribution cos(px) from exp(ipx), does not describe the motion x=vt which is also real.The problem, as we show, is that G(px) cannot be real. This forces one to adopt a two dimensional probability and ultimately obtain exp(ipx) i.e. the two dimensional vector one might expect.
In a previous note (1) we considered a free particle entropy based on the real part of the wavefunction exp(ipx) i.e. |cos(px)|. |cos(px|) may be normalized over a certain length, just as exp(ipx) is. The idea is that there exists an uncertainty in space within a wavelength which should be reflected in an entropy expression. As p becomes very large, cos(px) becomes very narrow and has many peaks in L. Thus probability →0 so p ln(p)--> 0 using l’Hopital’s rule. This coincides with a free classical particle having zero entropy. In this note, we revisit arguments presented previously, suggesting that A= -Et+px, a Lorentz invariant and also the free particle classical relativistic and nonrelativistic action for v=x/t We suggested in the past that A is responsible for both the wavelength hbar/p and the time period hbar/E. Here we suggest uncertainty within a time period should be considered together with uncertainty within a wavelength. For instance, recent experiments (2) have shown that one may have quantum interference in time as well as space. Thus there is uncertainty for a free particle in both space and time. We add the entropies for these two. Next, we consider the case of a bound system. Such a system creates a crest-trough pattern (with different heights and widths) for its spatial wavefunction, but E, average energy is constant and there is the free particle type exp(-iEnt). We suggest that the system is characterized by uncertainty in space and time with En being a known constant at each x. This suggests that various free particle p (momentum) values combine to create the wavefunction such that -1/2m d/dx d/dx W/W is average kinetic energy which in turn adds to V(x) to create a constant En. We are thus concerned with uncertainty in time and space and not in momentum. In the limit of high energy values, a crest fits within a tiny dx and prms is essentially the classical p value which is considered certain, even though in quantum mechanics the range of a(p) weights hold for all n (W(x)=Sum over p a(p)exp(ipx)). Thus we treat a high n limit as being completely certain and not having entropy. This differs from the approach of using C/v(x), where v(x) is velocity given by .5mv(x)v(x)+V(x)=En, as a probability and calculating Shannon’s entropy using this.
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.
For a Maxwell-Boltzmann gas (MB) it is known that T (temperature) is a statistical parameter related to the average energy. It appears in f(ei) = C exp(-ei/T) so T has the same units as ei, a single particle energy. If T increases, ei increases identically if ei/T is treated as constant. Thinking in terms of ei=pp/2m, this means the momentum spread (i.e. variance of the Gaussian increases) and one expects entropy to increase In (1) we argued that exp(-ei/T) for an MB gas and exp(ipx) for a quantum free particle (constant p), follow from the same statistical rule P(a1)P(a2)=P(a1+a2), but ei is a scalar and p a vector. Furthermore exp(ipx) is associated with spatial invariance and the generator of translations d/dx. In this note, we try to argue that a constant p may act as a statistical parameter just as T, although in this case hbar/p has the behaviour of a length i.e. the wavelength. If p increases (just like T is increased), the wavelength decreases meaning there are more peaks in a fixed length or more entropy, we argue. Thus there seems to be a statistical mechanical similarity to increasing T for an MB gas and to increasing p for a single free quantum particle. The association of x and p in exp(ipx) has implications, not only for the resolution of x space, but the increased entropy suggested by more peaks of cos(px). In addition, this means more peak places for impulses to occur. Thus a large p not only represents a large impulse, but is associated with more peak points for this impulse to be delivered and more entropy. In (2) we argued that the humps of cos(px) encode information about the impulse/force used to create p and which may be delivered if p hits an object. Thus there is a somewhat unusual relationship between p and x that one does not find in Newtonian mechanics, but has its roots in special relativity i.e. in the Lorentz invariant -Et+px. We suggest that just like exp(-V(x)/T) in an MB gas means that for a large force=-dV/dx, exp(-V/T) changes rapidly in x, a large p, associated with a large force should also imply that the humps in exp(ipx) rise (and fall) sharply, thus leading to sharp peaks. Thus sharp peaks are indicative of high internal force and increased entropy for a free particle exp(ipx), we argue.