Classical mechanics was developed in the 1600s and is considered a complete theory in that it matches observables in the macroscopic world. In the 1920s, however, it was found that different mechanical rules seem to apply to the electron in an atom. In this note we try to argue that classical mechanics is a subset of quantum mechanics i.e. it is based on partial information or the dropping of information. In (1) we noted that a classical spatial density Cdx/v(x,) where v(x) is velocit, is sometimes presented in the literature (2). This expression is proportional to dt associated with a fixed length dx so the density is a relative one. If one considers two bound classical systems (a particle in a box) with different constant speeds (in the interior) then dt is different for each, but Cdx/v(x) does not distinguish. Thus we argue that information has been lost. The higher v particle not only has a higher impulse, but an increased frequency and we argued in (1) that quantum mechanics takes note of this extra frequency by increasing the number of crests associated with higher constant velocities. In this note, we point out that quantum mechanics itself is linked to the dropping of information. If one considers free particle wavefunctions W(x) for different momenta (i.e velocities if mo is the same), one has exp(i p1 x), exp(i p2 x) etc. These clearly have different information. They describe direction, but more importantly they have an associated wavelength proportional to 1/p. P(x)=W*(x)W(x)=constant and so this information, which is physical, is lost . P(x) is related to the question: What is the probability for a particle to be at x? If one considers cos(px) as a nonlocal spatial distribution which shows motion i.e. negative and positive parts of cos(px), then sin(px) is a spatially shifted distribution which helps to indicate direction, but also shows that all x points are the same. One has a peak of cos(px) at x=0 and a lower value at x=delta, but due to the motion, the peak may later be at delta just like a classical wave. This is what W*(x)W(x)=constant captures. It shows that even though cos(px) maps to x in a certain way with sin(px) in an extra dimension, overall the free particle has no preference for one x point versus another. There are two points we wish to make. First, classical mechanics seems to be based on P(x) i.e. it drops information about a wavelength. Space is broken into tiny dx units and the particle has a constant velocity in each even if it is accelerating. This is the Newtonian picture and leads to dp/dt = Force or pp/2m + V(x) = E. This information, however, is contained completely in quantum mechanics because for a bound state with V(x) one has: -1/2m d/dx dW/dx / W + V(x) = En. -1/2m d/dx dW/dx / W is exactly KE(x) classical. Quantum mechanics, however, displays more information than the classical picture. For example, only certain En are allowed and W*(x)W(x) leads to a spatial density which does not appear in classical mechanics. This leads to a second point. exp(ipx) contains information about a wavelength which is lost if one only considers P(x) for a free particle. If there is an interaction (two electrons shifted in space, a V(x) or a single or double slits etc) exp(ipx) acts like a probability and adds in OR or interaction situations. In such a case the information regarding the wavelength becomes physically crucial. In other words for an interaction this information does not drop out as it does in the classical scenario of pp/2m + V(x) = E. Thus we argue that classical mechanics is a subset of quantum mechanics. Wavelengths are tiny in the macroscopic world and so are simply not observed, allowing the subset scenario of pp/2m + V(x)=E (nonrelativistic) to be an excellent approximation. Indeed for every energy level of a quantum bound state this relationship also holds. It is the spatial density which is altered.
Classical mechanics was developed in the 1600s and is considered a complete theory in that it matches observables in the macroscopic world. In the 1920s, however, it was found that different mechanical rules seem to apply to the electron in an atom. In this note we try to argue that classical mechanics is a subset of quantum mechanics i.e. it is based on partial information or the dropping of information. In (1) we noted that a classical spatial density Cdx/v(x,) where v(x) is velocit, is sometimes presented in the literature (2). This expression is proportional to dt associated with a fixed length dx so the density is a relative one. If one considers two bound classical systems (a particle in a box) with different constant speeds (in the interior) then dt is different for each, but Cdx/v(x) does not distinguish. Thus we argue that information has been lost. The higher v particle not only has a higher impulse, but an increased frequency and we argued in (1) that quantum mechanics takes note of this extra frequency by increasing the number of crests associated with higher constant velocities. In this note, we point out that quantum mechanics itself is linked to the dropping of information. If one considers free particle wavefunctions W(x) for different momenta (i.e velocities if mo is the same), one has exp(i p1 x), exp(i p2 x) etc. These clearly have different information. They describe direction, but more importantly they have an associated wavelength proportional to 1/p. P(x)=W*(x)W(x)=constant and so this information, which is physical, is lost . P(x) is related to the question: What is the probability for a particle to be at x? If one considers cos(px) as a nonlocal spatial distribution which shows motion i.e. negative and positive parts of cos(px), then sin(px) is a spatially shifted distribution which helps to indicate direction, but also shows that all x points are the same. One has a peak of cos(px) at x=0 and a lower value at x=delta, but due to the motion, the peak may later be at delta just like a classical wave. This is what W*(x)W(x)=constant captures. It shows that even though cos(px) maps to x in a certain way with sin(px) in an extra dimension, overall the free particle has no preference for one x point versus another. There are two points we wish to make. First, classical mechanics seems to be based on P(x) i.e. it drops information about a wavelength. Space is broken into tiny dx units and the particle has a constant velocity in each even if it is accelerating. This is the Newtonian picture and leads to dp/dt = Force or pp/2m + V(x) = E. This information, however, is contained completely in quantum mechanics because for a bound state with V(x) one has: -1/2m d/dx dW/dx / W + V(x) = En. -1/2m d/dx dW/dx / W is exactly KE(x) classical. Quantum mechanics, however, displays more information than the classical picture. For example, only certain En are allowed and W*(x)W(x) leads to a spatial density which does not appear in classical mechanics. This leads to a second point. exp(ipx) contains information about a wavelength which is lost if one only considers P(x) for a free particle. If there is an interaction (two electrons shifted in space, a V(x) or a single or double slits etc) exp(ipx) acts like a probability and adds in OR or interaction situations. In such a case the information regarding the wavelength becomes physically crucial. In other words for an interaction this information does not drop out as it does in the classical scenario of pp/2m + V(x) = E. Thus we argue that classical mechanics is a subset of quantum mechanics. Wavelengths are tiny in the macroscopic world and so are simply not observed, allowing the subset scenario of pp/2m + V(x)=E (nonrelativistic) to be an excellent approximation. Indeed for every energy level of a quantum bound state this relationship also holds. It is the spatial density which is altered.
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.
Velocity plays a central role in classical mechanics, yet quantum mechanics deals with the variables E,t,p and x. P nonrelativistically is mo v, but a given p value corresponds to various m1v1=m2v2=m3v3 and so does not determine v uniquely. (A p, E pair or a p,mo pair is needed for this.). Here we consider a quantum particle in a square well potential. If detection of a particle occurs through impulse= Integral F dt, then m1v1=p and m2v2=p yield the same impulse although the velocities may be completely different. If one seeks a “time-independent” scenario then one would expect the two to yield the same spatial probability distribution even though E is different for each. In such a case E1 and E2 distinguish between v1 and v2. For a particle in an infinite well v1 and v2 represent different frequencies and so this “frequency” information role is taken on by energy. What happens if one considers two different average momenta levels for the same single particle? In such a case mo is fixed and p1 and p2 correspond to different v1 and v2 i.e. to different frequencies. There is no time in a time-independent scenario so how does one represent the relative frequencies? One may do this spatially, we argue. If p1=n1 3.14/L and p2=n2 3.14/L where L is the well size then p1 has n1 crests and p2, n2 crests. Thus in one case relative velocity for the same p is distinguished by E1 and E2, but not by any spatial distribution. In the second for the same mo and different v1 and v2, there is both a different spatial distribution describing the relative frequencies, but also different E’s which also distinguish and are linked to frequency. This leads to the idea of probability crests with certain base widths proportional to 1/p. What happens in the case of a free particle? Such a particle should also be associated with crests linked to 1/p, but the particle is traveling through space. It is not restricted to L and so its probability density should mimic classical mechanics i.e. be a constant everywhere. To allow for this one considers a probability distribution of crests and troughs together with a second spatially shifted distribution indicating the direction and nature motion , but with the constraint that the modulus is 1. This suggests cos(px) and sin(px) for the two distributions. This is very different from the classical mechanical scenario of a bound state spatial density (1) which is proportional to dt with fixed dx i.e. Cdx/v(x) where v(x) is velocity. Such a picture only accounts for relative densities in space. Thus for two different constant velocities v1 and v2 which represent different frequencies of traversing a distance L, the spatial density is the same classically. Thus important information is lost and we argue that Cdx/v(x) is incomplete. In quantum mechanics the envelope of P(x)=W*(x)W(x)dx approaches the classical value for high n (energy level), but quantum mechanics contains much more information, namely the idea that different v for the same mo have different frequencies which must be manifested spatially.
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 correspondence principle states that classical mechanics emerges from quantum mechanics in the appropriate limits. However, beyond this heuristic rule, an information-theoretic perspective reveals that classical mechanics is a compressed, lower-information representation of quantum reality. Quantum mechanics encodes significantly more information through superposition, entanglement, and phase coherence, which are lost due to decoherence, phase averaging, and measurement, reducing the system to a classical probability distribution. This transition is quantified using Kolmogorov complexity, where classical systems require O(N) bits of information, while quantum descriptions require O(2N), showing an exponential reduction in complexity. Further justification comes from Ehrenfest's theorem, which ensures that quantum expectation values obey Newton's laws, and path integral suppression, which eliminates non-classical trajectories when S≫ℏ. Thus, rather than viewing quantum mechanics as an extension of classical mechanics, we argue that classical mechanics is a lossy, computationally reduced encoding of quantum physics, emerging from a systematic loss of quantum correlations.
It was argued in (1) that Fermat’s least time principle for light (in a two dimensional (x,y) reflection or refraction scenario) minimizes time by actually varying it in terms of a spatially invariant variable i.e. x (i.e. taking d/dx) with the flat face of the mirror/medium along the x axis at y=0. This is equivalent to conserving momentum in the x direction. Thus d/dx is associated with (px) which may be associated with the Lorentz invariant -Et+p dot r. In other words, d/dx emerges as a momentum operator (up to a factor) and d/dt as an energy operator, yet both are generators in time and space linked with invariance of constant values i.e. conservation. Furthermore an eigenvalue equation: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt) automatically (by the nature of eigenvalue) leads to orthogonal eigenfunctions linked to conservation of energy and momentum. In special relativity rest mass mo represents a constant in time which may exist at any point x. In fact, it is often thought of as a point, for example a center of mass point. The actual object with mo, however, may be large with moving parts all within a “container”. What then does x,t in -Et+px mean for constant E,p? It seems that x,t are center-of-mass coordinates for the overall system. We argued in previous notes that 1/p is proportional to a wavelength so for p→0, mo may be at any x point with the same probability. Within the system with moving parts, however, there is motion and an x scale for each moving piece. For a single particle bound system, there is a single energy at each x which is like mo and a potential V(x) and KE(x). In (2) we argued that given the free particle relationship -EE+pp= -momo (c=1) one may retain a constant E value by replacing E with E-V(x) and pp with <pp> (and not ). How do the probabilities appear? They are linked to -Et+px with frequency hbar/E and wavelength hbar/p and are specifically manifested by exp(-iEt) and exp(ipx). For and <pp> we use spatial probabilities i.e. W(x) = Sum over p a(p)exp(ipx). Each exp(ipx) is linked to the operator -id/dx and invariance, but pp is linked with the operator -d/dx d/dx and one does not use an eigenfunction of this operator. Thus the notion of conservation of momentum is retained in the exp(ipx)s and their linear combination W(x) yields a function such that an average kinetic energy exists at each x point i.e KE(x)=KEclassical = -1/2m d/dx dW/dx / W. This is combined with a V(x) to ensure constant E at each x point i.e. create a kind of mo piece linked with id/dt. We argue that E being constant at each x is a statistical equilibrium type of result. Each exp(ipx) is associated with an average motion of x=p/m t and given that exp(ipx) and exp(-ipx) are included, there is also the notion of which is positive and negative in different regions and integrates to 0. prms(x) from <pp> becomes the classical p(x). Thus bound state quantum mechanics seems to be based on a free particle treatment which follows from the special relativistic Lorentz invariants -EE+pp=-momo and -Et+px = constant, directly linked to conservation ideas. Conservation of momentum of an internal bound particle is maintained through exp(ipx), while <pp> by itself is not linked with conservation as there is no eigenstate, but is combined with a V(x) to create a constant En linked with the entire system (and P=0) which then becomes associated with invariance in time and the generator id/dt. In this note, we try to argue that the notion of W*(x)W(x) (where W(x)=wavefunction= Sum over p a(p)exp(ipx)) is spatial density and P(p)=a*(p)a(p) based on spatial invariance and conservation of momentum for a single particle bound state even though there seems to be no spatial invariance or notion of conservation of momentum in W(x) which is simply a solution of the time-independent Schrodinger equation. One may move from a W(x) scenario (i.e. Schrodinger equation) to a W*(x)W(x) picture, by trying to “pick out” invariant (p only terms) using the orthogonality of the exp(ipx)s. Thus W*(x)W(x) (spatial density) emerges naturally in this approach.
In the 17th century, it was thought that motion of macroscopic and microscopic objects could be well-expressed by using the laws of classical mechanics. But towards the end of 19th century, experimental studies showed that classical mechanics failed when applied to microscopic objects. This was mainly because classical mechanics did not take into account the concept of de-Broglie’s dual nature of matter and Heisenberg’s uncertainty principle. Also, classical mechanics does not put any restrictions on the values of dynamical properties (likes position, momentum, energy etc.) calculated for microscopic object. According to classical mechanics, microscopic objects can have any value of dynamical properties, ( exact, large or small is permissible). Also any number of properties can be calculated simultaneously for a system.