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.
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.
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.
Traditionally, Newtonian mechanics is recognized as an approximate theory which is said to arise from quantum mechanics (apparently a more detailed theory) in the large energy level limit (correspondence principle). We argue that both quantum mechanics and Newtonian mechanics are both approximate and both complement each other at any energy level. In the Newtonian case, quantum effects are so small that they are not visible, but in the quantum case, Newtonian effects are visible, we argue. We first note that exp(-iEt+ipx), the free particle wavefunction (complex probability) does not exist without the Newtonian expression x=vt. x=vt implies it is possible for dx->0, dt->0, while hbar/p and hbar/E show that there are limitations with respect to interactions. The only way to ascertain x=vt is through measurement and if these measurements have dt=hbar/E and dx=hbar/p uncertainties, one cannot have dx->0, dt->0 resolution. Nevertheless, the quantum free particle does move as x=vt (as long as one respects the uncertainties of dt,dx).The quantum exp(-iEt+ipx) does not exist without the Newtonian x=vt. In a single particle quantum bound state (time-independent), we suggest there are significant approximations. First, a conservation of energy equation exists in Newtonian physics, namely pp/2m + V(x) = E ((1)). This equation does not contain time and does not explicitly show x(t), but that does not mean that such motion does not physically exist. The quantum bound state solution follows from ((1)) which is an equation which does not display full information, even in the classical case, and so this loss of information carries down to the quantum solution. In particular, in the quantum case one solves a time-independent problem and so uses a(p)exp(ipx) + a(-p)exp(-ipx) with a(p)=a(-p) or a(p)= -a(-p). These conditions are equivalent to insisting that the local momentum average at each x (-i W* grad W + i W grad W) is 0. The fact that one is solving a quantum bound state using only energy considerations means that one averages out time which means averaging momentum. Thus, W*(x)W(x) describes a resolution spatial pattern for interaction, but W = W(x) exp(-iEt+ip x) where p=0 at each x. In other words, just because one does not see forward/backward motion in a particular quantum solution (based on energy alone) does not mean that it does not exist and that it cannot be measured, we suggest. For high energy levels, one should be able to measure such motion. The factor exp(-iEt+ipx) is a further approximation of the bound state which may be used if one wishes to describe a particle of mo in a potential V(x) created by a particle of mass M as an M+m+E rest mass system. Thus, one may see that various levels of approximation are used in the same quantum bound state solution. Furthermore, classical and Newtonian mechanics are still linked at any energy level because the quantum average KE (in the nonrelativistic case) must equal the classical kinetic energy. Thus, the notion of x= vrms(x) t is present and if hbar/(m vrms) << dx (where dx <<L with L being the system size), then one again has exp(-ipx+i KE t) and x=vrms(x) within each tiny dx, as an approximation.
Netwonian mechanics and quantum mechanics seem to be very different theories, even though Newtonian concepts like momentum, energy and their conservatioon carry into quantum mechanics. The difference is not even due so much to the case that Newtonian mechanics is deterministic and quantum mechanics, statistical, we argue. We suggest that quantum mechanics utilizes a turly dynamic probability which treats forwards and backwards motion differently thorugh exp(ipx) and exp(-ipx), while Newtonian mechanics treats them in essentially the same manner, with the exception of the direction. We thus suggest that Newtonian mechanics is really based on the notion of stationary objects. In particular, x(t) means that a particle is at x in dx at t, as if it were stationary with the exception that it is translating with a fixed velocity. This stationary viewpoint might describe why the P(x)=1/L probability for stationary objects is sometimes also applied to objects moving with constant speed. The x(t) approach really means at t, the particle is frozen at x for dt before appearing frozen at x+dx at t+dt with a new velocity. There is of course the notion of acceleration, but for a particle to be at x at t, one must measure it and this means that it is as if it is stationary for an instant. The velocity is then considered to be constant at x (i.e. a translation) with a new translation value at x+dx. As a result, the notion of quantum mechanics does not really follow directly from this Newtonian stationary viewpoint. One has to think in terms of a very different idea, namely that exp(ipx and exp(-ipx) are not merely translations in different directions. A truly dynamic theory is needed. As we have argued in previous notes, P(x)=1/L represents a stationary object and so if one had an AND condition for p and -p (i.e not speed), then exp(ipx)/sqrt(L) exp(-ipx)/sqrt(L) = 1/L. Freezing out speed using quantum dynamic probability exp(ipx) (exp(-ipx)) leads to a stationary picture, but the actual physics of exp(ipx) is very different from anything in Newtonian mechanics. This leads to the question: What happens in the high energy (i.e. classical limit) if Newtonian mechanics and quantum mechanics are so fundamentally different? Quantum mechanics differs from classical physics in that there are fixed uncertainties of dt = hbar/E and dx=hbar/p for a particle with E,p, while Newtonian mechanics uses infinite resolution. One may note that exp(ipx) and exp(-ipx) are very different, but that is only if one can resolve x regions on the order of hbar/p. If one has worse resolution, one would not notice that exp(ipx) and exp(-ipx) differ. They are both possibly present in a dx which is the resolution limit. Thus, it is actually not the infinite resolution of Newtonian mechanics that is manifest, but rather the approximate finite resolution which means that on classical scales, a tiny dx is actually larger than hbar/prms where prms is the root means square momentum of a quantum bound solution. The quantum bound state solution is based on finding energy levels and a wavefunction W(x) = Sum over p a(p)exp(ipx). It includes exp(ipx) and exp(-ipx), but physically in the classical limit, the particle moves to the right for some time and then later to the left. Quantum mechanics cannot change this measured situation. As noted, exp(ipx)*exp(-ipx) =1 and this is linked with a stationary object even though exp(ipx) and exp(-ipx) both represent dynamic probabilities. The dynamic probabilities allow one to correctly solve problems at scales of hbar/p. W*(x)W(x) is an attempt to create a stationary distribution which is consistent with the kind of measurements made in the classical world. It is W(x), however, that is actually used in solving problems. As the energy level increases, the peaks of W*(x)W(x) become increasingly narrow and so even this stationary-quantum approach is replaced by the simpler, approximate exp(i prms x) * exp(-i prms x), leading to a stationary free particle distribution. This is why we suggest that Newtonian mechanics is really a stationary free particle solution with the particle then translating from one x (within dx) to the next in dt. In other words, x(t) means that the particle is at rest in t (for an instant as it must be measured). The v(x) = dx(t)/dt is called the constant speed in dx (as it must be measured) even though there is acceleration occurring. It seems that Newtonian mechanics is a static kind of viewpoint which is very different from the dynamic exp(ipx) and exp(-ipx).
Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. In classical mechanics, one has the conservation equation: p(x)*p(x)/2m + V(x) = E. In quantum bound systems one has: KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component p*p/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that: p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x) ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)
Both quantum mechanics and classical statistical mechanics predict the same momentum and spatial density forms forms for the solution of an oscillator. In the case of quantum mechanics, the ground state is considered. One may ask whether this is a coincidence, and if not, why two seemingly different theories should lead to the same result for the oscillator case. We argue in this note that it is the form of the conservation equation of the oscillator which leads to x and p having the same probability distribution.Furthermore, given that both quantum mechanics and classical statistical mechanics are of a statistical nature, one may inquire into the actual nature of the statistics being used, the meaning behind averages being used. For example, the classical deterministic solution of an oscillator can be written in statistical terms by removing time dependence and using density = dx/v. This does not mean, however, that the actual physical motion is no longer deterministic. In addition, we try to argue that quantum mechanics may be a more detailed statistical model including dynamics which satisfies the more static statistical result of spatial density for classical statistical mechanics.
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.