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.
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.
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.
Momentum and velocity are two well-known physical observables in classical physics (and quantum physics as well because quantum particles have impulse and move through space in a certain amount of time based on an external clock and ruler). They are, however, measured in different ways. Velocity describes motion through x in some time t, while p is measured through impulses. Thus one does not need a clock to measure momentum. It may be measured in terms of the damage done to a foil, compression of a spring etc. Thus p=m1v1=m2v2 etc., i.e. a numerical number associated with momentum does not automatically yield the velocity unless one knows rest mass. In many situations rest mass is known (i.e. one deals with an electron etc) and so there seems to be a tendency to consider momentum and velocity to be almost the same thing when they are distinctly different. In fact, we argue that the notion of wavefunction W(x) and density P(x)=W*(x)W(x) are associated with momentum and velocity respectively. In fact, for a free particle these two measurables are linked with completely different spatial probabilities. Constant velocity impies P(x)=constant while exp(ipx) describes the two-dimensional x-probability associated with p which is based on a constant mo and v. In special relativity for a free particle (i.e. p constant, v constant), there are two well known Lorentz invariants: EE = pp + momo (c=1) and A = -Et + px. One may note that v does not appear explicitly even though one may write E and p in terms of v i.e. E = mo/sqrt(1-vv) and p=Ev. The first invariant describes E in terms of rest mass and p (measured through impulse). The second involves p,E and x,t with x and t appearing as independent even though ultimately x(t) and v=dx/dt=constant = x/t if x and t both start at 0. This independence is crucial because it means one may consider fluctuations which keep A constant separately. For example, if t→ t+ hbar/E and x→ x+hbar/p, then A remains constant. This leads to the idea of the quantum wavelength and a spatial probability independent of t, namely exp(ipx). One average, however, x=vt, but how does this appear in Lorentz invariants? There is no explicit v. We argue that the Lorentz invariant A = -Et + px must convert E from a function of p to one in v and that this occurs through a Legendre transformation. Thus the form of the invariant is itself a Legendre transformation with E being transformed into L, the Lagrangian. There is, however, a second important feature because p=dL/dv (partial) and given that p is constant, dp/dt=0 so d/dt dL/dv partial =0 which is in the form of an extremization. We argue that this is not an accident. dL/dv=p follows from E being constant, so, in other words, it follows from the px term. This same term leads to exp(ipx) which means that one does not have smooth x=vt motion within a wavelength. Thus, we argue that when one transforms to the second measurable v, the notion of this fluctuation behaviour must also be manifested. In other words, v=constant or dv/dt is not enough. The extremization behaviour of the Lagrangian, we argue, is thus not a mathematical convenience, but gives an indication that even though the extreme path x=vt is followed on average, other paths are sampled. We argue that this idea is incorporated into special relativity, in fact within a single Lorentz invariant A = -Et + px.
The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.
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.
In classical physics, velocity follows from exact measurements of space and time intervals (i.e perfect resolution of both). Inertial mass follows from a resistance of an object to a force (push or pull). Momentum, however, which nonrelativistically is mv and should be associated with ideas of inertial mass, and space time resolution may be instead linked to the physical idea of an impulse. An impulse seems to be like a hit and does not appear to be linked directly to space time physically, although its consequences are i.e. a particle receiving the impulse moves in space and time. In classical mechanics, x(t) and so one considers either equally spaced dt intervals and corresponding dx(t) or vice versa (constant dx) if an integral only involves space dependent values. In quantum mechanics, one has two uncertainty principles, one related to E and t and the other to p and x. We argue that p is linked to the notion of an impulse which does not on the surface involve space or time. Given that p, even in nonrelativistic quantum mechanics, is mv there should at least be some built in notion of dx and dt. From the uncertainty principles one sees that dx (i.e. the wavelength) is linked to hbar/p and dt to hbar/E. Thus neither dt nor dx are constants, but depend on the motion. In the high energy quantum limit dt —> W(x)W(x) dx →1/v(x) dx where v(x) is classical velocity and W(x) the bound state wavefunction. The bound state wavefunction, however, applies to the envelope of WW (not the actual crests and troughs) and so the dx-dt scheme which emerges is a smoothed over one and not the actual dx -dp of quantum mechanics. We try to examine some of these ideas in this note.
In physics (including classical physics), conservation of momentum is always considered, but one must be more careful with conservation of energy as it may include forms like sound etc. As a result, there is a tendency to focus on conservation of momentum all of the time, and conservation of particle energy only in elastic collisions. In addition to this, one may think of a particle striking a target, in which case two equal momentum (but different energies due to different v speed and mo rest mass) cause the same impact. In the case of an interaction with a potential, however, it is strictly energy considerations which are important classically, i.e one uses p dot p /2m + V(x) = E. Knowing momentum without rest mass is not sufficient. One may consider it unusual that one may consider two different scenarios, one which focuses only on momentum (impulse) or conservation of momentum in collisions while only energy is the focus in classical bound states or motion in V(x). Given the classical focus on momentum, imagine that one wishes to describe all free particles with different momenta as a uniform probability distribution such that a product of two probabilities for p1,p2 is the same as for p3,p4 if p1+p2=p3+p4. (We use 1-dimension for simplicity.). One cannot use P= 1/n because p is absent and n->infinite. In one sense, all particles must have the same probability, but at the same time must contain extra information related to p for the conservation of p part of the problem. The notion of a uniform distribution is independent of conservation of momentum, hence of p momentum. Conservation of momentum requires an explicit presence of p. As a result, there seems to be a paradox. One may, however, resolve this paradox by arguing that a probability is a mathematical way of presenting a subset of the information present. If one argues for a uniform probability, then it does not need to contain p, and one would have 1 as an unnormalized probability for any free particle p. A different probability (representing a different subset of information) which respects all free particles as having the same weight, but includes p can also be constructed, namely exp(i C p). C is a constant introduced for unit issues. In this case, the modulus gives all free particle probabilities with p the same weight, but the presence of p allows one to have conservation of momentum. Given that momentum is the key variable that is conserved in classical physics, one should be satisfied with exp(i C p). One might argue that there is no need to consider energy E. We note, however, that if one has p, E1 and -p, E2 in a frame, then in a frame moving with -v, one has gp+gvE1= p1’ and p2’ = -gp + gvE2. Adding yields gv(E1+E2) and so energy automatically appears and cannot be discarded. Furthermore for p, E1 and -p, E2, exp(i C 0) is exp(i C gv(E1+E2)) as seen in the moving frame and so one is forced to consider conservation of energy together with momentum and include both p and E in the more detailed probability. In other words, special relativity automatically links p and E in a 4-vector, leading to the idea that both are important. We argue that one cannot simply consider exp(i C p) due to special relativity. Both E and p must be considered together and this is a key idea, we argue, because if one wishes to have a Lorentz invariant probability, it must be in E and p, i.e. -Et+px. Instead of having a probability for an interaction with p given by exp(i C p) , one has exp(-iEt+ipx) and we argue that this must have the same value in all frames for a particular interaction. One may note that E and p are, on the one hand, separated allowing for impulse hit versus E related interactions, but that overall the probability for interaction includes both and it is the full -Et+ipx which must be invariant. This further suggests that if one does not artificially force an interaction to occur at the center-of-mass, then x,t and x+hbar/p, t+hbar/E are equally good points for the interaction. We argue that there is no need to force x=vt because exp(-iEt+ipx) * exp(iEt+ipx) already shows that one has uniformity, not only for any p,E, but also for any x, t, as there are now four variables. Thus, there is no reason to enforce x=vt in exp(-iEt+ipx), when considering an interaction, we argue. This means that one has two possible interactions at different x,t values, i.e. x,t and x+hbar/p, t+hbar/E. If one is unable to follow the interaction in time, but may follow it in space as in the case of 2-slits about hbar/p apart or for a potential V(x), then one may use the exp(ipx) part of the interaction probability piece by itself. Thus, we argue that special relativity does not allow one to treat p independently of E. For example, for p values that are the same in one frame, but have E1, E2, will be different in another frame. One must consider E and p together and so instead of creating a probability for p alone, one must create one for E,p and then impose Lorentz invariance which leads to different x,t pairs yielding the same probability value for an interaction, i.e. free particle quantum mechanics.