In Newtonian mechanics, one often thinks in terms of x(t) i.e. an association between x and t. Although this is part of Newtonian mechanics, the notion of motion always being linked to time (and so an internal clock being required) is not always the case. For example, two particles with different masses and different velocities such that their momenta are the same may strike a target and reflect with the same magnitude of momenta. The impulse is 2p for both even though they have different speeds. One does not need to consider their speed to discuss the impulse change and so an external clock is not needed. This is further emphasized by the Lorentz invariant A = -Et+px which separates x and t as if they were independent variables (even though implicitly a speed exists in the problem i.e. v=dx/dt.). The point is that there exists a subset of physics (i.e. interactions) which is not dependent on velocity and hence the presence of a clock. The impulse hit discussed above is one example, but interference through 2 slits etc is another. Two quantum particles with the same momentum, but different rest masses and velocities produce the same interference patterns. A particle with momentum, however, has energy and so both time and space are running variables, but there are problems in which one considers p without a clock. Interestingly, even if one thinks in terms of impulses without using a clock, if there is an x interval between impulses then the notion of velocity comes into play even without an external clock because there will be more impulses imparted by a fixed p with a higher velocity than a lower one in the same fixed time period. Without using the idea of a clock, this give rise to the observable called pressure in classical physics, but is also present in a quantum bound state, we argue. Pressure in classical physics is proportional to temperature which in turn is proportional to energy (and a running time). In a quantum bound state, the equivalent to pressure (namely impulse hits with a potential) also require an energy viewpoint which is linked to running time i..e exp(-i En t) where En is the bound energy.
In Newtonian mechanics, one often thinks in terms of x(t) i.e. an association between x and t. Although this is part of Newtonian mechanics, the notion of motion always being linked to time (and so an internal clock being required) is not always the case. For example, two particles with different masses and different velocities such that their momenta are the same may strike a target and reflect with the same magnitude of momenta. The impulse is 2p for both even though they have different speeds. One does not need to consider their speed to discuss the impulse change and so an external clock is not needed. This is further emphasized by the Lorentz invariant A = -Et+px which separates x and t as if they were independent variables (even though implicitly a speed exists in the problem i.e. v=dx/dt.). The point is that there exists a subset of physics (i.e. interactions) which is not dependent on velocity and hence the presence of a clock. The impulse hit discussed above is one example, but interference through 2 slits etc is another. Two quantum particles with the same momentum, but different rest masses and velocities produce the same interference patterns. A particle with momentum, however, has energy and so both time and space are running variables, but there are problems in which one considers p without a clock. Interestingly, even if one thinks in terms of impulses without using a clock, if there is an x interval between impulses then the notion of velocity comes into play even without an external clock because there will be more impulses imparted by a fixed p with a higher velocity than a lower one in the same fixed time period. Without using the idea of a clock, this give rise to the observable called pressure in classical physics, but is also present in a quantum bound state, we argue. Pressure in classical physics is proportional to temperature which in turn is proportional to energy (and a running time). In a quantum bound state, the equivalent to pressure (namely impulse hits with a potential) also require an energy viewpoint which is linked to running time i..e exp(-i En t) where En is the bound energy.
The idea of impulse, i.e. Change in momentum = Integral dt Force appears in Newtonian mechanics. Impulse is considered to be a physical observable, for example it may be linked to the damage done by a particle hitting a target. Impulse, however, only depends on momentum and so two particles with the same momentum, but different masses and velocities may cause the same damage i.e. represent the same physical observable. Momentum in nonrelativistic Newtonian mechanics is m v where m is rest mass and v velocity. Velocity requires a clock in order to measure it, yet two particles with the same momentum and different velocities are considered equivalent in terms of the damage done upon collision with a target i.e the physical impulse. Thus we suggest that time must formally disappear from mv, but this implies that rest mass must behave as t or dt. Furthermore, given Kinetic energy = .5mvv, it must behave as 1/dt. In other words, the reciprocal of kinetic energy behaves as a gradation of time. This is not a standard clock as the gradation is based on a specific kinetic energy and is different for another. In other words, two particles with the same p and different v, have different time gradations. This, however, is a quantum mechanical result. Thus we argue that quantum mechanics may be responsible for the idea of p yielding the same impulse hit for two different v. We further argue that these ideas may be extended to include rest mass with v=0 and to indicate that 1/p is linked to dx, a ruler gradation which is again a quantum mechanical result. Lastly, we argue that these ideas lead to the special relativistic result EE = ppcc + mmcccc and so argue that free particle quantum mechanics and special relativity are closely linked and that classical impulse follows from free particle quantum mechanics.
In special relativity, energy and momentum are part of a 4-vector, but are also properties of a particle. X and t also form a 4-vector, but are not properties of a particle, but rather are measured relative to an external ruler or clock. Shifting the origin of the ruler/clock changes their value, but does not change the value of velocity and hence momentum.In (1), we called x and t running variables. Special relativity deals with observations based on measurers in different frames moving at constant velocities. The ruler or clocks used by these people, however, may be shifted so there is no absolute property of time or x in this case either. Shifting a ruler or clock does not change the value of velocity, it is a property of the particle (assuming there are equally space gradations and clock intervals). In order to make x and t properties of a particle, we argue (as in (1)) that one may use A= -Et+px, a Lorentz invariant. For -EE+pp, (c=1), another Lorentz invariant, there is a link between all frames with rest mass squared (c=1). Within a specific frame (E,p exact numbers), -Et+px creates a link between special degenerate (in terms of A) x and t points, i.e. x+ n constant/p and t + n constant/E. These keep A constant due to the minus sign. Thus E and p create an internal clock and ruler for a particle as we have argued in (1). This means that n constant/p are special “x” points (and the same with n*constant /E and time) and suggests a probability distribution in x with a period of constant/p. Thus E and p interacting with x,t allow the latter to be properties of a particle, but at the cost of introducing probability. Given the notion of probability, this suggests that entropy should come into play. One may ask: Is entropy maximized subject to a constraint? If so, what constraint should one use? We argue for a constraint of ipx associated with the period constant /p, the “i” being associated with periodicity. In such a case, the Lorentz invariant -Et+px leads to exp(ipx) as a dynamic probability without the use of an eigenvalue equation.
Newton’s second law dp/dt = F(x)=-dV/dx may be written in terms of x alone i.e. dp/dx v(x) = -dV/dx or 1/(2m) d/dx pp = -dV/dx leading to the idea of kinetic energy pp/2m and potential energy V(x). In (1) we argued that there is velocity, rest mass mo, and momentum in quantum mechanics. A free particle, however, does not undergo smooth acceleration as in classical mechanics, and we argued there exists a momentum space wave exp(ipx) accompanying the moving particle. In such a scenario, force is considered to be probabilistic momentum hits of the form Vk exp(ikx) leading to a free particle momentum distribution W(x)= Sum over p a(p) exp(ipx). Both of these expressions involve interference due to momentum waves which do not track a particle in time. A mass flux mov or energy flux p=Ev relativistically seems to be the main consideration. Nevertheless, summing Vkexp(ikx) over k to obtain V(x) yields a function varying in x alone. This is a kind of average based on interference and Sum k Vk exp(ikx) is an average linked to momentum hits. We speculate that average momentum hits i.e. -d/dx V(x) are linked to momentum flux changes. In other words, even though an impulse is based on p and does not distinguish v i.e. p=m1v1=m2v2, momentum flux is velocity dependent and appears for an average momentum hit i.e. -dV/dx. In such a case (pave+dpave)(vave + dv ave) = 2 pave dv(ave) . This expression, however, contains both forward and backward motion which is part of quantum mechanics. Moving to an average picture seems to introduce time and direction i.e one uses pave d v(ave). This is already a change from a quantum to a classical scenario albeit for averages (which have not been specifically defined). Thus we consider pave dv(ave). If this is the case, then: d/dx pave 1/m space = -dV/dx or 1/(2m) pave pave = -d/dx V(x). Instead of treating pave pave as two separate averages one considers pave=prms or pavepave=<pp>. Thus the quantum bound picture (at any energy level) becomes a classical type of scenario which includes time and velocity. Note that for a two slit interference pattern, one does not need velocity and time because the interaction with the potentials from both slits is based on exp(ipx). One does not even need the form of the potential. Interference is the same for p=m1v1=m2v2 (even though the process takes place during different time intervals, the pattern is not affected). In other words, we argue that Newton’s second law might follow from a picture of momentum hits in which a flux i.e. p=mov is treated as a quantity which in an average scenario leads to a second flux i.e. an average momentum flux which reintroduces direction, time and velocity.. This is somewhat like a hydrodynamics scenario.
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.
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.
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.