Quantum mechanics is often compared with classical mechanics, but it seems that the momentum/spatial treatment of bound quantum mechanics provides for a particle-wave duality with both features having an experimental basis. In bound state quantum mechanics, one has a density pattern often similar to a two slit pattern i.e. with humps, yet at the same time one may knock a particle out of the bound state and obtain a probability distribution a(p)a(p) of momentum states. Describing a particle simply by its momentum even in the bound state seems to be classical as does using pp/2m to describe the kinetic energy of a plane wave in the decomposition of W(x), the wavefunction. Wave aspects are linked to classical particle features like momentum and do not change the particle energy pp/2m. The wave, we argue, may be related to some type of periodic cyclic forward backward motion which creates a spatial probability needed to estables resolution on the order of a wavelength proportional to 1/p. It is often not seen unless there is a potential or other probability waves which may interfere with it In this note, we consider the scenario that momentum (i.e. mov in the nonrelativistic case) is directly linked to a kind of forward-backward cyclic motion of a quantum particle regardless of its spin. The fact that spin exists suggests some kind of internal motion. In the case of a photon, it may be directly linked to angular motion of energy, although for spin ½ matters are more complicated. This motion again does not change classical energy pp/2m unless spin interacts with a magnetic field etc. We argue that a bound state is stochastic and so various free particle momentum states exist without being restricted to an x position. This is the particle scenario. The internal cycling motion, however, gives rise to a resolution of the order of 1/p suggesting a spatial distribution to account for such a resolution. Furthermore, the internal cycling (i.e. forward and backwards motion) even for a particle traveling to the right requires two dimensions to describe overall forwards and backwards motion. The spatial density of the cycling motion may be associated with internal changes in velocity, but overall there is a particle with mass mo moving with v (nonrelativistic case) and kinetic energy pp/2m. This holds for a free particle, but also for a particle in an infinite potential well with a single average |p|. An issue occurs, however, if one wishes to form an average of different p values at the same point x. In such a case the spatial probability density exp(i pj x) differs for each p at x and so the average [p1 exp(i p1 xo) + p2 exp(i p2 xo) ] / [ exp(i p1 xo) + exp(i p2 xo)] becomes relevant. In other words, one takes the average momentum, say p1, and links it to the spatial probability i.e. multiplies two averages and then divides by spatial probability to find a momentum flow at a specific point within a wavelength region. For the particle aspect, such a resolution is non-issue. The particle is the same at each x point in space.
Quantum mechanics is often compared with classical mechanics, but it seems that the momentum/spatial treatment of bound quantum mechanics provides for a particle-wave duality with both features having an experimental basis. In bound state quantum mechanics, one has a density pattern often similar to a two slit pattern i.e. with humps, yet at the same time one may knock a particle out of the bound state and obtain a probability distribution a(p)a(p) of momentum states. Describing a particle simply by its momentum even in the bound state seems to be classical as does using pp/2m to describe the kinetic energy of a plane wave in the decomposition of W(x), the wavefunction. Wave aspects are linked to classical particle features like momentum and do not change the particle energy pp/2m. The wave, we argue, may be related to some type of periodic cyclic forward backward motion which creates a spatial probability needed to estables resolution on the order of a wavelength proportional to 1/p. It is often not seen unless there is a potential or other probability waves which may interfere with it In this note, we consider the scenario that momentum (i.e. mov in the nonrelativistic case) is directly linked to a kind of forward-backward cyclic motion of a quantum particle regardless of its spin. The fact that spin exists suggests some kind of internal motion. In the case of a photon, it may be directly linked to angular motion of energy, although for spin ½ matters are more complicated. This motion again does not change classical energy pp/2m unless spin interacts with a magnetic field etc. We argue that a bound state is stochastic and so various free particle momentum states exist without being restricted to an x position. This is the particle scenario. The internal cycling motion, however, gives rise to a resolution of the order of 1/p suggesting a spatial distribution to account for such a resolution. Furthermore, the internal cycling (i.e. forward and backwards motion) even for a particle traveling to the right requires two dimensions to describe overall forwards and backwards motion. The spatial density of the cycling motion may be associated with internal changes in velocity, but overall there is a particle with mass mo moving with v (nonrelativistic case) and kinetic energy pp/2m. This holds for a free particle, but also for a particle in an infinite potential well with a single average |p|. An issue occurs, however, if one wishes to form an average of different p values at the same point x. In such a case the spatial probability density exp(i pj x) differs for each p at x and so the average [p1 exp(i p1 xo) + p2 exp(i p2 xo) ] / [ exp(i p1 xo) + exp(i p2 xo)] becomes relevant. In other words, one takes the average momentum, say p1, and links it to the spatial probability i.e. multiplies two averages and then divides by spatial probability to find a momentum flow at a specific point within a wavelength region. For the particle aspect, such a resolution is non-issue. The particle is the same at each x point in space.
Early experiments linked to bound state quantum mechanics noticed discrete energy E levels. (Ignoring the issue of radiation from accelerating charges), classical physics predicts continuous E levels. If one tries a statistical approach i.e. thinks in terms of a “perceived” density dt=C dx/v(x) where v(x) is velocity, one again has continuous energy levels. Thus, experimental results may suggest a non-classical condition which affects both acceptable densities and energy levels. Bypassing Newtonian mechanics which traces a particle through x(t), we note that even classically a potential interacts with p and not v i.e. pp/2m - popo/2m = -V(x)+V(xo). This equation holds for p= mo v = (.5mo) 2v etc. Thus, energy flow i.e. p is of interest and if one does not trace the particle as x(t) or consider acceleration (which is not considered statistically), one may view energy flux p as being “spread” over the x axis, it seems. The length 1/p then corresponds to one unit of energy flux for the case of p. In other words, each p is associated with a length 1/p. If this length may be physically resolved, one immediate trial result suggests a spatial probability which varies over 1/p. In other words one may have a periodic function such as cos(px). This physical scenario seems to arise only when the quantum particle interacts with a potential. Calling 2*3.14/p = wavelength, this length is not associated with the velocity of the particle, but with energy flow. Each particle has an identical type of cycling within its own wavelength i.e. cos(px) behaviour as each wavelength represents one unit of energy flux. The wavelength limits resolution and leads to a probability scheme. In a bound quantum system dx associated with V(x) is usually much smaller than the larger wavelengths and so classical ideas such as p and pp/2m still apply, but are modified i.e. multiplied by probabilities which may be positive or negative. Having a spatial probability linked to density which also creates a probability scheme used to calculate average kinetic energy at each x based on 1/p type wavelengths is independent of classical physics. Trying to force average kinetic energy at x in the wavelength scheme to match KE(x) classical leads to discrete E values.
A quantum particle is often described as exhibiting dual particle-wave nature. A particle seems to be an object which obeys Newton’s laws/special relativity with a wave also being a phenomenon existing in Newtonian mechanics (special relativity). In other words the particle and wave are governed by x-t external variables which have infinite resolution. It was originally thought that x and t in addition to being external were absolute, but special relativity changed this idea. We have suggested in previous notes that a free quantum particle with energy E and momentum p creates its own space time i.e time intervals are proportional to 1/E and space to 1/p. This means no external clock is needed and different internal rulers and clocks exist for different E,p. It also suggests finite time and space intervals. In fact, an absolute is introduced namely A=0 = -Et+px with t and x being independent. This suggests that t→b/E and x→b/p. A is the free particle Lagrangian (both relativistic and nonrelativistic) with x and t being considered independent. Thus we suggest that the external x ruler and t clock of Newtonian mechanics represent an average of the internal picture and are also described by A, but with x/t=v and not x and t being treated as independent. In (1) we showed that having discrete x spacing with a running time (i.e. Netwon’s time variable) and insisting on spatial invariance leads to dA/dx partial = p. Thus instead of considering particle wave duality we suggest an underlying wave structure which creates its own spacetime intervals (1/E, 1/p), but may be mapped in certain situations to the external running x and t of Newton’s theory. For example, consider an electron (a quantum particle) with a fixed p moving over a distance much greater than its wavelength. It may be mapped to Newton’s external running x,t values with x=vt, but this is an average mapping into continuous x and t. In reality the situation is more complicated because there may be a Bohm-Aharonov effect occurring. Thus the underlying wave picture is in fact physically important and linked to interactions. Thus a self-ruler is not simply a measuring device - interactions may occur unlike in the case of the external ruler. Given that p defines a kind of spatial invariance approximated by a periodic function exp(ipx) (eigenfunction of -id/dx) and that Newton’s running time is not needed in a quantum bound state equilibrium, one may wonder what one should do in the case of a particle bound in a V(x) potential. In Newtonian mechanics V(x) suggests non-spatial invariance and v(x) velocity changes with x as well in contrast with exp(ipx) which is periodically invariant. In fact Newtonian mechanics (and Lagrangian/Hamiltonian as well) focus on d/dt i.e x(t). We have argued above that time intervals 1/E are independent of 1/p spatial ones. It may be noticed, however, that different p values lead to different wavelengths and so even though there is a kind of spatial invariance for each, when compared to each other this invariance is broken and superpositions of exp(ipx) may create local x probability densities. At the same they create spatial momentum densities because -d/dx (d/dx) 1/2m yields a kind of average kinetic energy when normalized. This seems to describe the x-p uncertainty relationship. In order to create spatial resolution (other than the built-in one in the wavelength) one needs to superpose different momenta, each of which are spatially invariant according to their periodic scheme exp(ipx). Thus one may create an average 1/2m <pp> and obtain a vrms. This may be described in terms of Newtonian external running x and t variables which have infinite resolution, but this vrms does not map to a physical state i.e to exp(i (mvrms) x) localized at x. Rather the various exp(ipx)’s which create vrms are the physical entities as may be seen by knocking a particle out of a bound state. The physical exp(ipx) momenta states have spatial invariance and so cannot be associated with a specific position in space. Only their superposition creates the notion of specific position in space, but then there is a distribution of p. This seems to be the basis for the x-p uncertainty principle. Thus we suggest that the particle of particle-wave duality is created through a mapping of a physical wave scenario to a continuous (infinite resolution) x-t scheme which in reality is an abstraction because each p,E free particle state has it sown internal clock-ruler i.e. 1/E and 1/p time and space units. Given that 1/E and 1/p have finite values the idea of dt and dx tending to 0 only makes sense as an abstraction which only holds in certain situations.
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.
It is sometimes stated in the literature (1) that quantum mechanics is based on conditional probabilities. If this is the case, one would expect such conditional probabilities to appear in a single particle bound state. A free quantum particle is associated with a probability to interact of exp(ipx) (linked to 2-slit interference, 1-D reflection-refraction at an n1-n2 junction etc), but the kinetic energy and momentum are pp/2m (nonrelativistic) and p at all x even though each x has a different phase according to exp(ipx). The momentum is p at x, but the probability for an impulse hit at x is not the same even though its modulus is. When different probabilities exp(ipx) add in an OR situation, this leads to a spatial pattern which no longer has a modulus of 1, i.e. translational invariance is broken. Such a pattern also occurs for a single particle bound state W(x)=Sum over p a(p)exp(ipx). W(x) can have 0 points, but the average kinetic energy -1/2m d/dx dW/dx / W is not 0 at such a point. One has, on average (a statistical calculation), no average interaction at such an x point. One might ask why this should have anything to do with average kinetic energy? If one writes a conservation law: KEave(x) + V(x) = 0, then one considers kinetic energy interacting with potential energy. If there is no interaction at some x point for a given p (not W(x) on average), then it should not be part of the conservation of energy at this point. One is only interested in kinetic energy which interacts with potential energy at each x point. Thus, a(p) exp(ipx) + a(-p) exp(-ipx) (with a(p)=a(-p)) may have a 0 point at some x and so it does not have an average interaction at such a point, i.e. pp/2m (a(p)exp(ipx) + a(-p)exp(-ipx)) = 0. As a result, in a bound state, one considers kinetic energy per p, -p multiplied by the average probability (p, -p). Different a(p)exp(ipx) + a(-p)exp(-ipx) (a(p)=a(-p)) weights in this average picture indicate the extent to which the particle interacts. This idea holds for a single p,-p pair, but one must normalize over all such pairs, leading to the idea of a conditional probability. Given that hbar/p must be “created physically” in space so to speak, one must have interaction probability which creates this physical situation even in cos(px)+ i sin(px). This leads to the idea of interaction probability being 0, positive, but <1, or even negative and it is the interaction which allows one to consider conservation of energy with a potential V(x) in the first place. As a result, one cannot include a full pp/2m if there is negative, 0 or only particle interaction. Thus, pp/2m is multiplied by 2 a(p)cos(px) /W(x) for a(p)=a(-p).
A classical bound state does indicate the direction of motion if one only considers kinetic energy pp/2m. It is the vector quantities, p and x, which indicate direction. In quantum mechanics, a free particle interaction probability (for impulse hits) is directional, i.e. exp(ipx) not= exp(-ipx). Unless one has a single value of pp/2m, KE+= sum over p>0 a(p)exp(ipx) pp/2m / Sum over p>0 a(p)exp(ipx) is not equal to KE- even if a(p)=a(-p). In order to use conservation of energy in quantum mechanics, one is forced to consider both a(p)exp(ipx) +a(p)exp(-ipx) with a(p)=a(-p) which is the correct statistical weighting for a bound state. One may note that exp(ipx) is a probability for interaction and is separate from x=p/m t even though the notion of this velocity is tied to the fact that nonrelativistic kinetic energy is pp/2m. As a result, one must use an average scenario for calculating kinetic energy which seems to be tied to an ensemble average. Such a notion should hold for all energy levels. If one considers the classical limit, then one has p(x)p(x)/2m in a tiny dx. If one were to simply use 1/sqrt(p)exp(ip(x) x)/sqrt(dx) in dx, then Integral W*W dx in dx = C/p. The relative weight of two adjacent dx regions is: p1/ p2 = p1 / (p1+dp). This is equivalent to probabilities proportional to dt= dx/v, where p=mv. Thus, the area represented by W*W captures the number of possible interactions in dx if attempted hits have a fixed period. We argue, however, that given the bound state conservation recipe, one must use a(p) (exp(ipx)+exp(-ipx)) to describe a dx (with p=p(x)), even in the high energy limit. exp(ipx) is simply an interaction pattern (and is a probability) and so if the particle is bound, the correct statistical thing to do for an ensemble seems to be to combine exp(ipx) with exp(-ipx). We then show that quantum mechanics lowers the number of interactions in an area with increased speed more than the classical case. In particular, we find instead of p/(p+dp), 1- dp/p - 2dp/(pp).
In previous notes, we argued one may formulate bound state quantum mechanics by assuming a stochastic potential which delivers “hits” of momentum k V(k) exp(ikx), while having an average form of V(x). A main constraint seems to be that the different k V(k) exp(ikx) hits may occur at any x point and that V(x) be of the form Sum over k b(kx) so that -d/dx b(kx) = -k b(kx). In other words, conservation of energy exists at the level of each hit as well as for the overall average. To force all of these constraints, a solution of b(kx)=V(k) exp(ikx) emerges which introduces periodicity and complex numbers. It should be noted that V(x)= Sum V(k) exp(ikx) so each term is actually a “piece of potential energy”. Furthermore, we argue that V(x) is decomposed for physical reasons i.e. V(x) is a low resolution result which incorporates all of the “pieces”. In a quantum mechanical situation, a “piece” V(k)exp(ikx) may act at t1 and V(k2)exp(i k2x) at t2. We argue that exp(ikx) should have physical meaning and leads to sin(kx) terms for the “k” and “-k” pieces. Thus, there is forward and backward motion which “cancel” i.e. have nothing to do with V(x). For a particle, however, in this region, there may be forward and backward motion, both of which have kinetic energy which should not be part of the average kinetic energy at x. The various physical (resolved) pieces of the potential lead to momentum distribution at each x point with P(p/x)= a(p) exp(ipx)/ W(x) (W(x)=wavefunction). In this case, one has a probability, not “pieces” of KE(x). At first, one may be alarmed by complex probabilities, but they are necessary to allow for cancellation of kinetic energy pieces due to the positive/negative values combined V(k)exp(ikx) pieces. Thus, even though KE(x) (average) matches classical results, the actual motion is more complicated. The potential seems to be the driver of the ensemble picture and should be linked to the microscopic physical nature of the potential. In this note, we wish to examine features and consequences of such a stochastic potential, including the idea of tunneling and infinite potential walls.