Free quantum particles may be represented by a wavefunction of exp(ipx) where “px” suggests a resolution scale L with p=2(3.14)hbar/L. In this note, we suggest that such a resolution may already be present in classical mechanics. exp(ipx) is a periodic form which incorporates this resolution and through a modulus of 1 ensures equal probability to be at any point x. In order to use exp(ipx), it seems that there must be a physical mechanics associated with the resolution. Given that exp(ipx)=cos(px) + isin(px), we argue that two different probability cycling or flows exist, one related to energy (pp/2m) and time and the other to flux- momentum and space i.e. sin(px). The link between the two is: cos(px)cos(px)+sin(px)sin(px) =1 i.e. the two are out of phase. This appears to be similar to the two dimensions x and ict. Each cycle or flow has its own frequency with positive and negative mutually exclusive subdivisions within resolution pockets, 1/E for time and L with p=2(3.14)hbar/L for space. The frequencies are associated by positive and negative probabilities (linked perhaps with forward and backward motions etc) and lead to interference if one considers different momentum values. We attmept to consider these issues in this note. In addition, we examine ln(W) representing information, with W(x), being the wavefunction . The change in information is -i dx . Thus, it seems that momentum flow carries information in a quantum bound state.
Free quantum particles may be represented by a wavefunction of exp(ipx) where “px” suggests a resolution scale L with p=2(3.14)hbar/L. In this note, we suggest that such a resolution may already be present in classical mechanics. exp(ipx) is a periodic form which incorporates this resolution and through a modulus of 1 ensures equal probability to be at any point x. In order to use exp(ipx), it seems that there must be a physical mechanics associated with the resolution. Given that exp(ipx)=cos(px) + isin(px), we argue that two different probability cycling or flows exist, one related to energy (pp/2m) and time and the other to flux- momentum and space i.e. sin(px). The link between the two is: cos(px)cos(px)+sin(px)sin(px) =1 i.e. the two are out of phase. This appears to be similar to the two dimensions x and ict. Each cycle or flow has its own frequency with positive and negative mutually exclusive subdivisions within resolution pockets, 1/E for time and L with p=2(3.14)hbar/L for space. The frequencies are associated by positive and negative probabilities (linked perhaps with forward and backward motions etc) and lead to interference if one considers different momentum values. We attmept to consider these issues in this note. In addition, we examine ln(W) representing information, with W(x), being the wavefunction . The change in information is -i dx . Thus, it seems that momentum flow carries information in a quantum bound state.
In a previous note, we argued that a quantum wavefunction W(x) is a relative conditional probability at x such that: W(x+dx)=W(x)exp[i dx d lnW/dx (1/i)] ((1)) where = -i d/dx lnW and <pp ave at x>= - d/dx d/dx W/W. The latter may be utilized in an energy conservation equation to yield either the Schrodinger or Klein-Gordon equation. W(x) in general is periodic with positive and negative values. Alternatively, one may write: W(x)=Sum over p a(p)exp(ipx) which implies a particle may be in an instantaneous free momentum p state. These states “interfere” not just at the exp(ipx) level, but also when forming averages of p or pp/2m because exp(ipx)=cos(px) + i sin(px) and both functions take on positive and negative values. One may argue that “coarse graining” occurs removing extra information contained in the exp(ipx)’s. We wish to investigate this further in this note. In particular, we wish to show that ((1)) together with KEave(x) + V(x) = E seems to force interference upon exp(ipx) components.
In Part III we considered the quantum free particle wavefunction exp(ipx) as consisting of two parts cos(px) and sin(px) treated as two different dimensions. We also argued that each, is a probability distribution in space and so associated with an internal force. One may notice that exp(ipx) does not contain time, but still predicts the direction of motion. Given that one cannot consider x(t) within a wavelength hbar/p even though on average the particle moves as x=p/m t, there is no real sense of time within a wavelength. Extending this idea, one repeats cos(px) humps. Given that the particle is moving, one must somehow juxtapose two spatial probability densities. The derivative of the distribution is proportional to average momentum and so one may see the notion of action-reaction occurring between cos(px) and sin(px) and within cos(px) and sin(px) accounting for the sign changes with x. We argue that d/dx cos(px) regions which change sign imply that the impulse direction changes. Furthermore, cos(px) has regions near the endpoints of the wavelength which have very low values (close to zero) which is fine if one has two juxtapositioned distributions. If, however, one creates a single particle bound state, exp(ipx) and exp(-ipx) for a(p)=a(-p) (a specific parity) give the appearance of a cancelation leaving the impression of 2cos(px) i.e a single “static” distribution. This occurs because there is no internal time in the picture and so p and -p probabilities add. The so-called canceled pieces have not really disappeared. as will be argued, they simply do not contribute to an average impulse which is what experiments apparently detect. Thus “negative” quantum probability is linked to impulse in the opposite direction, we argue. Thus interference is really linked to the notion of probabilistic impulses i.e. p exp(ipx) even though exp(ipx) is used as a kind of probability. exp(ipx) contains internal action-reaction impulse hits which have experimental consequences as we argue is shown in a reflection/refraction problem, but are also linked with impulse hits associated with p. Thus there are two sets of forces to deal with. This, we argue, explains why one writes the energy conservation equation AA/c1=BB/c1+CC/c2 for reflection/refraction where A,B,C represent the incident, reflected and refracted rays and AA simply shows that intensity/flux is positive as two equations: A+B=C and Ap - pB = p2C with E=pc. The two equations,, needed to solve for B,C in terms of A,, describe the internal force in exp(ipx) and the probabilistic impulse p exp(ipx). For a single particle bound state, one considers a linear combination of free particle states indicating that impulse hits represent the underlying physics. Given the apparent spatial invariance of a free particle exp(ipx), one may wonder how a kinetic energy profile appears. We argue that different p values yield different peak points for maximum impulse hits and these are out of synch leading to a spatial distribution of KE(x) or d/dx KE(x) which is equivalent to -dV/dx =Force where V(x) is the potential.
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
Interference between exp(ipx) momentum eigenstates in W(x)=wavefunction= Sum over p a(p)exp(ipx) or bound eigenstates in W(x,t)=Sum an(En)exp(-iEnt) Wn(x) are well known in quantum mechanics. The resulting interference (seen in W*(x,t)W(x,t)) seems to be the effect of an OR probability condition i.e. adding complex probabilities with both the real and imaginary terms being periodic and containing both positive and negative values. (Here the wavefunction is treated as a kind of probability (i.e. a conditional probability.) In this note, we wish to argue that interference does not always occur. In particular, a free particle has a wavefunction of exp(-iE(p)t) exp(ipx where E=pp/2m, but only exp(ipx) terms interfere in W(x)=Sum a(p)exp(ipx). exp(-iE(p)t) terms do not interfere, rather a new time cycling resonance is created described by an overall exp(-iEt) where E represents overall energy (kinetic and potential). In other words it seems that a new kind of energy is created namely potential energy which must be considered together with kinetic energy. A particle in a bound state, treated as a free particle for statistical purposes, maintains a kind of identity and energy flow independent of the potential. Its energy identity, however, described by exp(-iE(p)t) where E(p)=pp/2m is transformed as it now interacts with exp(ikx) terms of V(x) to contribute to an overall new cycling resonance of exp(-iEt). Thus, exp(iE(p)t) terms do not interfere. For a linear combination of bound states (e.g. in a kind of heat reservoir) W(x,t)= Sum over n an(En)exp(-iEnt) Wn(x) and a new cycling resonance is not created, rather the reservoir kicks one n state to another and exp(-iEnt) terms interfere. Interference gives rise to the humps or troughs found in the original exp(ipx) or exp(-iEnt), but no longer yields a modulus of 1 at each x or t point. Thus interference shows the nonlocality in the problem. exp(ipx) on the other hand contains both locality and nonlocality. There is a wavelength proportional to 1/p and cos(px) and sin(px) are both nonlocal, but the modulus of exp(ipx) is 1 which is a local property. Quantum mechanics allows both for interference which yields nonlocality and noninterference, e.g. exp(-iEnt) which contains both locality and nonlocality. We investigate this idea in this note.
In a previous note (1) we argued that a quantum free particle, represented by exp(ipx), carries the information of the force that created it and which it may also impart when striking a target in the distributions cos(px), sin(px) of its wavefunction exp(ipx). In this note, we try to quantify this argument by first examining a second collective force in a quantum bound state other than -dV(x)/dx, namely KE(x) (-dW/dx / W) where W is the wavefunction. We argue this force should be present in a particle in a box with infinite walls and ultimately in a quantum free particle.Thus -id//dx, related to the translational operator, is linked to momentum and its conservation, but also ultimately to impulse carried by such a momentum, namely p. The force is found from pp/2m p where pp/2m is the frequency. Thus p is enough information for both momentum and force.. A high resolution (i.e. small wavelength) of a high momentum particle is directly linked to the high force it may impart when striking a target. Alternatively, the notion that an impulse Integral F dt occurs at a point x, as in two-body scattering in a Maxwell-Boltzmann gas, is only achieved in the high p limit, but an MB case is considered to be a high energy, high p limit i.e. a classical situation.
Quantum mechanics seems to function at two levels in terms of probability. One level is represented by classical probabilities P(x)=W*(x)W(x) and P(p)=a*(p)a(p) where W(x)=wavefunction= Sum over p a(p)exp(ipx). For a free particle this level maps into continuous space and time i.e. exp(-iEt+ipx)*exp(-iEt+ipx)=1, but at a more fundamental level (the wavefunction level) there seems to be spatial and time resolution. Spatial resolution for a free particle is based on exp(ipx) i.e. a resolution pattern cos(px) and its space shifted sin(px) representing motion. This resolution carries over to bound states for which P(x) is not 1, but is represented by crests and troughs. exp(-iEt) represents resolution in time, we argue and is linked to 1/E where E is an energy associated with an interaction. As a result a bound system may have an overall exp(-i (M1 + m2 + En) t) where En is binding energy and an overall frequency (or spatial resolution of 1/(M1 + m2 + En), but exp(-iEnt) represents a frequency of 1/En which is linked to the interaction between M1 and m2. Thus the time resolution seems to be associated with energy and there may be various kinds of energy in a system (e.g. rest mass energy, binding energy) etc. These appear to be linked to different time resolutions. If En becomes 0, i.e. M1 and m2 no longer interact, then M1 has a time resolution proportional to 1/M1 and m2, to 1/m2. Thus at the wavefunction level it does not seem there is an absolute time, rather there are various time resolutions associated with different energies.