The wavefunction for a quantum free particle is exp(-iEt + ipx) In this case, E and p are linked through either E=pp/2m (nonrelativistic case) or EE =pp + momo (relativistic case c=1). We argue that there are different interactions involved with E and p. P (momentum) is energy flow and is involved in impulse type interactions with a potential usually V(x). For V(x,t) matters are more complicated, but will also be discussed. The impulse hits are associated with exp(ipx) and lead to interfering plane waves and an overall crest-trough spatial scenario. There is a kinetic energy (nonrelativistic case) associated with each p, but a new overall energy En (n= eigenlevel) is created which yields the free form exp(-iEnt). This energy is linked to the cycling of exp(ipx)’s. It is also part of the new mass of the bound state system i.e. mo+En. This new system could be considered as an overall free particle and obtain a new overall momentum. (The initial bound state has 0 overall momentum.) Thus, time and space parts of the wavefunction behave in very different manners due to the nature of the interaction even though there is a link between 1/E and cycling of exp(ipx) terms. We call this case an impulse or space based type of interaction. It is also possible to have a time based interaction i.e. the absorption of a photon, phonon etc. In such a case, there is momentum conservation and so the whole system receives a new momentum, but this is usually ignored. The main effect is for the system to move from one bound state E(n1) to another E(n2). This has consequences on the spatial density and cycling of exp(ipx) terms in the wavefunction, but is not an impulse interaction in space. In the case of a system: W(x,t)= Sum over n an(En) exp(-iEnt) Wn(x), one has time based interactions which knock the system from one bound state to another (as in heat bath interaction). Again there are links between each En and the spatial density, but there is an overall spatial density(x,t) and average energy(x,t) which depends on the interference of exp(-iEnt) terms. (This disappears if one integrates over all x.) This is similar to the diffraction/interference spatial pattern, but in time. In the impulse scenario for a(p)=a(-p) one has cross terms in the spatial density of cos( (pi-pj)x) which create the crest/trough pattern seen. For the heat bath time type interaction, there exist cos( (Ei-Ej)t) type terms if an(En)=a*n(En) which are linked to the energy Ei-Ej of the intermediary photon/phonon etc and its time period. Thus in both cases, the quantum crest/trough pattern (in space or time) seems to be linked to the momentum or energy transfer. The energy transfer is known to be linked to periodic objects (photons/phonons) and it seems the momentum transfer pi-pj also has a similar periodicity, but in space. It is almost as if these “entities” are creating the unusual quantum crest/trough patterns.
The wavefunction for a quantum free particle is exp(-iEt + ipx) In this case, E and p are linked through either E=pp/2m (nonrelativistic case) or EE =pp + momo (relativistic case c=1). We argue that there are different interactions involved with E and p. P (momentum) is energy flow and is involved in impulse type interactions with a potential usually V(x). For V(x,t) matters are more complicated, but will also be discussed. The impulse hits are associated with exp(ipx) and lead to interfering plane waves and an overall crest-trough spatial scenario. There is a kinetic energy (nonrelativistic case) associated with each p, but a new overall energy En (n= eigenlevel) is created which yields the free form exp(-iEnt). This energy is linked to the cycling of exp(ipx)’s. It is also part of the new mass of the bound state system i.e. mo+En. This new system could be considered as an overall free particle and obtain a new overall momentum. (The initial bound state has 0 overall momentum.) Thus, time and space parts of the wavefunction behave in very different manners due to the nature of the interaction even though there is a link between 1/E and cycling of exp(ipx) terms. We call this case an impulse or space based type of interaction. It is also possible to have a time based interaction i.e. the absorption of a photon, phonon etc. In such a case, there is momentum conservation and so the whole system receives a new momentum, but this is usually ignored. The main effect is for the system to move from one bound state E(n1) to another E(n2). This has consequences on the spatial density and cycling of exp(ipx) terms in the wavefunction, but is not an impulse interaction in space. In the case of a system: W(x,t)= Sum over n an(En) exp(-iEnt) Wn(x), one has time based interactions which knock the system from one bound state to another (as in heat bath interaction). Again there are links between each En and the spatial density, but there is an overall spatial density(x,t) and average energy(x,t) which depends on the interference of exp(-iEnt) terms. (This disappears if one integrates over all x.) This is similar to the diffraction/interference spatial pattern, but in time. In the impulse scenario for a(p)=a(-p) one has cross terms in the spatial density of cos( (pi-pj)x) which create the crest/trough pattern seen. For the heat bath time type interaction, there exist cos( (Ei-Ej)t) type terms if an(En)=a*n(En) which are linked to the energy Ei-Ej of the intermediary photon/phonon etc and its time period. Thus in both cases, the quantum crest/trough pattern (in space or time) seems to be linked to the momentum or energy transfer. The energy transfer is known to be linked to periodic objects (photons/phonons) and it seems the momentum transfer pi-pj also has a similar periodicity, but in space. It is almost as if these “entities” are creating the unusual quantum crest/trough patterns.
In a previous note (1) we considered a free particle entropy based on the real part of the wavefunction exp(ipx) i.e. |cos(px)|. |cos(px|) may be normalized over a certain length, just as exp(ipx) is. The idea is that there exists an uncertainty in space within a wavelength which should be reflected in an entropy expression. As p becomes very large, cos(px) becomes very narrow and has many peaks in L. Thus probability →0 so p ln(p)--> 0 using l’Hopital’s rule. This coincides with a free classical particle having zero entropy. In this note, we revisit arguments presented previously, suggesting that A= -Et+px, a Lorentz invariant and also the free particle classical relativistic and nonrelativistic action for v=x/t We suggested in the past that A is responsible for both the wavelength hbar/p and the time period hbar/E. Here we suggest uncertainty within a time period should be considered together with uncertainty within a wavelength. For instance, recent experiments (2) have shown that one may have quantum interference in time as well as space. Thus there is uncertainty for a free particle in both space and time. We add the entropies for these two. Next, we consider the case of a bound system. Such a system creates a crest-trough pattern (with different heights and widths) for its spatial wavefunction, but E, average energy is constant and there is the free particle type exp(-iEnt). We suggest that the system is characterized by uncertainty in space and time with En being a known constant at each x. This suggests that various free particle p (momentum) values combine to create the wavefunction such that -1/2m d/dx d/dx W/W is average kinetic energy which in turn adds to V(x) to create a constant En. We are thus concerned with uncertainty in time and space and not in momentum. In the limit of high energy values, a crest fits within a tiny dx and prms is essentially the classical p value which is considered certain, even though in quantum mechanics the range of a(p) weights hold for all n (W(x)=Sum over p a(p)exp(ipx)). Thus we treat a high n limit as being completely certain and not having entropy. This differs from the approach of using C/v(x), where v(x) is velocity given by .5mv(x)v(x)+V(x)=En, as a probability and calculating Shannon’s entropy using this.
In classical physics, a particle traveling to the right is the mirror image of one traveling to the left. In terms of a spatial density (the time spent in each constant dx), the value is the same, as is the energy. A similar notion of such a spatial invariance is in the classical conservation of energy equation for a bound particle: p(x)p(x)/2m + V(x) = E. Quantum mechanics seems to be different because exp(ipx) and exp(-ipx) are distinctly different spatial probabilities, i.e. they are dynamic. It is not just that case that a particle moves to the right for exp(ipx) and to the left for exp(-ipx). The impulse hit probabilities (represented by exp(ipx) and exp(-ipx)) are distinctly different. If one wishes to create an OR situation and write: W(x) = Sum over p a(p)exp(ipx) and use this in a conservation of energy equation, then simply including only positive or negative p values in W(x) leads to a complex average kinetic energy -1/2m d/dx dW/dx / W which cannot be used with V(x) and E. Furthermore, W_ (only negative p) and W+ (only positive p) values do not yield the same average kinetic energy. Fortunately, given that this is a statistical average for a localized object (somewhat analogous to a gas), one may combine W+ + W- and obtain a real kinetic energy. We have argued in previous notes that there exists a quantum-classical duality. For example, for a free particle one has two independent notions. One is x=vt which describes the center-of-mass of the particle. We take this to represent a state description which treats v and -v as mirror images. exp(ipx), on the other hand, represents probabilistic impulse hits and does not define the state by itself, we argue, only its interactions. Interactions are physical and define what happens in an interacting system. One may, however, measure a system, i.e interact with exp(ipx), and this should define a measurement state ( a second state ensemble average). For example, exp(ipx)exp(-ipx) = 1 which is equivalent to the constant dt in each dx associated with x=vt. Thus, the two states coincide, but exp(ipx)exp(-ipx) means that one may place the 2-slit apparatus at any x point and is not about the center-of-mass, as x=vt is. In some cases, the two states are quite different. W*(x)W(x) represents an interaction state, but vrms(x) from .5mvrms(x)vrms(x) = -1/2m d/dx dW/dx / W represents a center-of-mass state description. (Note: in these cases a state means an average state, i.e. an average over many ensemble entries.) If one takes W*(x)W(x) and integrates over x, then interference terms vanish and one obtains a classical counting of distinct p value states with their weight a*(p)a(p).This, we argue, leads to the notion of orthogonality of probabilities. In classical probability, there is no such notion, but there is the notion of AND (multiply) probabilities. Given exp(ipx) and exp(-ipx), if one multiplies one removes p, i.e. the interactions and only sees the state left. W*(x)W(x) yields a set of peaks which represent a pattern of probabilistic hits in space which define a state which one would measure in space. The interference pattern, however, must be one such that if one integrates over x, one only obtains a classical sum of probabilities a*(p)a(p)’s as mentioned above. This leads to the notion of orthogonal exp(ipx)’s (which follows mathematically anyway). If one considers different bound En states, then these two are orthogonal because W*n1(x)Wn2(x) mean n1 and n2 are associated with different energies and integrating over x cannot show a result which looks like a state result, where W*n(x)Wn(x) shows a state result. In order to achieve this orthogonality (linked directly to the notion of a state) one must have that Wn(2) usually has 0 points and is positive and negative with the possible exception of a ground state. In other words, if Wn1*(x)Wn2(x) integrated over x is not 0, one violates conservation of energy. As a result, we argue an average probability for a bound state must statistically include +p and -p contributions and must create a function which has positive and negative peaks (except possibly for the ground state), i.e. W(x) must have 0 points. This W(x) is linked to a measurement state W*(x)W(x), but this is strictly considering exp(ipx)s. We argue that there are two pieces of information for a free particle, x=vt and exp(ipx). There must also be an average state linked to x=vt and this given by KEave(x) = .5m vrms(x)vrms(x) + V(x) = E which shows that the particle may exist and have average kinetic energy even at W(x)=0 points. When measuring a system, one interacts with the exp(ipx)s and so the W*(x)W(x) state is the one which is traditionally presented, but whether one measures the center-of-mass motion or not, one also has the notion of the x=vt, i.e. vrms(x) state (average) as well, we argue.
Traditionally, a quantum bound state is said to approach classical behaviour for large n (from En=energy). The problem is that peak and valley spatial density regions still exist although an envelope passing roughly through the peaks of Wn(x)Wn(x) (where Wn is the wavefunction) matches C/v(x) where v(x) is the classical spatial density. This is proportional to delta t the time a particle spends within constant sized delta x pieces of a bound path. Alternatively, a wave packet is frequently used, but its width tends to increase in time. In a previous note (1), we argued that one may write the free particle classical action (relativistic or nonrelativistic) as A= -Et+px with t and x being independent. This leads to dA/dx partial = p and dA/dt partial = -E. Creating eigenvalue equations -i d/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = Eexp(iA), one may see that exp(-iEt) is periodic in two dimensions in time and exp(ipx) in space. In this note we examine whether the separation of x and t suggests two schemes for the spatial region. The first scheme is linked to exp(-iEt) where E is average energy at each x. In such a case E=pp/2m (for a free particle) at each x and x=p/m t. exp(ipx) on the other hand demonstrates fluctuations in two dimensions within this region which would not normally be seen in the classical realm due to lack of resolution. If one considers a bound system, these two schemes become clearer. exp(-En t) is associated with an energy En equal to KEclassical(x) + V(x). On the other hand there is the quantum momentum distribution Wn(x)=Sum over p a(p)exp(ipx) such that: [Sum over p pp/2m a(p)exp(ipx)]/Wn(x) + V(x) = En ((A)) As a result, the momentum distribution Wn(x) is linked to En at each x through ((A)). The two dimensional waves exp(ipx) interfere to create an rms v which is classical. This classical motion exists for all n. Wn(x)Wn(x) on the other hand demonstrates quantum fluctuations, but these include also an rms probability of C/v(x) where v(x) is the classical velocity. For large n, C/v(x) becomes the envelope of Wn(x)Wn(x) and so quantum fluctuations are within the envelope. For lower n we argue that C/v(x) must be “removed” from Wn(x)Wn(x) to see the full quantum fluctuations, but that C/v(x) still applies to the classical rms motion which is present and associated with exp(-iEnt). In other words we use neither wavepackets or the idea that classical mechanics is involved only for large n. Classical mechanics is present for all n in ((A)).
Quantum mechanics utilizes a wavefunction W(x)=Sum over p a(p)exp(ipx) (in time independent cases) with exp(ipx) being what we call a free particle complex probability. In such a case, W(x) is an OR scenario of these various probabilities. We have argued in previous notes that exp(ipx) is the probability for a p impulse hit at x, even if the center-of-mass of the particle is not at x. This means that impulse hits may occur probabilistically in space as in the 2-slit example which requires adding two probabilities as an OR case for each slit. The point then seems to be that W(x) (or even its momentum frozen form W*(x)W(x) gives a profile in space of the number of interactions (various p impulse hits) at each x. Classically, the kinetic energy (nonrelativistic) is pp/2m. In a classical bound problem this kinetic energy is considered to exist (in calculations) at the center-of-mass, but forces are also said to be rendered at this point so pp/2m, the center-of-mass and the force hit point all coincide. In quantum mechanics, a p impulse hit and the center-of-mass do not coincide. If p is said to hit at a probabilistic x, then it seems that pp/2m should also be seen to exist at this same hit point. A physical interaction point verifies in a sense the location of pp/2m in a probabilistic manner, i.e. a(p) exp(ipx)/W(x) is the probability used to calculate KEaverage and this is based on impulse hit locations. By normalizing using W(x), one removes the relative number of average hits at each x and obtains a pure energy value at x. (This may then be used in a KEave(x)+V(x)=E equation.) On average W(x) has zero points, which means that if one considers W*(x)W(x) to represent the motion-frozen-out spatial distribution of hits, then no hits are rendered at certain x points (on average). One might argue then that the particle cannot exist at x, but the center-of-mass can exist anywhere. One might next argue that there is no kinetic energy at x, but that cannot be true because 1/2m d/dx d/dx W has the same 0 points as W and the two divide. Thus, there is average kinetic energy at all bound points and this represents an average based on impulse hits (exp(ipx)). Classically, if one has many particles in a potential, one may write density(x)V(x) to find the potential energy of the particles. This implies that the particles interact with V(x), e.g. neutrinos usually do not interact. Density(x) in a classical sense means that the center-of-mass and force hit point are both at x. In quantum mechanics, exp(ipx) represents the probability for a force hit point, W(x) represents an average and W*(x)W(x) represents a motion-frozen average hit point. W*(x)W(x)V(x) seems to represent the average potential energy of the particle at x because W*(x)W(x) is linked with interaction. Even if a particle is not present, however, there is energy V(x) in space. It seems that if W*(x)W(x)=0 at some x, then there is no average particle interaction at this point, but there is still average kinetic energy Sum over p a(p)pp/2m exp(ipx) / W(x) based on the probability for average probability hits. The key idea is that one normalizes probability by dividing by W(x) in the average kinetic energy calculation. W*(x)W(x) is normalized not at x, but integrating over dx and setting this value to 1. One has relative hit at x values represented by W*(x)W(x). For exp(-ipx)exp(ipx)=1, each dx has the same average hit value (with one not being able to distinguish details in hbar/p, i.e. this is a motion frozen result. The question then becomes: Why would one write: W*(x)W(x) KEave(x)? One might argue that this term occurs if one takes the quantum mechanical equation: KEave(x) +V(x) = En and multiplies by W*(x)W(x). One may ask: Does V(x)W*(x)W(x) have physical meaning? V(x) is the potential energy at x. The average interaction probability with V(x) at x should be W(x)V(x) and not W*(x)W(x)V(x). The latter is a classical mechanical idealization. If W*(x)W(x) is 3 at x1 and 2 at x2, then it is as if the particle interacts 1.5 times more at x1 and so if V(x1)=V(x2), then this approach assumes 1.5 the interaction energy at x1. KEave(x), however, is simply free particle energy (average) at x and one does not need to take this average value and multiply it by the relative number of interactions at x. W(x) has zero points meaning that there is no interaction at such a point on average. KEave(x) on the other hand is the average energy at x based on the particle kinetic energy presence pp/2m being at the point of an impulse hit. This does not mean that an interaction is occurring with V(x) at this point in this formalism. Even a free particle has an exp(ipx) probability when it is not interacting. The sum Sum over p a(p)exp(ipx) means that all of the interactions linked with x have occurred and one is making an OR calculation. KEave(x) is free particle kinetic energy at x and V(x) is potential energy (not interaction potential energy with the particle). Multiplying by W*(x)W(x) is a classical minded approach to creating a classical equation: density(x) (KEave(x)+V(x)) = E density(x) ((1)) Here KEave(x) would represent the average kinetic energy of a single particle and density(x) would be the average number of particles at x. density(x)KE(x) has a clear meaning. W*(x)W(x) KEave(x) on the other hand is different. KEave(x) is actually the classical kinetic energy value at x and so it is not clear why one would multiply it by W*(x)W(x). W*(x)W(x) represents the probability for an impulse hit at x in a p-frozen out picture. The notion of exp(ipx) (i.e. probability for a p hit at x) is already buried within the quantum calculation of KEave(x) (which yields the classical value). In order to find the physical meaning of W*(x)W(x) multiplied by KEave(x) or V(x) it seems that one has broken dx into equally sized segments which correspond to different detection hits of the particle. This is then somehow equated with the presence of the particle per unit time in dx which is then somehow rendered as the average energy at x multiplied by the detected presence at x through a steady rate of measurement/time. Thus, multiplying by W*(x)W(x) is not multiplying by a physical density.
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.
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.