In Part 1, we tried to create a toy model involving a rotating-translating coin. We examined what kind of interaction is needed to create a typical 2-slit interference pattern. Here we examine a physical object, namely a photon, which has a built-in periodic cycle for its electric and magnetic fields. These are perpendicular and in plane perpendicular to motion and both change as cos(-Et/hbar + px/hbar). We point out that the electric and magnetic fields are both real, they are not described by exp(-iEt+ipx) and furthermore, they are deterministic, not probabilistic. We try to see how such a picture can lead to quantum two slit interference. Basically, we argue that if one only considers the electric field, long range interactions from atoms around both slits interact with a photon heading towards the center between the slits. As a result, the physical interaction must involve both slits, but one has no idea through which slit the photon passes. There is probability in the problem, but at the same time interactions with both slits before the photon passes through and also after. One needs a model which captures both the notion of probability and interaction with both slits. For probability this should be an OR situation (one slit or the other), but it cannot be the usual classical probability scheme which implies that moving through one slit only involves interaction with that slit. Interactions with both slits occur. This seems like a contradiction because one applies OR probability to one slit or another. Given that the interaction with the electric field occurs in a periodic manner cos(-Et+px) hbar=1, we suggest that exp(-iEt+ipx) with its modulus of 1 captures the idea that any electric field position as the same probability, but that there is periodic motion linked with interaction in space. . As a result, one may keep the mathematical notion of an OR probability scheme by introducing an interacting probability exp(ipx). This means that the probaiblities from each slit in the OR (add) scheme interfere or interact which is what is required because an interaction with both slits occurs. exp(ipx) captures the periodicity and also enforces conservation of momentum (exp(ip1x)exp(ip2x) = exp(I (p1+p2)x) and it seems that this is the only information needed.
In Part 1, we tried to create a toy model involving a rotating-translating coin. We examined what kind of interaction is needed to create a typical 2-slit interference pattern. Here we examine a physical object, namely a photon, which has a built-in periodic cycle for its electric and magnetic fields. These are perpendicular and in plane perpendicular to motion and both change as cos(-Et/hbar + px/hbar). We point out that the electric and magnetic fields are both real, they are not described by exp(-iEt+ipx) and furthermore, they are deterministic, not probabilistic. We try to see how such a picture can lead to quantum two slit interference. Basically, we argue that if one only considers the electric field, long range interactions from atoms around both slits interact with a photon heading towards the center between the slits. As a result, the physical interaction must involve both slits, but one has no idea through which slit the photon passes. There is probability in the problem, but at the same time interactions with both slits before the photon passes through and also after. One needs a model which captures both the notion of probability and interaction with both slits. For probability this should be an OR situation (one slit or the other), but it cannot be the usual classical probability scheme which implies that moving through one slit only involves interaction with that slit. Interactions with both slits occur. This seems like a contradiction because one applies OR probability to one slit or another. Given that the interaction with the electric field occurs in a periodic manner cos(-Et+px) hbar=1, we suggest that exp(-iEt+ipx) with its modulus of 1 captures the idea that any electric field position as the same probability, but that there is periodic motion linked with interaction in space. . As a result, one may keep the mathematical notion of an OR probability scheme by introducing an interacting probability exp(ipx). This means that the probaiblities from each slit in the OR (add) scheme interfere or interact which is what is required because an interaction with both slits occurs. exp(ipx) captures the periodicity and also enforces conservation of momentum (exp(ip1x)exp(ip2x) = exp(I (p1+p2)x) and it seems that this is the only information needed.
In Part 1, we argued that it is no coincidence that the photon free particle wavefunction exp(-iEt+ipx), which we call a complex probability that conserves energy and momentum, has a real part which matches the electric field of a photon. In particular, we argued that even if one did not know Maxwell’s electromagnetic laws, one would postulate a force linked to a photon which behaved as cos(-Et+px). Here, we reexamine Maxwell’s electromagnetic equations. These are presumably deterministic differential equations and the photon solution with perpendicular electric and magnetic fields in a plane perpendicular to motion with each field changing as cos(-Et+px) for motion in the x direction, a seemingly deterministic solution. This deterministic solution, however, seems to be problematic for the following reason. If one considers one-dimensional reflection-refraction according to the prescription of (1), one writes a continuity equation for the electric field of an incoming photon with a reflected and refracted one, and another continuity equation for the first derivative. Physically, this appears to be a steady stream approach as well as a mathematical continuity one. The solution in (1) is obtained without any notion of probability until the last line: 1= Probability(reflect)+Probability(refract). The situation, however, is completely probabilistic from the beginning we argue and so one should understand how a deterministic electric (and magnetic field) is actually associated with probability. (Furthermore, photon interference and diffraction are also probabilistic.). We have argued in previous notes that examination of special relativity, in particular, the Lorentz invariant -Et+px = constant(x,t), leads to a probability for interaction of exp(-iEt+ipx) for both a particle with rest mass and a photon. This probability conserves E,p, but at first glance does display the inherent force in the problem. It is the inherent force which delivers E,p. Probabilistically, there is an uncertainty associated with the interaction time of hbar/E and with interaction position, hbar/p. These should be associated with the inherent force. As a result, the photon electric field, which acts as a force, cannot be precisely as described by cos(-Et+px) because this violates the notion of hbar/p and hbar/E. Rather, this formula is an expression of probability of the action of the force. One ultimately does not have a deterministic situation, we argue. In Part 1, we noted that a rest mass mo is associated with a force (inertia) as well as energy and momentum, which are proportional to mo. A photon has E and p and so should be associated with a force as it may render its energy to an object. The interesting thing about the photon is that it has no rest frame, even for the force. Given that exp(-imocc t) describes the probability associated with a rest force mo, we suggest that for a force linked with a moving photon, one should consider cos(-Et+px). This was already proposed in Part 1. This means that instead of considering the momentum as delivering the force, as it is proportional to mo for a particle with rest mass, for a photon, the electric field represents probabilistic force rendering, even though it is real and appears to be deterministic. Given that the electric field (inherent force) changes as cos(-Et+px), one may see that this force may change sign, but momentum must be in one direction. This suggests the presence of a second vector, together with the electric field in a quadratic arrangement with both becoming negative at the same time, i.e. both following cos(-Et+px). One may then create energy and momentum using Electric field dot Electric field and second field dot second field and electric field x second field for momentum. In such a case, energy and momentum would be cycle averages as the electric field changes as cos(-Et+px).
Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis.
In classical physics, interactions appear to be of the form of cause and effect, as in Newton’s second law is: F = dp/dt. The force F governs the interaction and dp/dt is the effect. In quantum mechanics, there is a probability for an impulse hit interaction by a particle given by exp(ipx) within a wavelength region of size hbar/p. As a result, instead of an interaction occurring within dx->0, the particle itself dictates an interaction region of the size hbar/p. Consequently, in a 2-slit interaction, the slits should be about a wavelength hbar/p apart. In classical physics, an interaction occurs within the interaction region (dx->0), but in the 2-slit case the interaction region includes the space from the first slit to the second, so to ask: What is happening as the particle interacts within this region? means that a particle can pass through one slit and inteact with the other, or it can interact with the slit through which it passes. The interaction region is the smallest region in which the interaction occurs (as if it were a black box). The particle physically passes through one slit or the other, but interacts either way with both slits probabilisitcally because otherwise the interaction region would only be one slit or the other contradicting the probabilistic range hbar/p of interaction. As a result, a two-slit calculation is really a calculation of an interaction within an hbar/p region including both slits. This interaction is probabilistic due to the exp(ipx) interaction hit nature of the particle and so it is the particle which dictates the interaction to a large degree. The fact that the particle must pass through one slit or the other, together with a probabilistic interaction over the hbar/p region means that two key notions are at work, one being the physical location of the particle (passing through one slit or the other) which governs how the probabilistic interaction (due to the particle) operates. In particular, one draws two parallel rays, one from the center of each slit towards a fixed point on a screen far away. The interaction is based on the exp(ipx) probabilistic interaction of the particle and the separation of the two slits because the particle interacts with both as it passes through only one. In other words, reality is not simply based on what one can measure during an interaction. The fact that one does measure through which slit the particle passes does not mean that the positions of the slits cannot stil helpl lead to determinate statements such as zero particles arrive at certain x points on the screen. The slit separation is part of the physical reality of the problem. (Note: there is also diffraction from the two edges of each slit, i.e. diffraction, which follows the same ideas and so one has a combination of diffraction from each slit together with interference.) As a result, we suggest that one consider the entire region from one slit to the other as a quantum interaction region similar to the dx->0 used in Newtonian mechanics. Quantum mechanics then provides a probabilistic, not deterministic, descripition of possible interactions and their probability weights, unlike the cause (force) and effect (deterministic change in p of the particle) of Newtonian mechanics. The exp(ipx), hbar/p gradation form of probability, allows for probabilities to add or subtract at different x points. The walls of the 2-slit apparatus are linked with probabilistic momentum angle changing, with each angle having the same probability (uniform distribution), but otherwise much of the interaction is dictated by the particle, unlike in Netwonian mechanics. For a V(x) = Sum over i Vk exp(ikx), both exp(ipx) and exp(ikx) are one a similar footing.
In Part 1, we argued that the Heisenberg uncertainty principle (standard deviation p) * (standard deviation x) >= hbar/2 only applies when a measurement (i.e. interaction) has occurred. Otherwise, the notion of standard deviation does not exist. Thus, if one has a Gaussian wavepacket, it is the measuring device which has created it and constrained is standard deviations of momentum and position. A free particle, which has not been measured, is described by exp(-iEt+ipx), as argued in Part 1, and not by a wavepacket. Here we ask: How does one interpret a wavepacket calculation? In (1), we argued that exp(-iEt) is a free particle energy conserving probability and exp(ipx), a momentum conserving one. Then, a sum of exp(ipx - iEt) represents an OR situation in probability theory. There is superposition and so one obtains a weighting of possible outcomes which is the probabilistic interpretation of the scenario. If one considers two slit interference, the viewpoint is the same. exp(ipx) is a probability which allows for momentum hits of p in a range of dx=hbar/p outside the center-of-mass, which follows x=vt. This built-in stochasticity of impulse delivery leads to the notion that probabilistically, a particle may interact with both slits if their separation is about hbar/p. One writes an OR probability statement: W(x)= exp(i p dot r1) + exp(i p dot r2), where r1 is the position vector from the center of slit 1 to a point far away on a screen, and r2, a vector from the center of slit 2 to the same point. The result, W*(x)W(x) is a probabilistic expression which describes the possible outcomes probabilistically. (1). Seen in this way, the Gaussian wavepacket should be interpreted in a similar manner and not as a physical description of the particle which moves freely. A single bound state in a potential V(x) is localized by a potential and one may consider only OR cases of exp(ipx), i.e W(x) = Sum over p a(p)exp(ipx). In the measurement case, there is no quasi-permanent localization and one uses a linear superposition of exp(-iEt+ipx)s. Nevertheless for a Gaussian wavepacket, two probability distributions describing possible outcomes are created. One has W*(x)W(x)= exp(- (x-u)(x-u) / (2 sigma(t) sigma(t))) where sigma is the x standard deviation. In addition, W(x) = Sum over p a(p) exp(ipx), leading to a momentum outcome distribution of a*(p)a(p). As written, sigma(t) for x which means that the Gaussian is expanding. We called this unphysical if one considers the wavefunction as representing a description of a single physical entity (2) at times beyond t=0. Like in the 2-slit case, we argue that it represents an outcome distribution. Thus, one would not consider any more time values than t=0, at which time the measurement is completed. An exp(-iEt+ipx), describing a free particle, is measured by a specific device which creates the Gaussian form and gives it a specific standard deviation in space and momentum. The measurement is then done and W*(x,t=0)W(x, t=0) represents the possible outcome positions of the particle at t=0 (i.e. probabilities for x portions of a phase shift). The momentum distribution yields possible p outcome values. As soon as the measurement is finished, the particle moves as exp(-iEt+ipx), with p being probabilistically linked to a*(p)a(p) (a Gaussian as well), where W(x,t) = Sum over p a(p)exp(-iEt+ipx).If one has a second measuring device which creates a tiny momentum standard deviation, one would be able to carry out experiments which would determine roughly the standard deviation of p created by the first measuring device. Furthermore, one would be able to measure phase shifts as well by comparing a free particle measured by a device with a tiny p standard deviation and a large x one with free particles which were not measured.
In Part II of this note, we argued that if one describes a bound particle in a potential in a statistical manner, like classical statistical mechanics, one is led to a complex, periodic space invariant probability, exp(ipx), for a free particle. We showed that exp(ipx) represents a two vector solution to the vector operated on by a rotation matrix (using cos(px), -sin(px)) set equal to a translation operator 1+delta X d/dx operating on the same vector. In this note, we examine the electromagnetic scenario. The idea of a “wave” describing a photon first emerged with Maxwell and his equations in the 1800s, with interference/diffraction and the speed of light both appearing. It was later found (Einstein and the photoelectric effect) that electromagnetic energy is not spread out all over space like a wave, but is concentrated. Furthermore, if one uses probabilistic ideas to obtain exp(ipx) for a quantum particle, like an electron, which undergoes two slit interference, then why would not two slit interference for photons also be considered in terms of probability? In this note, we examine these ideas.
In classical statistical mechanics, although one has averages of a system, a single particle within the system has properties which are sharply defined at any x,t, even though these properties change stochastically, i.e. through interactions. A quantum free particle, on the other hand, has information based on the Lorentz invariant -Et+px, multiplied by i to create periodicity which indicates an inherent information, not one based on the outcome of trial runs or interactions. The probability is exp(-iEt+ipx). If E and p are considered as constant, then there are independent probabilistic fluctuations in x and t, which means there may be corresponding probabilistic interactions, both impulse ones (one dimensional reflection/refraction, n-slits) and temporal ones (Rabi frequency). In such a case, one has interference and the form exp(ipx) or exp(-iEt) governing the statistics, not a maximization of entropy linked to interaction/outcome results. Nevertheless, the quantum particle interacts with fields, it seems, which give rise to p or E changes. It seems the key idea is that one has stochastic uncertainty before an interaction occurs and hence interference. The alternative is that such stochastic uncertainty does not exist before a specific interaction, i.e. the inherent probability exp(-iEt) or exp(ipx) has no effect on the interaction. This is not to say that there cannot be probabilities before the interaction. In a Maxwell-Boltzmann gas, there are probabilities P(e1) and P(e2) for particles with e1 and e2, but a specific e1 and e2 have no uncertainty. The uncertainty is in the outcome of the reaction. Another example is a two state electron in an atom which absorbs and emits photons. The rules for time evolution are that the ground state is represented by exp(-iEot), but at some stochastic t value, it changes to exp(-iE1t) and vice versa. The ground state absorbs a photon and the excited emits. The two states are clearly defined by the differing interactions which govern their time dependence which is taken to be simply the changing of Eo to E1 and vice versa. This problem is then based on probabilities based on outcomes of an interaction and maximization of entropy subject to a constraint is employed. There is no initial interference.