Addendum July 19, 2021 The Fermi-Dirac and Bose-Einstein statistical cases, like the Maxwell-Boltzmann case, may be formulated in terms of a reaction balance between a regular two body elastic collision and its time reversed reaction: e1+e2 = e3 + e4 and f(e1)[1-/+f(e3)]f(e2)[1-/+f(e4)] = f(e3)[1-/+ f(e1)] f(e4)[1-/+ f(e2)]. Both the Fermi-Dirac and Bose-Einstein cases reduce to the Maxwell-Boltzmann case under certain conditions. An unusual feature of bound state quantum mechanics is that there is a spatial density (or probability to be at x = P(x)), just as in classical statistical mechanics, but also a wavefunction W(x) such that P(x)=W(x)W(x). In other words, W(x) for the bound state is like a square root of probability. Furthermore, there is a P(p)=a(p)a(p) where W(x)=Sum over p a(p)exp(ipx). In this note, we try to argue that P(p), which is based on the measurement of momenta of particles knocked out of an ensemble of identical single particle bound states, does not have physical meaning within the bound state, unlike P(p,x) which has physical meaning for a Maxwell-Boltzmann gas both inside the gas and for particles released from a small hole in the gas wall. This forces one to create another function to “link” the two physically observed probabilities P(x) and P(p).
Addendum July 19, 2021 The Fermi-Dirac and Bose-Einstein statistical cases, like the Maxwell-Boltzmann case, may be formulated in terms of a reaction balance between a regular two body elastic collision and its time reversed reaction: e1+e2 = e3 + e4 and f(e1)[1-/+f(e3)]f(e2)[1-/+f(e4)] = f(e3)[1-/+ f(e1)] f(e4)[1-/+ f(e2)]. Both the Fermi-Dirac and Bose-Einstein cases reduce to the Maxwell-Boltzmann case under certain conditions. An unusual feature of bound state quantum mechanics is that there is a spatial density (or probability to be at x = P(x)), just as in classical statistical mechanics, but also a wavefunction W(x) such that P(x)=W(x)W(x). In other words, W(x) for the bound state is like a square root of probability. Furthermore, there is a P(p)=a(p)a(p) where W(x)=Sum over p a(p)exp(ipx). In this note, we try to argue that P(p), which is based on the measurement of momenta of particles knocked out of an ensemble of identical single particle bound states, does not have physical meaning within the bound state, unlike P(p,x) which has physical meaning for a Maxwell-Boltzmann gas both inside the gas and for particles released from a small hole in the gas wall. This forces one to create another function to “link” the two physically observed probabilities P(x) and P(p).
For quantum bound states, classical style probabilities P(p)=a(p)a(p) and P(x)=W(x)W(x) [where W(x)= the wavefunction = Sum over p a(p) exp(ipx)] exist. These are real and normalized to one, but unusually, they separate p and x completely. For example, P(p) is the probability to find a momentum p anywhere in the system - no information about x is known. This seems to suggest that the P(p/x) is not defined in a classical sense. (Similar arguments hold for P(x).) We have argued in a previous note (1), that due to a lack of time reversal balance, there is no clear classical notion of P(p/x) in quantum mechanics while in classical statistical mechanics, there is a time reversal balance of two-body collisions and so P(p/x) is well-defined. In particular, a momentum p collides with a pj to form another momentum pk. There is ambiguity if p is changing, but if at the same time a pk is changing (colliding) back into a p at x, then the ambiguity is removed. In quantum mechanics, the ambiguity is not removed at x, but one may discern one p state from another throughout the entire x space. This suggests orthonormal probabilities e.g. say exp(ip1 x) and exp(i p2 x). These exp(i p x) probabilities comprise the wavefunction which is linked to P(p/x) = a(p) exp(ipx)/W(x), but although Sum over p P(p/x)=1, P(p/x) is complex with the real and complex portions taking on positive and negative values. Nevertheless, it is W(x) which is the solution of the Schrodinger (and other) equations. It is known that solutions Wn(x) of the Schrodinger equation have discrete En (energy) values. It is possible to have [H, Operator]=0 and possibly to have H written in terms of this operator. In such a case, Wn(x) might be both an eigenfunction of H and the Operator. Alternatively, a linear combination of Operator eigenfunctions may be used. If En is discrete, one may expect the operator eigenfunctions to also be discrete, leading to quantization of the Operator eigenvalues. In some cases, one may have two operators which commute with H, e.g. LL and Lz (related to angular momentum). Lz eigenfunctions might not be linked to H so one might ask why Lz should be quantized. Another H, however, may distinguish between Lz eigenfunctions and so they also have discrete eigenvalues. In this note, we try to consider why one should have orthonormal eigenfunctions of these commuting operators which make up the wavefunction and try to interpret their physical meaning.
A Maxwell-Boltzmann gas consists of particles which undergo physical collisions which are associated with the idea of causality, but we argue that reaction balance: e1+e2=e3+e4 and n(e1)n(e2)= n(e3)n(e4) represent steady stream energy-probability conservation and so one need not actually think in terms of direct collisions and causality i.e. one may just as well write: n(e1)n(e2)=n(e1+e2). In the case of quantum mechanics, we argue, as we have done in previous notes, that the free particle form follows from the Lorentz invariant A= -Et+px. In a rest frame, there is energy, but no net momentum. We thus argue for a physical separation of the -Et and px terms. We further suggest that energy is associated with a resonance in time and p momentum with one in space i.e exp(-iEt) and exp(ipx). Given that A=-Et+px is also the relativistic/nonrelativistic classical action, we suggest that these represent probability resonances associated with constant p motion. As a result, given that p is associated with impulse, we suggest that an interaction means forming a new resonance i.e. p1->p2 implies exp(ip1 x)--> exp(ip2 x). As a result, interactions take place in length regions of hbar/p. If this value is very small, as it is in the classical world, then it appears that the resonances occur at points. In the quantum world, it is the opposite. If one wishes to treat quantum mechanics in a statistical manner, one needs to establish an energy-probability balance. Even in classical mechanics, impulses occur that change p, but now one has resonances which occur over a wavelength long piece of space. Furthermore a given p may be anywhere, so one thinks in terms of a(p)exp(ipx) probabilities. We try to show that a probability-energy balance is equivalent to the Schrodinger equation. Thus we argue that quantum mechanics uses a p-probability-energy balance for the time-independent Schrodinger equation and so does not necessarily address causality and physical dynamics.
In Part I of this note, we argued that in classical statistical mechanics, the probability P(p), where p is momentum, is well defined because there is a balance of two-body elastic collisions and their time-reversed reactions. In particular, this may be applied to a point x where one imagines a tiny box with many particles. In quantum bound states, collisions are with a potential and one does not have a balance of reactions and their time reversed cases. Thus, the notion of p at x does not seem to be clearly defined. A particle may be a x with momentum p, but at the same time may be reacting with V(x) and transforming into another pj. Thus, there is an ambiguity which does not exist in classical statistical mechanics. There, a particle may be colliding and so a p appears to disappear, but a new p is created by a time reversed reaction. P(p), we argue, must hold over time in an equilibrium. Due to this issue, it appears that the best one may do is to define P(p) for the whole x region in quantum mechanics. This P(p) is measurable by taking an ensemble of identical single particles, knocking them out and measuring their momentum. As a result, knowing “p” exactly means one does not know “x”. As a result, a different view must be taken to dealing with probabilities in a quantum bound state. We use the idea of invariance in x and argue that a wave-like behaviour in probability may be used to describe results. This is unusual because it appears as if particles are appearing or disappearing at different x points. If one has periodic motion, a question which arises is: What is the spatial frequency? We argue that the answer follows from classical physics. There: delta p = delta t (-dV/dx). If p initial =0 and delta t is constant, one sees that a large p final corresponds to a large change in V(x) in a tiny dx or that there is very good resolution in x. Thus a large p (linear) implies high resolution in x. Thus, we use px as the argument of the periodic probability function.
If one tosses a die or coin, different outcomes occur with a probability p(i) where i is the outcome. This probability may be measured by performing trial runs. For N trial runs, where N is very large, one might expect Np(outcome i) results of an outcome i. In other words, p(outcome i) is both the probability and describes the trial run. Even in the Maxwell-Boltzmann case, one has a probability p(ei) where ei is kinetic energy (no V(x)) and reaction balance p(e1)p(e2) = p(e3)p(e4) represents trial runs. Thus the trial run is again characterized by p(ei) as ei’s contribution. It is, however, possible to have coupling of probabilities in certain physical situations even though p(outcome i) still exists. In such a case, the number of outcome i results for N large trials is not N p(outcome i). In other words, p(i) does not fully describe the trial run. There may be other physical rules in place. If elastic scattering of particles is the way trial runs occur, e1+e2 = e3 + e4 represents the salient information in the problem and even if there is nonindependence of p(outcome i) in the scattering, one should at least obtain a function Q(p(e1)) such that Q(p(e1)) = -e1/T. Q(p(ei)), however, need not be p(ei). We consider the physical rule given by Pauli stating that only one particle may fit in an energy level (no spin). Thus there is either 0 or 1 particle in the level. This should inhibit scattering, so the trial run is no longer simply described by p(ei). Rather the probability to scatter into level ej is (1-n(ej)) and so one has: n(e1)[1-n(e3)] n(e2)[1-n(e4)]= n(e3)[1-n(e1)]n(e4)[1-n(e2)] where n(ej) is the average occupancy, a number between 0 and 1. The opposite of only 0 or 1 particle in a state ei, is to have any possible number of particles in this state and an enhancement, not a hindrance given by 1+p(ej). This is only a guess, but experiment shows that bosons follow this rule. Given a definition of trial runs in terms of the inhibition and enhancement factors, one may do two things. First, since the trial rule seems to involve different numbers of particles in a state associated with a p(ei), one may consider sums of p(ei) (power number of particles) when describing the system statistically (i.e. grand canonical scheme). This approach gives rise to linear terms (a+bp(ei)) which may in turn be formulated in terms of a factorial/arrangement equation which may be maximized subject to the constraints Sum over i ei n(ei) and Sum over i n(ei). We note that this maximization is not simply in terms of all degrees of freedom in the system, but is specifically linked with the rules associated with the trial runs and the constraints. We note, however, that issues seem to arise with the factorial scheme in the boson case. Thus, overall we argue that a single particle probability and its trial run scheme are key to establishing the statistical behaviour of a system. (We also note that single particle independent probability, which is an approximation, also affects the final results.)
Given a single particle bound wavefunction, W(x)=Sum over p a(p)exp(ipx), it is traditionally stated that W(x)W(x) is the probability to find a particle at x. We have argued that W(x)W(x) is actually a resolution profile or a probability for interaction and does not indicate whether a particle is at x or not. (We discuss this briefly in the note.) It is linked to an observer trying to find the particle, on average at x, not if the particle is present there and interacting with V(x). If one normalizes Integral dx W(x)W(x)=1, then it follows that Sum over p a(p)a(p) = 1. Thus, a(p)a(p) has the appearance of a probability as well and one might argue that it is the probability to have a momentum p. The catch is that with W(x)W(x) one considers spatial resolution/interaction without knowing p and with a(p)a(p), one knows nothing about x. Given exp(ipx), the probability for a free particle impulse hit at x, one knows both p and x and may use this probability to calculate KEave(x) = {Sum over p a(p)pp/2m exp(ipx)} / W(x). We note that this probability does not have 0 points when W(x)=0. It shows the particle does not avoid W(x)=0 points in internal interactions with V(x), we argue. a(p)exp(ipx) / W(x) is a third probability. Given that it holds at a fixed x and ranges over p (it is sometimes called P(p/x)) it should be linked to a(p)a(p), but it considers spatial issues. We suggest that it should be combined with W(x)W(x) as this demonstrates resolution or measuring interaction hits and is linked to an external observer. The product probability, however, imposes the 0 points of W(x). This simply means, we argue, that a person making measurements does not interact with the particle at these W(x)=0 points, not that the particle does not exist there. Using this probability, as we have shown before, yields an average of the Shannon’s entropies created using W(x)W(x) and a(p)a(p) as probabilities. Here we try to understand the relationship between these three probabilities more closely without using the usual P(x)=W(x)W(x), P(p) = a(p)a(p) and P(p/x) = a(p)exp(ipx)/W.
Both quantum bound state equilibrium and classical statistical mechanical equilibrium are characterized by d/dt partial density(x,t)=0. For a quantum particle, W(x,t)=exp(-iEt)W(x) and d(x,t)=W*W. Density is linked to fluid flow and arguments of pressure balance such that the overall average fluid flow is zero are made. This pressure-force balance is common in classical statistical mechanics and used to calculate the variation of density with x i.e. obtain a differential equation with d d(x,t) /dt partial. A similar balance equation is used in (1) as the authors try to derive the Schrodinger equation using a fluid dynamical approach. In this note, we try to argue that one may think of the two equilibria in terms of an energy quantity which remains constant throughout space. This energy quantity depends directly on the type of interactions in the system at any point x. For a classical ideal gas, balanced two body time reversal scattering represents the interaction. V(x), a potential, does not change this fundamental manner of interacting at any given point, although it does affect density. As a result the energy related constant at each x is T, related to average kinetic energy. We argue that this already points to the dominant type of interaction creating the statistical interaction i.e. one that depends only on p in interactions which conserve kinetic energy. This “happens” to be equivalent to maximizing Shannon’s entropy, but we argue this is particular to this scenario in the sense that not all interacting systems will have this equivalence. In other words, we argue that one must be careful with regards to the maximization of entropy as a fundamental rule. In particular, we consider the single particle bound quantum state. The interaction exists between the particle and the potential V(x), but one may consider a distribution of free particle momentum probabilities i.e W(x)=wavefunction = Sum over p a(p)exp(ipx). The form exp(ipx) comes from separate arguments as is discussed in the note, but the point is that there are no two body scattering reactions occurring at a given x, neither is there acceleration from on x to another (except for a vrms mathematical quantity). Nevertheless the “free particle” exp(ipx) interacts at each x with V(x). As a result of this interaction, one has free particles exp(ipx) which are no longer associated with pp/2m, but rather with an Eave which is a constant for every p. In other words the identity of exp(ipx) is unusual because it is linked to exp(ipx) and pp/2m, but also to Vk1 a(p-k1), Vk2 a(p-k2) etc where V(x)=Sum over k Vk exp(ikx). Thus each x point is associated with a constant Eave and this defines the equilibrium. (It is also equivalent to conservation of energy at each x.) In such a case, one does not maximize entropy to find a(p) values. In the two cases, the common point seems to be considering the interactions at each x and finding a constant “energy” type quantity which does not vary with x. This we argue appears to define equilibrium for both the classical statistical and quantum bound states. Instead of considering density which does not change in time at each x, there is constant energy which not only does not change in time, but has a constant value at each x which seems to be an alternative rule.