In Newtonian mechanics there exists a continuous x i.e. for a conservative force: .5mv(x)v(x) + V(x) = E with x taking any value. This is associated with the idea of dx→0. As a result one may choose an x and x+dx with dx→0 and v(x) and v(x+dx) become arbitrarily close in value. It is almost as if their identities merge. In (1) entropy and average energy are calculated using the Maxwell-Boltzmann MB factor exp(-e/T) and a continuous energy. The results E= NT (k=1) and S = -ln(T/To) with To being an integration constant show that one must have T>To because T→0 does not lead to S→0. Thus there is a problem if one uses continuous energy. In other words continuous energy is rejected.. (1) then considers an equally spaced energy i.e. e=bn, n=0,1,2,3 … and calculates energy and entropy again. In this case S→0 as T→0. Thus (1) concludes that it is entropy which leads to the discretization of energy In this note we try to argue that discretization of energy is already implied by the MB factor. The MB factor exp(-energy/T) where T is a math parameter called temperature represents a probability. In other words, different energies have different probabilities i.e. one must distinguish between them. Having two energy values merge is a problem now because one has a second parameter T which may also go to zero i.e. e2-e1→0 and T→0 leads to trouble because exp(-e2/T) does not tend to zero as it should if e1=0. We argue that the underlying reason for e1 and e2 being discretely different is that one cannot use the probability/T formalism i.e. MB factor unless interactions occur. We argue that such two body interactions cannot ultimately transfer or remove a de→0 and keep e1 and e2 distinct. In other words the interaction/collision is ultimately linked to a finite energy transfer even though T→0. As a result energy should be discretized in an equilibrium scenario. This argument does not make use of entropy explicitly although implicitly having exp(-e2/T)-->0 as T→0 (say for a 2 level system) means that the particle is in state e1 which may be associated with an entropy of 0 if one ignores internal features of this state. We also note that the continuous energy result E=NT, and not just the entropy, leads to trouble because as T becomes extremely tiny E which is an average of say e1 and e2 with e1=0 as an example approaches T in magnitude. The smaller T becomes the smaller e2 becomes which e2 and e1 no longer have separate identities, but in such a case one does not have an equilibrium scenario either. In other words exp(-e/T) implies interactions between distinguishable states with respect to energy.
In Newtonian mechanics there exists a continuous x i.e. for a conservative force: .5mv(x)v(x) + V(x) = E with x taking any value. This is associated with the idea of dx→0. As a result one may choose an x and x+dx with dx→0 and v(x) and v(x+dx) become arbitrarily close in value. It is almost as if their identities merge. In (1) entropy and average energy are calculated using the Maxwell-Boltzmann MB factor exp(-e/T) and a continuous energy. The results E= NT (k=1) and S = -ln(T/To) with To being an integration constant show that one must have T>To because T→0 does not lead to S→0. Thus there is a problem if one uses continuous energy. In other words continuous energy is rejected.. (1) then considers an equally spaced energy i.e. e=bn, n=0,1,2,3 … and calculates energy and entropy again. In this case S→0 as T→0. Thus (1) concludes that it is entropy which leads to the discretization of energy In this note we try to argue that discretization of energy is already implied by the MB factor. The MB factor exp(-energy/T) where T is a math parameter called temperature represents a probability. In other words, different energies have different probabilities i.e. one must distinguish between them. Having two energy values merge is a problem now because one has a second parameter T which may also go to zero i.e. e2-e1→0 and T→0 leads to trouble because exp(-e2/T) does not tend to zero as it should if e1=0. We argue that the underlying reason for e1 and e2 being discretely different is that one cannot use the probability/T formalism i.e. MB factor unless interactions occur. We argue that such two body interactions cannot ultimately transfer or remove a de→0 and keep e1 and e2 distinct. In other words the interaction/collision is ultimately linked to a finite energy transfer even though T→0. As a result energy should be discretized in an equilibrium scenario. This argument does not make use of entropy explicitly although implicitly having exp(-e2/T)-->0 as T→0 (say for a 2 level system) means that the particle is in state e1 which may be associated with an entropy of 0 if one ignores internal features of this state. We also note that the continuous energy result E=NT, and not just the entropy, leads to trouble because as T becomes extremely tiny E which is an average of say e1 and e2 with e1=0 as an example approaches T in magnitude. The smaller T becomes the smaller e2 becomes which e2 and e1 no longer have separate identities, but in such a case one does not have an equilibrium scenario either. In other words exp(-e/T) implies interactions between distinguishable states with respect to energy.
Note Nov. 12, 2022 In (1) the discrete energy is e=bn b=constant and n=0,1,2,3... not n=1,2,3 .. as listed below. Ideas of entropy appeared before quantum mechanics was developed. In particular Planck introduced discrete energy levels: en where e= constant and n=1,2,3 to describe allowed energies of an oscillator representing allowed light in a blackbody as shown in (1). In such a case average energy and entropy may be calculated using formulas from Boltzmann. In (1) it is shown that if continuous energy is used entropy is undefined as T=temperature→0 while entropy→0 as T→0 if there are discrete energy levels. As noted this problem applies to light and uses an oscillator model, but leads to the notion of discrete energy levels. In this note we suggest using a particle in a box instead and consider only two levels for simplicity. We still use Boltzmann’s probability factor exp(- energy(i)/T) where i=1,2. In such a case the two probabilities are 1-exp(-e(i)/T) and exp(-e(i)/T). If there is a gap between e(1) and e(2) then as T→0 the particle is in state e(1) i.e. has an entropy of 0. If there is no gap then one has a problem as e(2)-e(1)-->0, but T→0 as well. Thus it seems that a gap must be introduced in order that entropy behave properly as already argued in (1). This gap for the particle in a box, however, violates Newtonian mechanics which allows all momentum p values. As a result given that a gap is needed it may also be the case that e1 does not correspond to p1=0 as in Newtonian mechanics. We used the term box above and argue that the concept of equilibrium and T seem to apply to a confined system i.e one with a length constraint for a 1D example.. For example, the Boltzmann factor exp(-e/T) may be derived considering probability densities n(e(i)) with e1+e2=e3+e4 and n(e1)n(e2) = n(e3)n(e4). This is the reaction balance approach, but it seems that the particle must be confined for such an approach to even be defined. Thus we argue that there is something “special” occurring due to this constraint i.e. confinement in space. If energy has gaps then for a particle in a box: energy = pp/2m so p=momentum itself must have gaps. Furthermore these gaps must be somehow linked to the length of the box. Thus momentum is being linked (and energy too) with length. In fact ,given that one has gaps and uniformity throughout the 1D box, one may consider the length being broken into integer equal sized portions L/n. This is already a crude notion of a wavelength and leads to the idea of allowed p momentum values being associated with L/n in some manner. Finally we note that Fermat’s principle already links an invariant space variable with momentum conservation in the same direction. The classical analogue is Maupertuis’ principle for particles with rest mass. In both cases one may link these approaches with the Lorentz invariant A=-Et+px which happens to be both the relativistic and nonrelativistic classical actions of a free particle. dA/dx partial = p so an eigenvalue equation is: -id/dx exp(ipx) = p exp(ipx) with b/p representing the wavelength. Thus one has a frequency proportional to 1/E and a wavelength proportional to 1/p. Any p, however, is allowed so there is no gap. There is, however, an association of length with p. Thus if a box with a T has a length constraint one may suggest fitting integer numbers of sine wave crests into the box to create the “gap” required by entropy considerations.
Newtonian mechanics deals with two types of force delivery schemes, one linked to the variable x and the other to t. They may be written as: Integral Fdt (impulse) and Integral Fdx (work) and appear to be symmetric in the two variables. The problem is that expressed in terms of dp one sees an immediate difference linked to discrete versus continuous change. In particular: dp= Fdt ((1a)) and vdp = Fdx ((1a)). ((1a)) allows for a discrete change in p in a tiny interval of time. “X” does not come into play at all in the equation and one might think of the impulse occurring at a point x. Due to the discrete nature of integral dp changes, one may think in terms of probabilities because a given pA may have been obtained from discrete impulses: p1→p2→p3 or p4+p5. etc. The final p state is the same and so not only does there exist probability but also the notion of product probabilities such that exp( pi b) exp( pj b) = exp( (pi+pj) b) where b is a constant. Given that exp(pb) increases exponentially, one might rather use the periodic probability exp(i p b). One may note how impulse type hits and their discrete momentum results lead to stochasticity in a Maxwell-Boltzmann gas. The form vdp = Fdx is very different in that it does not allow discrete jumps of p because v=dx/dt. If one assumes a very large F at x1 and dx–0 such that the product is not 0, one might consider dp changing discretely, but what is then used for v? Instead one considers a range of x with a corresponding range in t such that x(t) in order to compute v. Thus this is a deterministic situation because x and t cannot be decoupled. It leads to the notion of a kinetic energy pp/2m associated with p. This kinetic energy is calculated by considering p change from 0 to p. An interesting feature arises because impulse = Integral Fdt may be thought of as discrete allowing p1 to jump to p2. Nevertheless, the deterministic approach states there is a kinetic energy associated with p1 and p2, namely p1p1/2m, p2p2/2m, even though no deterministic path in time and space was taken. The impulse was said to occur at x in a very tiny time range. Thus there seems to be a contradiction. pp/2m requires a range of x and t with x(t). The impulse only occurs in time, with x decoupled but there must be an associated x range which is then not deterministic i.e. a decoupled x range. This suggests a characteristic length associated with a momentum p in which one does not follow time deterministically. If one does not follow x in t in this special length, then one has a probability distribution. A similar argument occurs for time. Consider Fdx with time absent. If dx becomes very tiny and F(x) very large , one may think that work is done at an instant in time, but there must be a time range for pp/2m. Thus there is a special time period not associated with x in which one has probabilistic behaviour i.e. not x(t). One may note from the notion of impulse that a given p may be obtained through different series of smaller impulses, e.g. {p1,p2,p3} and {p4,p5}. These are associated with kinetic energy series {p1p1/2m, p2p2/2m,p3p3/2m} and {p4p4/2m, p5p5/2m}. The final result is the same. Thus the discrete nature of changes leads to a probability scheme such that P(p1)P(p2) = P(p1+p2) and P(e1)P(e2)=P(e1+e2). This we argue leads to: P(p) = exp(i p b1) and P(e) = exp(i e b2) where b1 and b2 at first seem to be constant. We noted, however, that an impulse implies a probabilistic range in x and Fdx→finite when dx→0 a probabilistic range in t. Thus b1 may be replaced with x and b2 with t. At any given point in time or space, x and t are constants i.e. b1 and b2. Next, we consider the situation of special relativity i.e. a Lorentz boost. We consider an initial state of energy= mo, x=0, t=to and view it from a frame moving at a constant velocity v. Although one may use infinitesimal velocities, a main point we wish to make is that special relativity allows for the use of finite velocities which give the sense of discrete changes like in the impulse case.This should again be associated with probabilities. Special relativity yields the Lorentz invariant A = -Et+px where v=x/t. Nevertheless, x and t are written in the invariant as if they were independent. It may be noted that dA=0 if t changes on its own as hbar/E and x on its own as hbar/p. Thus one may obtain specific x and t probability ranges.This is again consistent with the probabilistic picture or exp(-iEt) and exp(-ipx). It may be noted that even with the probabilistic picture, a particle with constant p moves determinisitcally on average as x= p/m t. If x = hbar/p, then 1/t is not hbar/E and if t=hbar/E then x is not hbar/p. Thus the values hbar/E and hbar/p seem not to be linked with the current deterministic motion, but rather with the creation of p in the first place. E=pp/2m and this is obtained by considering p changing from 0 to p. Thus the “frequency” E seems to capture the information of the creation of p and we argue that the wavelength captures the creation of x as a probability distribution in x which one would usually associate with force/acceleration.
We have argued in previous notes that C(T) exp(-ei/T) (the Maxwell-Boltzmann probability) and exp(-iEt) exp(ipx) (the free particle quantum probability called a wavefunction) have the same informational structure in that product probabilities for a given E1=Ei+Ej for the MB case and Ei,pi and Ej, pj for the quantum case lead to uniform distributions, i.e. ones with minimal information if energy and momentum are conserved. We suggest here that the root for this may be same. We have argued in previous notes that dp/dt = Force cannot have dt be infinitesimally small. This means that there exists a scale at which dt is not 0 or even tiny. We also suggested that the force (even though obeying action-reaction in collisions) may strike at some probabilistic time within dt. We then used special relativity to find that dt =hbar/E and dp=hbar/p. A probability which demonstrates these gradations is exp(-iEt+ipx) (and also Lorentz invariant). This is akin to the notion that p(e1)p(e2)=p(e3)p(e4) and p(p1)p(p2) = p(p3)p(p4) (one dimension for simplicity). In other words, the stochasticity introduced is one of a uniform distribution for product probabilities. One could obtain the same result by starting with this assumption and then suggesting Lorentz invariance. We suggest that this quantum uniform distribution product probability also holds for the Maxwell-Boltzmann case. There, the probability is not complex because p(ei) must be a real number linked to the number of particles with ei. Nevertheless, in an elastic collision of ei and ej, we suggest that the quantum result (and even dp/dt=F) suggest that there is stochasticity in x and t and that one cannot deterministically predict the outcome. Any outcome which conserves energy and momentum has the same likelihood. We suggest that it is the quantum mechanical assumption which is identical to the ideal gas case (which uses quantum mechanical gas molecules albeit at high temperatures). The high temperatures mean that one can only use the conservation laws and the associated stochasticity. We suggest that the stochasticity of interaction inherent in quantum mechanics carries over to collisions in the classical gas. It is not simply that one is approximating the situation in a gas by throwing away information. If one were to try to be as deterministic as possible at some scale (the quantum one) stochasticity arises and there is no full determinism in a collision. The best one can do is use conservation of momentum and sometimes conservation of energy in certain situations like high temperature. If this is the case, one may ask: Why is it possible to perform deterministic classical calculations such as particles bouncing off walls or mirrors or interacting with a potential V(x) in a completely deterministic manner. We suggest that these involve constrained interactions, i.e. the wall or V(x) is associated with a huge mass compared to the particle, unlike in a collision of two identical particles. Action-reaction in the former case is associated with the large mass and so always the same approximately 0 kinetic energy and hence no p(ei)p(ej) =p(ek)p(el), but in the latter, with a mass the same size and so the stochasticity may be seen clearly.
The Maxwell-Boltzmann MB distribution is often derived by maximizing the number of possible arrangements of particles with different energies subject to the constraint of a fixed number of particles and fixed energy. Using the canonical approximation that only average energy is conserved and using Stirling’s approximation for large number factorials leads to the Maxwell-Boltzmann distribution. The ln of total number of arrangements using Stirling’s approximation is linked to a math form called Shannon’s entropy i.e. - Sum over i f(ei) ln(f(ei)). Interestingly this same MB result may be obtained through reaction balance which makes no use of the idea of arrangements or entropy or maximization. It is concerned with including all possible physical reactions (for example two body reactions between all energy particles) subject to conservation of energy. In this sense it deals directly with the dynamics of the problem while the arrangement approach does not. The dynamical approach considers all possible reactions with notions of independence or lack of information and ensures the system remains unchanged. Why are they the same? We argue that it is because the “recipe” for the probability of two body reactions namely n(e1)n(e2) is of the same form as the leading factor of n(e1)! n(e2)!. In the case of quantum mechanics one may apply the dynamical approach to obtain the time-independent Schrodinger equation. In such a case one does not need to think in terms of “maximizing” any quantity although one considers all possible reactions with a potential V(x). a(p) factors in W(x) = wavefunction = Sum over p a(p)exp(ipx) are interrelated, but retain their identity after all possible interactions occur. This we argue is the idea for the eigenvalue equation. At the same time, Shannon’s entropy, which is directly linked to the notion of arrangements, is applied to a bound state and possibly linked to thermodynamics i.e. dE = TdS -PdV which may be applied to an adiabatic expansion. Shannon’s entropy is created from existing probabilities in this case which do not arise from explicitly maximizing entropy subject to constraints as the probabilities follow directly from the Schrodinger equation. One might ask why Shannon’s entropy has any relevance if it is the dynamics which determines the value of W(x) and a(p). It seems, however, that one may knock out a particle in a state p with probability P(p)= a*(p)a(p) or find a particle at x with probability P(x)=W*(x)W(x). These are classical type probabilities linked to the quantum dynamical probabilities a(p) and W(x). It seems Shannon’s entropy is only applied to P(x) and P(p) or a product P(x)P(p) which may give it physical meaning in a quantum bound state. We examine these ideas in this note.
Historically Newtonian mechanics appeared before the statistical mechanical treatment of a Maxwell-Boltzmann (MB) gas. In this note, however, we initially investigate a MB gas in terms of information which we define as -ln(density(x)). In equilibrium there are no macroscopic changes in time so there is only one variable namely x in one dimension. The density may be changed so it is variable while a potential V(x) is not. Thus we argue that there should be a fixed piece of information based on x in the system i.e. V(x). We then argue (as in (1)) that equilibrium involves minimizing the information so we suggest: -ln(density(x)) = C V(x). This ultimately leads to the form d(x) = C1 exp(-V(x)/T). Next we focus on a Newtonian system, but unlike the MB gas two variables x and t are needed to describe the macroscopic changes for the particle even though x suffices to describe V(x). {In particular one may send in a particle with an initial speed at xo and it will move through x,t in a certain manner. Sending another particle with the same initial speed at xo at a later time leads to the same motion. Thus we argue there is a given V(x) function in the system. } We also suggest there should exist a density d(x) and an information given by -ln(d(x)). One may see from (2) that a Newtonian system may be described by a density(x) even though it is usually only described by x(t), velocity, acceleration etc. The density is linked to a variable quantity of “information” in the system. Given the presence of V(x) we divide the x axis into equally spaced dx units. Thus d/dx the translation generator applies to V(x). For the moving object, however, one has a corresponding dt(x) for each equal dx and the dt’s are usually not the same. In other words the ratio dx/dt = function(x), but it is dt which changes. We place dt in the denominator so that it mimics the time translational generator d/dt. One cannot simply equate function(x) and C V(x) because function(x) does not change as d/dx, rather it changes because dt values change. Thus we suggest using d/dt to express its changes. As a result we propose: d/dt (dx/dt) = C dV/dx ((1)) as both sides of the equation must be vectors. Now d/dt= dx/dt d/dx so both sides of ((1)) may be expressed as functions of x. One then obtains function(x) d/dx function(x) = C dV/dx where function(x) = dx/dt. We call this v(x). v(x) is adjustable so we equate it with information i.e. -ln(d(x)) where d(x) is density as there is no reason to have two different functions of x. Thus d(x)=C/v(x). In other words density(x) is proportional to the “information” dt at each x. In (2) a density is ascribed to a Newtonian system using dt as the definition of density. Here we wish to link density to information= -ln(d(x)). As a result one finds that ((1)) involves information squared unlike the MB gas which involves information -ln(d(x)) to the first power. As a result: d/dx ( v(x)v(x)) = C dV/dx which may be integrated to yield Newton’s conservation of energy: .5v(x)v(x)+ C V(x) = constant. In other words we suggest it may be possible to obtain Newton’s second law using ideas of information. There is further loss of information as the constant applies to all x points so the the information from .5v(x)v(x) combines with CV(x) to eliminate x information. (x information, however, is present in each term i.e. .5v(x)v(x) and CV(x)).
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.