In Parts I and II we examined equilibrium situations based on an additive scalar (distributive variable) such as ei= pi dot pi/2m in a Maxwell-Boltzmann gas and a vector dynamic quantity such as the momentum vector in a reflection/refraction of light situation or a single particle quantum bound state. We argued that a loss of information in the form P(a1)P(a2)=P(a3) holds in both cases, but in the additive scalar situation this leads to P(a1) = C exp(-a1/T) while in the vector case to exp( -i a1 dot conjugate variable). For a one dimensional quantum bound state a1=p and the conjugate variable is x. In such a case, exp(ipx) is a periodic eigenstate of the translational generator multiplied by -i. Within the bound state, stochastic hits from V(x) the potential lead to an ensemble i..e W(x) = Sum over p a(p)exp(ipx) where a(p)’s are fixed by considering average energy at each x to be the same i.e. { Sum over p a(p) pp/2m exp(ipx) } / { Sum over p a(p)exp(ipx)} + V(x) = En. In this note, we consider the a(p)’s as also being probabilities like exp(ipx) and ask whether they could exhibit the same loss of information as in the vector p if one associate p in a(p) with the additive scalare pp/2m. We argue that such considerations lead to the ground state solution of a quantum oscillator.
In Parts I and II we examined equilibrium situations based on an additive scalar (distributive variable) such as ei= pi dot pi/2m in a Maxwell-Boltzmann gas and a vector dynamic quantity such as the momentum vector in a reflection/refraction of light situation or a single particle quantum bound state. We argued that a loss of information in the form P(a1)P(a2)=P(a3) holds in both cases, but in the additive scalar situation this leads to P(a1) = C exp(-a1/T) while in the vector case to exp( -i a1 dot conjugate variable). For a one dimensional quantum bound state a1=p and the conjugate variable is x. In such a case, exp(ipx) is a periodic eigenstate of the translational generator multiplied by -i. Within the bound state, stochastic hits from V(x) the potential lead to an ensemble i..e W(x) = Sum over p a(p)exp(ipx) where a(p)’s are fixed by considering average energy at each x to be the same i.e. { Sum over p a(p) pp/2m exp(ipx) } / { Sum over p a(p)exp(ipx)} + V(x) = En. In this note, we consider the a(p)’s as also being probabilities like exp(ipx) and ask whether they could exhibit the same loss of information as in the vector p if one associate p in a(p) with the additive scalare pp/2m. We argue that such considerations lead to the ground state solution of a quantum oscillator.
In Part I we considered two kinds of statistical processes. One is based on a scalar additive which changes in stochastic processes e.g. kinetic energy in two body elastic collisions or the spatial shell in a 3D random walk problem. We argued that in such a case, one considers a1+a2= a3 (where ai is the distributive scalar) and a product probability scheme i.e P(a1)P(a2) = P(a3) to find that P(ai) = C exp(-ai b) where b is a parameter which may be found if the average value of a is known. Furthermore, we argued that the distributive scalar process is identical to maximizing entropy subject to a constraint Sum over i ai P(ai). The second process is based on a dynamic vector changing in the stochastic process. We used the problem of light reflecting and refracting (which we argued is similar to quantum mechanics for a pure bound state) and noted that the p (momentum) vector changes i.e. an incoming p along the x axis in the positive direction becomes -p on reflection or p2 on refraction in 1 dimension. We suggested that a dynamic probability is needed in such a case, namely an eigenfunction of the generator associated with the dynamic variable, e.g. x for p. The eigenfunction is then periodic. Using exp(ipx)s (eigenfunctions of -id/dx) on may calculate average energy at x and fix weights i.e. {Sum over p pp/2m a(p)exp(ipx)} / { Sum over p a(p)exp(ipx)} + V(x) = En, but there is no maximization of entropy. At first the additive scalar (distributive variable) and dynamic vector approaches seem to be very different (for example, one maximizes entropy and the other does not), but in this note we argue there are a number of similarities. We focus on these similarities and suggest these may perhaps be key features that unite the distributive scalar approach (classical statistical mechanics, level jumping in quantum mechanics) with the dynamical vector approach (statistics of pure quantum bound state). In particular, both types of probabilities are linked with exponential functions although one is real and drops in value exp(-ei/T) and the other is complex and periodic exp(ipx) and P(a1)P(a2) =P(a3) in both cases. Also, both are eigenfunctions of generators. Thus we suggest that the similarities between the scalar additive approach (MB gas) and dynamical vector (quantum bound states) may be the key features of overall equilibrium type statistical mechanics.
It is sometimes stated in the literature (1) that quantum mechanics is based on conditional probabilities. If this is the case, one would expect such conditional probabilities to appear in a single particle bound state. A free quantum particle is associated with a probability to interact of exp(ipx) (linked to 2-slit interference, 1-D reflection-refraction at an n1-n2 junction etc), but the kinetic energy and momentum are pp/2m (nonrelativistic) and p at all x even though each x has a different phase according to exp(ipx). The momentum is p at x, but the probability for an impulse hit at x is not the same even though its modulus is. When different probabilities exp(ipx) add in an OR situation, this leads to a spatial pattern which no longer has a modulus of 1, i.e. translational invariance is broken. Such a pattern also occurs for a single particle bound state W(x)=Sum over p a(p)exp(ipx). W(x) can have 0 points, but the average kinetic energy -1/2m d/dx dW/dx / W is not 0 at such a point. One has, on average (a statistical calculation), no average interaction at such an x point. One might ask why this should have anything to do with average kinetic energy? If one writes a conservation law: KEave(x) + V(x) = 0, then one considers kinetic energy interacting with potential energy. If there is no interaction at some x point for a given p (not W(x) on average), then it should not be part of the conservation of energy at this point. One is only interested in kinetic energy which interacts with potential energy at each x point. Thus, a(p) exp(ipx) + a(-p) exp(-ipx) (with a(p)=a(-p)) may have a 0 point at some x and so it does not have an average interaction at such a point, i.e. pp/2m (a(p)exp(ipx) + a(-p)exp(-ipx)) = 0. As a result, in a bound state, one considers kinetic energy per p, -p multiplied by the average probability (p, -p). Different a(p)exp(ipx) + a(-p)exp(-ipx) (a(p)=a(-p)) weights in this average picture indicate the extent to which the particle interacts. This idea holds for a single p,-p pair, but one must normalize over all such pairs, leading to the idea of a conditional probability. Given that hbar/p must be “created physically” in space so to speak, one must have interaction probability which creates this physical situation even in cos(px)+ i sin(px). This leads to the idea of interaction probability being 0, positive, but <1, or even negative and it is the interaction which allows one to consider conservation of energy with a potential V(x) in the first place. As a result, one cannot include a full pp/2m if there is negative, 0 or only particle interaction. Thus, pp/2m is multiplied by 2 a(p)cos(px) /W(x) for a(p)=a(-p).
In classical statistical mechanics, e.g. a Maxwell-Boltzmann (MB) gas, one tries to find the probability distribution of a particular quantity such as particle energy pp/2m. Thus there is a distribution in a quantity, making it the stochastic variable. This quantity, however, is subject to a conservation law i.e. elastic two particle scattering. In a previous note (1), we considered the case of reflection/refraction and called momentum the stochastic variable and noted that it too was linked with conservation. In fact, we argued that both the MB gas and reflection/refraction followed the same stochastic law: P(variable1) P(variable 2) = P(variable 1 + variable 2) whether variable 1 is a scalar or vector. In (2), however, we noted that exp(ipx), a free particle wavefunction, treats p as a parameter with x being the “probability” variable. This seems a little unusual because p, which is normally a variable in an MB gas and even in a quantum bound state i.e. a*(p)a(p) where W(x)=wavefunction = Sum over p a(p) exp(ipx) takes on the role of a parameter i.e. p=hbar/wavelength. Why this duality? Why is p a parameter for “x” of the probability distribution and yet at the same time a variable associated with conservation of momentum? In this note, we argue that unlike the distribution of a defining property of a particle such as momentum or energy, one may have a distribution associated with probability itself. This we argue is the case for a reflection/refraction problem which applies to both light and a particle. In the case of light, there is an interface of two media with different indices of refraction and in the case of a particle an interface with two constant potentials. In both cases, total energy of the photon/particle is constant, but there are two choices or probabilities, reflection or refraction, but they are linked with different times. A single photon or particle can only undergo one at a time. One knows exactly the two values of momentum, but the probability of each is unknown. Thus unlike an MB gas where energy/momentum is being redistributed because the two particles of the outcome exist at the same time, here a probability of 1 must somehow be distributed to the two events.. Given that the incident momentum is p, the reflected -p and the refracted p2, only considering a specific time event i.e. reflection or refraction, does not conserve momentum. In the MB case, a single time event i.e. scattering conserves both momentum and energy. Thus it seems that one does not deal with events in time. There is a distribution of probability in such a way as to ensure conservation of momentum, but does this probability already exist or get created at the interface point? We argued in (3) that if the scenario were deterministic one would have to follow it in time and treat each event separately. Rather we suggest that a distribution of probabilities allows for conservation of momentum, but this means that all three momenta exist at the interface x point as one is not considering an event at a precise point in time. This then allows for each momentum to have its own x distribution based on momentum p so that one may impose conservation of momentum and not have a deterministic time associated with the interface point.. This last idea of conservation of momentum appearing is not discussed in (3) and allows for p to behave like a parameter for x while at the same time being a variable which is conserved. In short, in an MB gas one distributes energy between two colliding particles in order to conserve total energy while for a quantum particle or photon which reflects/refracts one distributes probability to the two scenarios (reflection or refraction) such that momentum is conserved. Thus the nature of what is “distributed” is very different.
Classical statistical equilibrium, for a Maxwell-Boltzmann gas for example, is based on localizing the gas to a certain volume and having no average changes in time. In other words, maximization of entropy subject to the average energy constraint sum over i ei P(ei) is linked with these two ideas. Ei (kinetic energy for no V(x)) is a conserved quantity in reactions and overall energy represents a quantity which does not change in time consistent with a center-of-mass which does not change and average values which do not change in time. This approach is associated with a specific kind of statistics, namely that of the constituents of the system. As a result, one obtains the Maxwell-Boltzmann distribution exp(-ei/T) where ei is an extensive property of a particle (or subset of the system) and T an intensive property linked with the whole system. Entire systems, however, may interact. For example, a N particle system with T1 may achieve equilibrium with a N system with T2 yielding a new average (T1+T2)/2. The probabilities exp(-ei/T), however, are for subcomponents of a system. One could argue that momentum is an additive (vector) quantity which is conserved in collisions, but the average momentum of the system is 0 which does not allow one to describe increases in average p and -p which may be described using energy, i.e. through T temperature. This begs the question whether momentum may be associated with maximization of entropy subject to a constraint. In (1), we argued that there were flow probabilities in quantum mechanics linked with exp(-iEt) and exp(ipx), which may be obtained formally through maximization of entropy. Here we argue that while an MB gas describes a localized volume with average values not changing in time, x and t state values change, even if p and pp/2m remain constant for a free particle and it may be possible to have a maximum entropy situation to describe the combination of px and -Et. In other words, we suggest localized versus nonlocalized equilibria. The latter represents a quantum free particle, but one may then try to localize such a particle in a potential. What happens in such a case? One does not switch to a localized maximization of entropy as in the MB gas case, because it is possible to have the potential leave the flow unbroken as argued in (1). In other words, the nonlocalized free particle equilibrium is akin to a particle bouncing back and forth in a box, which in classical mechanics is not considered a statistical process. Here we suggest that the underlying picture is statistical meaning that the classical mechanics approach seems to be an approximation. We consider both a quantum particle trapped in an infinite potential well and the ground state of an oscillator to examine ideas of maximizing entropy beyond exp(ipx). A key common feature of both types of equilibrium is the product rule P(ei)P(ej) = P(ei+ej) or exp(ip1x)exp(ip2x) = exp(i (p1+p2)x). We consider this feature as well.
Classical statistical mechanics is often presented in terms of the maximization of entropy (ln(# arrangements) or ln( total probability for a set of arrangements)) subject to a constraint. In a previous note (1), we argued that such a process seems to be associated with a distributive variable which is a scalar i.e.is additive such that a product of probabilities Product over i n(ei) of different values of this variable Sumover i ei = E is equivalent to a single probability of E i.e. n(E). We applied this idea to both a Maxwell-Boltzmann gas and a 3D random walk. This distributive variable changes in the stochastic process of the problem. For example for an MB gas, there are elastic collisions so energy is the distributive variable. For an MB gas in a potential V(x), there is potential and kinetic energy of a single particle, but this is not regarded as a stochastic process in the classical view. In (2) we argued that a distributed variable is not the only type of equilibrium which may exist, even in the classical world. We considered the example of a beam of light which reflects and refracts at x=0 (medium along the y axis). Energy is conserved even at the photon level, but there is an unknown probability namely that for reflection “a” because the refraction probability is then 1-a. Thus energy conservation is not enough. We argued that in order to solve for this problem in an equilibrium or steady state manner, one needed to deal with square roots of fluxes/intensities in order to obtain two equations to solve for B,C i.e. square root fluxes at x=0 for the reflected and refracted rays in terms of A, the square root flux for the incident beam. We argued these fluxes were associated with eigenfunctions of the translation generator i.e. exp(ipx). Thus a dynamic equilibrium is introduced (i.e. quantum mechanics) which does not use a distributed variable i.e. does not use maximization of entropy. Rather it is associated with eigenfunctions of the generator of the dynamical motion involved. It is this dynamical variable (e.g. p) which changes in this stochastic process even if in direction only. A question arises: When does one use one type of equilibrium approach versus the other? We attempt to answer this in this note.
We have argued in previous notes that both Newtonian mechanics and free particle quantum mechanics arise from special relativity. Thus, special relativity gives an expression for free particle energy and momentum and these are related to concepts of work. The Lorentz invariant -Et+px shows that x,t and x+hbar/p, t+hbar/E give the same solution to this potential-like term showing that there is stochasticity in the position and time delivery of interaction hits. In other words, even though the center-of-mass moves as x=vt, a momentum hit (proportional to rest mass mo) does not necessarily act at the center-of-mass. This gives the impression of shifts in probability, kinetic energy etc in the case of interference, i.e. interactions. For a single free particle, there is no interaction and -1/2m d/dx d/dx exp(ipx) = pp/2m exp(ipx). We argue that exp(ipx) is a probability and that in general one may add independent probabilities in an OR situation as in a 2-slit interference case. A bound state, however, is different because even though one adds weighted exp(ipx)s, they are part of a resonance or equilibrium and this is no longer an independent OR situation. We suggest that one uses a conditional probability when computing a Newtonian nonrelativistic conservation of energy equation to try to remove as much of the quantum features from this equation. As a result, one has: KEclassical(x) + V(x) = E = Sum over p pp/2m P(p/x) + V(x). One cannot bypass the fact that a quantum particle interacts differently from a classical one and that this involves a probabilistic granularity (wavelength), but classical notions of energy and energy conservation still exist. As a result, the bound state wavefunction Wn(x) = Sum over p an(p) exp(ipx) is not a usual OR sum, but is rather 1 over a normalization factor for an equilibrium picture. As a result, one cannot use Wn1(x)Wn2(x) = W(n1+n2) (x) as one would for independent probabilities. Furthermore, given that one has Wn(x) and Wn1(x)+Wn2(x) which are both of the form Sum over p a(p)exp(ipx), and both represent average energies, one needs a way to distinguish a resonance/equilibrium from a sum of resonance/equilibrium normalization factors. We suggest that the orthogonality of resonance Wn(x)’s provides this feature. The standard textbook explanation is based on Hermitian operators, but in the nonrelativistic case, we argue that one may obtain the result based on integration by parts based on the form of the p operator = -grad. Thus, it is ultimately the form exp(ipx) which leads to orthogonal bound state Wn(x)’s, together with the boundary condition that Wn(x)-->0 of x-> +infinite and -infinite.