Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. In classical mechanics, one has the conservation equation: p(x)*p(x)/2m + V(x) = E. In quantum bound systems one has: KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component p*p/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that: p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x) ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)
Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. In classical mechanics, one has the conservation equation: p(x)p(x)/2m + V(x) = E. In quantum bound systems one has: KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component pp/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that: p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x) ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)
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.
In previous notes, we argued one may formulate bound state quantum mechanics by assuming a stochastic potential which delivers “hits” of momentum k V(k) exp(ikx), while having an average form of V(x). A main constraint seems to be that the different k V(k) exp(ikx) hits may occur at any x point and that V(x) be of the form Sum over k b(kx) so that -d/dx b(kx) = -k b(kx). In other words, conservation of energy exists at the level of each hit as well as for the overall average. To force all of these constraints, a solution of b(kx)=V(k) exp(ikx) emerges which introduces periodicity and complex numbers. It should be noted that V(x)= Sum V(k) exp(ikx) so each term is actually a “piece of potential energy”. Furthermore, we argue that V(x) is decomposed for physical reasons i.e. V(x) is a low resolution result which incorporates all of the “pieces”. In a quantum mechanical situation, a “piece” V(k)exp(ikx) may act at t1 and V(k2)exp(i k2x) at t2. We argue that exp(ikx) should have physical meaning and leads to sin(kx) terms for the “k” and “-k” pieces. Thus, there is forward and backward motion which “cancel” i.e. have nothing to do with V(x). For a particle, however, in this region, there may be forward and backward motion, both of which have kinetic energy which should not be part of the average kinetic energy at x. The various physical (resolved) pieces of the potential lead to momentum distribution at each x point with P(p/x)= a(p) exp(ipx)/ W(x) (W(x)=wavefunction). In this case, one has a probability, not “pieces” of KE(x). At first, one may be alarmed by complex probabilities, but they are necessary to allow for cancellation of kinetic energy pieces due to the positive/negative values combined V(k)exp(ikx) pieces. Thus, even though KE(x) (average) matches classical results, the actual motion is more complicated. The potential seems to be the driver of the ensemble picture and should be linked to the microscopic physical nature of the potential. In this note, we wish to examine features and consequences of such a stochastic potential, including the idea of tunneling and infinite potential walls.
In classical statistical mechanics, C exp[ -( mv*v/2 + V(x))/T) represents density i.e. the probability for a particle to have a velocity v and be at x. If applies this probability to Shannon’s entropy formula, one obtains C/T ( mv*v/2 + V(x)) exp[ -( mv*v/2 + V(x))/T) + exp[ -( mv*v/2 + V(x))/T) ln(C). Thus, it seems energy times density + density times ln(C) is the entropy density, a well known result in classical statistical mechanics, although sometimes not explicitly stated. The first term integrated over x is essentially the kinetic energy, while the second integrated over velocity yields the potential energy term. Thus, the first term partial density seems to be a purely momentum dependent kinetic energy piece times probability and the second a purely x dependent potential energy piece. (There is still an integral over velocity for the first term and one over x for the second to obtain the full entropy.) In quantum mechanics, we argue that a(p1)exp(ip1 x) a(p2) exp(ip2 x) serves as the probability where W(x)=Sum over p a(p) exp(ipx) = wavefunction. This leads to an expression for entropy density with two terms, one of which may be immediately integrated over momentum and the other over space. If one wishes to make a link to the classical statistical mechanical case, one may ask what requirement is necessary for the momentum related piece to represent average kinetic energy and the spatial part, average potential energy. The requirement, we find, yields the wavefunction of the ground state harmonic oscillator. For other cases, the partial density is not equivalent to average energy. Thus, we try to establish a link between entropy density and energy density, but require the kinetic energy related term to be strictly a function of momentum and the potential related term strictly one of x. In quantum mechanics, one may form averages at a point x. For example, KEave(x)= [Sum over p p*p/2m a(p) exp(ipx)]/W(x). If one examines a p/-p pair with a(p)=a(-p) (for example), cos(px) terms arise which are positive at some x and negative at others. We argued in previous notes that this represents stochastic flow in the opposite direction from that expected by -dV/dx. In other words, -dV/dx is an average which consists of stochastic force addends which may give momentum hits to the right or left, but which add to yield -dV/dx. As a result, one may have negative contributions to the kinetic energy (and density as well). In this note, we argue one may also have negative contributions to the entropy density due to this phenomenon.
Quantum mechanics is often compared with classical mechanics because both may deal with a single particle subject to a potential. Classical mechanics is said to be deterministic describing x(t) and its derivatives i.e showing how a particle moves and accelerates while bound state quantum mechanics produces a real wavefunction W(x) such that W(x)W(x) yields a spatial density profile. The comparison is carried further with the correspondence principle which states that at high energy levels quantum mechanics should mimic classical mechanics. Classical mechanics does not simply describe x(t) and its derivatives i.e. deterministic motion. It also includes two body scattering, force and torque balances for a stationary object etc. In the case of the classical statistical mechanics of an equilibrium ideal gas, the Maxwell-Boltzmann (MB) distribution in the absence of a potential is C1 exp(- pp/(2mT) ). If a potential V(x) is added, one may find P(x) two different ways using Newtonian ideas. The first is to consider the gas as a collective entity like a tiny macroscopic classical object at x. Then one may argue that equilibrium means the system is stationary and apply Newtonian statics i.e. a balance of pressures and -dV/dx to obtain exp(-V(x)/T) as is often done in textbooks. Alternatively, one may consider a single particle and apply Newtonian mechanics differently, namely use a conservation of energy equation: p1p1/2m + V(x1) = p2p2/2m + V(x2). Again one obtains exp(-V(x)/T). The spatial part of probability may be based on Newtonian mechanics indicating that it is not stochastic. The stochastic function is C1 exp(-pp/2mT). Interestingly, however, the two Newtonian approaches are different with respect to a single particle. A collective nonmoving set of particles acts as a stationary object, yet within it each particle seems to be “bouncing” off of an imaginary wall creating pressure. Thus one has an impulse free particle picture yet obtains the correct result exp(-V(x)/T). Alternatively one may think of a single particle accelerating which it does if it is not colliding with other particles. This is a very different picture of the single particle than of one bouncing back and forth hitting an imaginary wall. In fact, there is usually a wall holding the gas together and pressure and bouncing particles are the real picture there, but within the gas one would normally think of a single particle accelerating. Nevertheless there are two scenarios which yield the same result. The idea of pressure balance is more in keeping with an overall statistical picture of the gas. In other words, one may actually ignore the idea of following a single particle as it accelerates. (That is not to say that single particle acceleration does not exist - it does.) Quantum mechanics we argue is a statistical theory. In the case of a single bound particle one has a momentum distribution W(x)=wavefunction = Sum over p a(p)exp(ipx). One may apply conservation of average energy at each x to obtain the time-independent Schrodinger equation. In such a picture, like with the statistical mechanical gas, we argue that one may imagine impulse hits with the potential instead of considering acceleration. The quantity vrms(x) from KEclassical= .5m vrms(x)vrms(x) = -1/2m d/dx dW/dx / W does display acceleration, but this is a mathematical average. As energy levels become higher the correspondence principle is said to apply, but a classical particle exhibits acceleration. The quantum bound state does not, but like in the case of the statistical mechanical gas it may be that two scenarios lead to the same result. In other words, the quantum bound W(x) still has crests and troughs, but with low resolution one may only see an envelope roughly touching W(x)W(x) which matches a classical C2/v(x) where v(x) is classical velocity. Given that quantum mechanics is a statistical theory it may not be necessary to have it explicitly display acceleration.
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.
In Part I, we considered the ideal gas law P= density(x) RT of a Maxwell-Botlzmann gas, which follows from the notion of impulse 2p multiplied by v (flux) and then averaged,and examined how it balanced (on two sides of a box) a force -dV(x)/dx multiplied by density. We argued that in a quantum bound state there is no notion of internal time and hence no flux and so considered an average impulse 2p multiplied by an overall internal time in the system, namely hbar/E. We found that this scenario balanced with -dVd/x matches the quantum oscillator ground state. In this note we ask: What about other potential V(x) situations? Quantum mechanics seems to be a mixture of statistical mechanics and Newtonian dynamics i.e. the time-independent Schrodinger equation is, on average, an energy conservation equation for all energy levels i.e. KE(x)+V(x) = En where KE = {Sum over p a(p)pp/2m exp(kpx)}/{Sum over p a(p)exp(ipx)}. In (1) we took d/dx to bring out a force term -dV/dx and showed if is linked to <ppp/2m> and <pp/2m> , but analyzed these in terms of fluxes. The notion of flux makes sense using external time, but internally there is no time and so here we try to examine the situation in terms of various impulses.