Classical statistical mechanical density maximizes entropy subject to constraints i.e. one has the Maxwell-Boltzmann (MB) distribution P(x)P(p) proportional to exp(- pp/2mT - V(x)/T). Quantum mechanics is also a statistical theory, but it is not based on the maximization of entropy. Rather it is based on a conditional probability P(p/x) = a(p)exp(ipx)/W(x) where W(x) =wavefunction = Sum over p a(p)exp(ipx) such that: Sum over p pp/2m P(p/x) + V(x) = E (bound states). As a result, it has classical densities P(x)=W(x)W(x) and P(p)=a(p)a(p), but these in general do not maximize entropy and so do not match the MB distribution. There is a special case, however, namely the ground state oscillator for which the quantum mechanical results P(x) and P(p) are both Gaussians. It is shown in (1) that for a fixed variance, a Gaussian maximizes entropy. Thus, the MB scenario must be a Gaussian for an oscillator as it is, but the quantum result is also a Gaussian and so matches the classical statistical mechanical result. In this note, we show that the time-independent Schrodinger equation collapses to the equality situation in Heisenberg’s uncertainty principle, suggesting that the solution also represents maximum entropy.
Classical statistical mechanical density maximizes entropy subject to constraints i.e. one has the Maxwell-Boltzmann (MB) distribution P(x)P(p) proportional to exp(- pp/2mT - V(x)/T). Quantum mechanics is also a statistical theory, but it is not based on the maximization of entropy. Rather it is based on a conditional probability P(p/x) = a(p)exp(ipx)/W(x) where W(x) =wavefunction = Sum over p a(p)exp(ipx) such that: Sum over p pp/2m P(p/x) + V(x) = E (bound states). As a result, it has classical densities P(x)=W(x)W(x) and P(p)=a(p)a(p), but these in general do not maximize entropy and so do not match the MB distribution. There is a special case, however, namely the ground state oscillator for which the quantum mechanical results P(x) and P(p) are both Gaussians. It is shown in (1) that for a fixed variance, a Gaussian maximizes entropy. Thus, the MB scenario must be a Gaussian for an oscillator as it is, but the quantum result is also a Gaussian and so matches the classical statistical mechanical result. In this note, we show that the time-independent Schrodinger equation collapses to the equality situation in Heisenberg’s uncertainty principle, suggesting that the solution also represents maximum entropy.
Extremization of entropy, usually subject to an average energy constraint,(Maxwell-Boltzmann gas) is often associated with maximization of the number of microstates N gas particles may represent. The math procedure of extremization (variational calculus) is then associated with the ln of a factorial expression which reduces to n(ei)ln(n(ei)) using Stirling’s approximation. The central limit theorem, on the other hand, states that the distribution of sample means of a random variable approaches a Gaussian, i.e exp(-pp/(2sig sig)). Here sig sig is the standard deviation. For a random variable, the important or characteristic information is the standard deviation associated with the average of the square pp. We suggest here that an alternative way of looking at maximization of entropy (which is a stability statement we argue) other than ln of permutations is to consider the stability of the variation of the variance of pp subject to a change in the probability, with pp written in terms of P(p). Average kinetic energy in a statistical problem represents such a variance and should be invariant under changes of the probability subject to an average kinetic energy constraint, we suggest. How does this idea apply to quantum mechanics? In quantum mechanics, one has a statistical function exp(ipx) and for bound states creates W(x)=wavefunction = Sum over p a(p)exp(ipx). a(p)a(p) is formally the probability for p, but this does not mean that a(p)a(p) is a random variable because one has the constraint: Sum over p a(p)a(p) pp2/m + Integral dx V(x) W(x)W(x) = En. In the case of a harmonic oscillator, however, one knows a priori that the two summands are each equal to En/2. Furthermore there are a set of En values, starting at a ground state and increasing. All En values, except the ground state, are metastable. Thus the ground state is a special state. Given that Sum over p a(p)a(p)pp2/m = Eo/2, a priori, one may consider that all of the information of a(p)a(p) is associated with Eo. This is exactly the case for a Gaussian from the central limit theorem. Alternatively, from information theory, information = ln(probability) which yields the same result. If this is the case, then pp/2m a(p)a(p) written strictly in terms of P(p)=a(p)a(p) should be stationary upon changing P(p) subject to the E/2 constraint. This special situation, however, only holds for the ground state oscillator. What about average potential energy for the ground state oscillator? If the probability for p (momentum) is a Gaussian due to the Central Limit theorem, this means that the standard deviation is the information (and is linked with randomness). Taking a Fourier transform of a(p) cannot add more information to the picture, we argue. W(x)W(x) should also be a Gaussian, as it is. As a result, xxW(x)W(x), written in terms of P(x)=W(x)W(x) subject to the same Eo/2 constraint should represent a maximization or stationary situation, we argue.
In (1), we noted that a Gaussian wavefunction may be a solution to the time-independent Schrodinger equation (e.g. the ground state solution of the quantum oscillator) and simultaneously a solution of maximizing spatial Shannon’s entropy - density(x) ln(density(x)) subject to the constraint V(x) xx constant, where density(x) =W*(x)W(x) with W(x) being the wavefunction. In (2), which deals strictly with classical probabilities and the Maxwell-Boltzmann factor n(ei) = exp(-ei/T), we argued that this function is equivalent to: ln(n(ei)) = -ei/T. This in turn is equivalent to the maximization of ln(n(ei)!) with the constraint ei n(ei) which is part of: ln( N!/ Product over i n(ei)!) with the full constraint Sum over i ei n(ei)=Eave. Here n(ei)=N p(ei) where p(ei) is a probability and N a very large number. Thus a function of exp( something) may indicate that “something” * density(x) is a constraint with -density ln(density) being Shannon’s entropy. Quantum mechanics for a bound system yields a real wavefunction W(x) such that density(x)=W(x)W(x)=probability(x). Thus maximization of Shannon’s entropy - density(x) ln (density) subject to a constraint: density(x)B(x) should correspond to a solution for density of exp( B(x)). B(x) as part of the constraint, however, should represent a physical constraint. Thus if there is a solution to the Schrodinger equation of the form exp(.5B(x)) and B(x)density(x) represents the physical constraint in the system, classical maximization of probability solves the quantum problem in such a case. The above ideas may be applied to the quantum oscillator ground state and essentially repeat what is done in (1). We, however, try to generalize them to see if they may apply to a free quantum particle. In this case the probability used is exp(ipx) which is complex. It corresponds to P(x)=exp(ipx)exp(-ipx)=1 which is a uniform distribution which has maximum lack of information.
Heisenberg’s uncertainty principle for position-momentum delta x delta p >= hbar/2 is considered as an important statement regarding quantum mechanics i.e. one cannot know both p and x exactly. Delta x and delta p stand out because in classical mechanics one has an eact p and x at each t. Mathematically, however, delta x and delta p are standard deviations which may also be calculated in classical statistical mechanics, for example with the Maxwell-Boltzmann distribution. In particular, if one considers the ground state of a simple harmonic oscillator, the quantum P(x) proportional to exp(-pp/a) and P(p) proportional to exp(-xxa) match the classical statistical probabilities P(x) and P(p) for a fixed value of T proportional to sqrt(k/m) where k is the spring constant. Thus for this special T, delta x delta p for a classical statistical mechanical gas is identical to delta x delta p for the ground state quantum oscillator. The probability is normalized the same way for both. The physics of the two situations, however, is completely different. Thus the Heisenberg uncertainty principle does not necessarily describe quantum mechanical behaviour alone. If one uses entropic uncertainty, i.e. Sp + Sx where S is Shannon’s entropy, once again one uses P(p)=a*(p)a(p) and P(x)=W*(x)W(x) where W(x)=wavefunction = Sum over p a(p)exp(ipx). In principle, one could also calculate such an entropy for a classical gas i.e. use P(x)P(p) in Shannon’s entropy. The result Sp+Sx does not imply one does not know about p if one knows x i.e. the usual quantum statement.
Classical statistical mechanics often utilizes the idea of maximizing entropy subject to a physical constraint For example, for a Maxwell-Boltzmann gas, this constraint is Sum over i ei p(ei) where ei=pp/2m (no V(x)). We have argued previously that there may be many possible constraints, i.e. an average of any power of ei could be a physical constraint. We stated that a conserved quantity associated with interactions present (creating equilibrium) should be used. Here we examine the constraint in terms of mechanical resistance i.e. pressure. We further consider the free particle quantum wavefunction exp(-iEt+ipx) in terms of resistances in time and space. We argue that the form exp(-ipx)exp(ipx) = 1 removes the stochasticity associated with motion due to the resistance (impulse) p Thus 1 or 1/L (L=length) seems to be a smoothed over picture associated with a constant flux and overlooks the inherent stochasticity associated with motion. This stochasticity is crucial when describing impulse hit reactions (n-slits, index of refraction interfaces, interactions with V(x)), but if a particle does not interact in space, then the W*(x)W(x) picture seems to be used to describe the v(average)=x/t scenario. The issue is that for a quantum bound state, there is also an rms velocity based on a wavefunction and one may create an average W(x)W(x) (W(x) is real for a bound state), but like exp(-ipx)exp(ipx), this bypasses some of the stochasticity in the problem, we argue. It only demonstrates stochasticity associated with differences in momentum.
In this note, we examine the behaviour of Shannon’s entropy density for some examples in quantum mechanics. We start with a quasiparticle probability P(x and p) = W(x) fp exp(ipx) (where W(x) is the wavefunction given by Sum over p fp exp(ipx)) and obtain a Shannon’s entropy (after summing on p) : Total entropy density = -P(x) ln(P(x)) - P(p) ln(P(p)) - W(x) x d/dx W(x) ((1)) It may be noted that if f(p)=f(-p) ((1)) may also be written as: Total entropy density = -P(x) ln(P(x)) - P(p) ln(P(p)) - fp p d/dp fp We consider the cases of a particle in a box with an infinite potential at the walls and the harmonic oscillator ground state. Finally we consider the influence of u(x)= [d/dx W(x)]/W(x) on the spatial change of Shannon’s spatial entropy as W(x) and u(x) are often periodic and this leads to a periodic entropy density. In general, entropy is obtained by integrating the entropy density and one may question whether one should examine the density at all. In (1), however, entropy density is considered and it seems that there are some interesting properties of Shannon’s entropy density in quantum systems. Furthermore, there has been some recent work in the literature which makes use of entropy density to create an extra potential term for the Schrodinger equation in cases of temperature.
In previous notes, we suggested that quantum bound states may be analysed in terms of the conditional probability P(p/x)= a(p)exp(ipx)/W(x) (where W(x) is the wavefunction) which allows for the calculation of average kinetic energy at each point x i.e. [Sum over p pp/2m P(p/x)]/ W(x). This added to V(x) yields E (average energy) at each x and yields the time-independent Schrodinger equation). We argued that probability is handled at the level of W(x)=Sum over p a(p) exp(ipx). Measurements (even internal) may knock a particle from one p1 state to another so another set of probabilities (P(x) and P(p)) arise, P(x) being based on: a(p1)exp(i p1x) exp(i p2x). Averages of f(p,x) follow from P(x) f(p,x) P(p/x). Thus, there are two sets of probabilities in quantum mechanics. Entropic dynamics (1) appears to begin with the probability P(x) and considers a transition probability P(x1 |x) with x1-x= an infinitesimal quantity. A product of transition probabilities is then taken to allow for finite x1-x differences. It is stated in (1) that there exists information about x but not p. The maximization of entropy density may be used to determine the form of P(x1 | x). A “drift velocity” is also included (which is later shown to be related to the phase of the wavefunction). Ultimately, the Schrodinger equation is also obtained. We argue one may use P(p/x) to also calculate P(x1 | x) and compare this result with that obtained through the maximization of entropy density. A type of drift velocity appears naturally in this approach. We argue that the form a(p)exp(ipx) already incorporates statistical ideas (i.e. maximum entropy) in that the modulus of exp(ipx) is identical for all x, endowing all x points with a “kind” of equivalent status. exp(ipx) is a function of p and x because one must calculate average kinetic energy at each x and this must change with x i.e. KEave(x) + V(x) = E. Thus, we think there are strong similarities between entropic dynamics and the approach of using P(p/x). There are also some differences in that the coefficient A of delta(x)delta(x) in P(x1 | x) = 1/Z exp{ A delta(x)delta(x) + B d Phi/dx delta(x)} is treated as independent of x in entropic dynamics, but is a specific function of x (related to d ln(W)/dx d ln(W)/dx + 1/W d/dx d/dx W) in the P(p/x) approach.