The spatial and momentum wavefunctions of a ground state quantum oscillator are Gausians in x and p respectively. A Gaussian is also the distribution which maximizes Shannon’s entropy - Sum over i P(i) ln(P(i)) for a fixed variance. Usually one would use average energy as a constraint, but for the nonrelativistic oscillator KE = pp/2m and PE=.5kxx so such a constraint is linked to the variance of x and p. In this note we consider the following questions? Why is the ground state of the quantum oscillator (i.e this form) the only quantum solution which matches a maximum entropy case (for a fixed variance)? Is there a link to a lack of information? Why do higher oscillator states represent non-maximum distributions for fixed variances? Can this also be linked to information? Here we apply the term information to actual information in the system not ln(P(i)).
The spatial and momentum wavefunctions of a ground state quantum oscillator are Gausians in x and p respectively. A Gaussian is also the distribution which maximizes Shannon’s entropy - Sum over i P(i) ln(P(i)) for a fixed variance. Usually one would use average energy as a constraint, but for the nonrelativistic oscillator KE = pp/2m and PE=.5kxx so such a constraint is linked to the variance of x and p. In this note we consider the following questions? Why is the ground state of the quantum oscillator (i.e this form) the only quantum solution which matches a maximum entropy case (for a fixed variance)? Is there a link to a lack of information? Why do higher oscillator states represent non-maximum distributions for fixed variances? Can this also be linked to information? Here we apply the term information to actual information in the system not ln(P(i)).
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.
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.
It is known that a quantum bound state wavefunction may have humps which are linked to the energy state n (e.g. a particle in a box with infinite walls or an oscillator). W(x)W(x) (bound) represents a probability distribution. For a fixed variance, a Gaussian distribution maximizes entropy. This is the wavefunction of the ground state of an oscillator, but adding structured phonons not only increases the energy of the state (i.e. moves it from the ground to an excited level), but gives structure (humps) to the distribution. We argue that these humps represent information about the way in which the state may decay. Thus quantum bound states differ from maximized entropy states subject to an overall constraint linked to say energy etc. There seems to be a correlation between higher energy and spatial information which ultimately follows form the momentum-spatial part of the classical action: px (from -Et+px) being held constant.
In the literature, one often finds a quantum expression for Shannon’s entropy of the form: - (W*W) ln(W*W) ((1)) where W*W is spatial density i.e. W(x)= the wavefunction. This implies W*(x)W(x) is an independent probability. Given that W(x)= Sum over p a(p) exp(ipx), and W*W= Sum over p1, p2 a(p1)a(p2) exp(ip1 x) exp(ip2 x), one may see that for p1/-p1 in the latter with a(p)=a(-p) leads to cos(p1x) terms. These are positive and negative in different x regions. In fact, in many cases, the overall spatial density exhibits peaks and valleys suggesting that W*(x)W(x) is correlated, i.e. not independent we argue. Furthermore, the form ((1)) uses only position when there is an expectation value for < p*p> an , both functions of x, where p is momentum. Thus, the variance of p-ensemble average may change with x, yet position only is used in the computation of Shannon’s entropy, which we argue seems unusual. In a previous note (1), we suggest using a(1)exp(ip1 x) a(p2)exp(ip2 x) as the probability for Shannon’s entropy. This leads to a spatial form 2W x d/dx W which matches ((1)) for the ground state of the quantum oscillator case. For the ground state oscillator, W*W=C exp(-ax*x) i.e. a Gaussian. This suggests that for this case only, W*(x)W(x) represents an independent probability which may be used in Shannon’s entropy, we argue. We attempt to consider these issues in this note.
For a classical ideal gas or single particle quantum system with energy levels {ei}, both average energy E=Sum over i ei P(ei/T) and Shannon’s entropy S= - Sum over i P(ei/T) ln(P(ei/T) change (for mechanical parameters kept fixed) in terms of d(P(ei/T)). In fact, in this case ln(P(ei/T)) appears as a “constant vector” in dS i.e. dS= - Sum over i ln(P(ei/T)) dP(ei/T). Both dE (mechanical parameter fixed) and dS change as dP(ei/T), thus there are two vectors {ei/T} and {ln(P(ei/T))} which represent constants over ei’s for a small range of dT where T is temperature. In this case, however, the two vectors are strongly linked as P(ei/T)=exp(-ei/T)/C(T) so ln(P) = -ei/T - ln(C). This is why the relationship E-TS = -T ln(C(T)) exists. For a quantum bound state one may write: e = Integral dp a(p)a(p) pp/2m + Integral dx V(x) W(x)W(x) where W(x)=Sum over p a(p)exp(ipx) is the wavefunction. Instead of having vectors of ei’s, one has two vectors, one over x and the other p (momenta) where a(p)a(p) = P(p) and P(x)=W(x)W(x) (bound states). Thus d e thermal = Integral dp pp/2m dPe(p) + Integral dx V(x) dPe(x) ((1)). In other words de thermal changes linearly in terms of dPe(p) and dPe(x) whereas for a classical ideal gas or many ei system it changes linearly in terms of dP(ei/T). Ei is the variable instead of x and p in that case. In other words {pp/2m} is a constant vector in p space and {V(x)} in x space. Mathematically Shannon’s entropy Sp = - Sum over p Pe(p) ln(Pe(p)) and Sx =- Sum over x Pe(x) ln(Pe(x)) also yield constant vectors in their respective spaces namely ln(Pe(p)) and ln(Pe(x)) which are called information. In particular dSp = - Sum over p ln(Pe(p)) dPe(p) then ln(Pe(p)) behaves like a constant vector within a region of small dPe(p) changes. Note: Sum over p dPe(p)=0. One may consider a set of single valued energy levels ei which are very close (imagine a particle in a box with infinite walls) as a region in which dPe(p) changes (or dPe(x) ones) are very small. In this region, ln(Pe(p)) and ln(Pe(x)), information, represents a constant vector statistically defining the small region of ei states. This we argue is the statistical significance of information ln(P). One may note that both de ((1)) adn dSp and dSx change linearly with dPe(p) and dPe(x), but the vectors {pp/2m} and {V(x)} over p and x space and ln(Pe(p)) and ln(Pe(x)) over the same space are constant vectors, but not equivalent except for the ground state of the quantum oscillator for which they become very similar because Pe(p)=exp(-app) and Pe(x)=exp(-xx/a) with a being proportional to 1/ e which plays the same role as temperature in the classical ideal gas case. For other quantum bound systems, the two sets of vectors are different, but the information vectors seem to give a kind of statistical identity to small regions containing very close ei levels for which dPe(p) and dPe(x) are small. Thus, though one may argue for the presence of ln(P) in Shannon’s entropy S=-Sum over i P(i)ln(P(i)) in many ways, it seems that ln(P(i)) is a constant vector in a certain space i for small dP(i) and gives a statistical identity to that small region.
Shannon’s entropy: -k Sum over i P(i) ln(P(i)) is expressed in terms of P(i) the probability for state i. If entropy exists as a concept separated from the first law of thermodynamics with dWork=0, then it is not clear what set of states i should be taken in quantum mechanics. In classical statistical mechanics, each particle is automatically described by its position and momentum vectors for no internal motion. For example a pure energy eigenstate wavefunction Wn(x) may be written in terms of many different orthonormal mathematical basis sets e.g. Wn(x)= Sum over i bi phi(i)(x) with probabilities bi*bi. Thus one may create many types of entropy expressions using Shannon’s form without any other physical considerations. For example, W(x)=Sum over p a(p)exp(ipx) and one may create Sp which is usually done in the literature. In previous notes we have argued that Sn = Sp(n)+Sx(n). For a particle in a box with infinite potential walls, Sx(n)=Sx. The sum of the two entropies ensures that the parameter L (box size) vanishes so that a quantum adiabatic transformation is isentropic. Thus a consideration other than Shannon’s form has been used to obtain Sn=Sp(n)+Sx for a particle in a box. Furthermore one may argue Shannon’s form even for a pure state based on S=S1+S2 where 1 and 2 represent two systems. A question still remains: Why are Sx and Sp entropies used and not some other set? Furthermore for a system with several energy levels ei one might ask why -Sum over i P(ei)ln(P(ei)) is used as in (1)? In a previous note (2) we considered that Shannon’s form of entropy follows from dE= TdS (dWork=0) with dE=Sum over i ei dP(ei). Furthermore we argued that P(ei) = C(T)exp(-ei/T) is a solution, but so is C(T) exp(-ei/T) exp(Si) where Si=Sx,i + Sp,i for the pure bound state. In other words, specific states related to the physical situation are chosen and entropy constructed from dE=TdS to reflect this. In this note we argue that entropy may be less related to counting arrangements than to considering the issues involved in energy transfers within a system.