In (1), a quantum particle in a box with infinite potential walls is described in terms of a p (momentum) distribution involving all p’s from minus infinite to infinite. The spatial wavefunction which is roughly of the form sin(kx) involves k which is the root mean square momentum. The energy E is an average energy which satisfies Integral dp p*p/2m fp*fp in one dimension. The condition of a zero wavefunction at the walls is also enforced. Here fp is the Fourier transform of W(x), the wavefunction, and is of the form cos(ap-pi/2 n)/ (E-p*p/2m) or sin(ap-pi/2 n)/(E-p*p/2m), where n is the energy level (1). In this note, we point out that fp may be obtained from a variational principle which uses W(a)=0 as a constraint. Next, we argue that there may somehow be a temperature in the wall which forces an average energy of Eave. Normally, one would treat such a system as a Fermi-Dirac gas or Bose-Einstein gas. Recently, however, (2), it has been suggested one may extremize Shannon’s spatial entropy to treat such a system. Here we extremize Shannon’s momentum entropy with the W(a)=0 constraint incorporated and compare with the case of (2).
In (1), a quantum particle in a box with infinite potential walls is described in terms of a p (momentum) distribution involving all p’s from minus infinite to infinite. The spatial wavefunction which is roughly of the form sin(kx) involves k which is the root mean square momentum. The energy E is an average energy which satisfies Integral dp pp/2m fpfp in one dimension. The condition of a zero wavefunction at the walls is also enforced. Here fp is the Fourier transform of W(x), the wavefunction, and is of the form cos(ap-pi/2 n)/ (E-pp/2m) or sin(ap-pi/2 n)/(E-pp/2m), where n is the energy level (1). In this note, we point out that fp may be obtained from a variational principle which uses W(a)=0 as a constraint. Next, we argue that there may somehow be a temperature in the wall which forces an average energy of Eave. Normally, one would treat such a system as a Fermi-Dirac gas or Bose-Einstein gas. Recently, however, (2), it has been suggested one may extremize Shannon’s spatial entropy to treat such a system. Here we extremize Shannon’s momentum entropy with the W(a)=0 constraint incorporated and compare with the case of (2).
The momentum wavefunction fp of a particle in a box (with infinite potential walls) has been described (1) (2) in terms of many plane waves (i.e. many fp’s) even though the energy E=(hbar k)/2/2m expresses only one k. Thus, the energy E appears to be an average. It seems one may have a plane wave exp(ipx) which collides inelastically with the wall and changes to exp(-ip1x) etc.; on average energy is conserved. In (1) and (2), the spatial wavefunction W(x)= Sum over p fp exp(ipx) is obtained by solving the time-independent Schrodinger equation, and fp from taking the Fourier transform of W(x). These are strictly math procedures it seems. In a previous note (3), the time independent Schrodinger equation is written with V(x) and W(x) both expressed in terms of Fourier transforms. This lead to an equation for each p of the form: Sum j+k=p Vk fk = (E-p*p/2m) fp ((1)). In such a case, V(x) exists throughout x. For the case of an infinite wall, there is still a potential, but only at the end walls. In this note, we suggest ((1)) may be used to obtain fp without taking the Fourier transform of W(x) or solving for Vk. This approach also shows cycling is involved, even for a particle in a box. The cycling occurs at the walls.
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.
Note March 9, 2023 At the same time that Fisher's information is linked with variance of an estimator, it is also linked with a classical notion of pressure for a particle in a box with infinite potential walls. Thus minimizing the variance subject to Integral P(x)dx=1 may also be thought of as extremizing pressure with respect to Integral P(x)dx=1. We argue that the cos(px), sin(px) distributions are linked with the impulse p and at the same time the particle moves with an average speed p/m (nonrelativistically). A quantum free particle is represented by the free particle wavefunction exp(ipx). This consists of two periodic distributions (in 2 dimensions) such that their modulus is 1. sin(px) and cos(px) are a possible solution set. sin(px) and cos(px) are, however, linked to a unit radius which rotates in 2 dimensions. Alternatively one may note that d/dx cos(px) = -p sin(px) and d/dx sin(px) = p cos(px). Thus one may determine the form of a free particle “probability” using the notion of periodicity, 2 dimensions and a constant modulus. No concept of extremization is needed. It is interesting to note, however, that d/dx d/dx cos(px) = pp cos(px) and d/dx d/dx sin(px) = pp sin(px). Each distribution sin(px), cos(px) by itself represents a solution of extremization problem, namely the extremization of Fisher’s information defined by I= Integral dx P(x) {d/dx ln(P)} {d/dx ln(P)} where P(x)=W(x)W(x) and W(x)=cos(px) or sin(px). Then I = Integral dx dW/dx dW/dx. Extremizing with respect to the constraint Integral P(x) = 1 yields: d/dx d/dxW = constant W. cos(px), sin(px) and exp(ipx) are solutions of this extremization equation. In order to have a solution which treats all x points the same in a certain sense, one may choose exp(ipx). The point is that even for a free particle, the two distributions involved are linked with extremization of Fisher’s information (with Integral W(x)W(x)dx=.5Length), although one may obtain exp(ipx) with no extremization and no knowledge of Fisher’s information. Why should Fisher’s information be extremized? Fisher’s information is a statistical quantity which appears in the Cramer-Rao bound: var(estimator) >= 1/ I. One may take the inverse of this equation and consider variance(estimator) to be linked to a spread or uncertainty in x. If one wishes to minimize this, one may maximize I subject to constraints. Thus there seems to be a purely statistical approach (i.e the extremization of Fisher’s information) even for a free particle. This extremization approach, however, is linked directly to the fact that one has two probability distributions in a 2-dimensional object with a constant modulus. This does not have the appearance of any extremization, but is in fact equivalent to one for each distribution function. In fact, we argue that if P(x)=W(x)W(x) with Integral P(x)dx=1, then if W(x) is cyclical/periodic i.e. d/dx d/dx W = constant W, then W may be obtained through the extremization of Fisher’s information subject to the constraint Integral P(x)dx =1. Thus periodicity and extremization are linked, but in a somewhat unusual manner.
Quantum mechanics and classical statistical mechanics/thermodynamics are both statistical theories, but have key differences. We argue that one such key difference is nonlocality in the quantum world i.e a wavelength proportional to 1/p and a probability of exp(ipx) associated with p versus locality in the classical case. Quantum nonlocality tends to lead to spatial densities with humps even for a particle with a fixed energy in an infinite potential box. Recently it has been suggested in (1) that one may think of a quantum particle in a box with infinite walls in terms of thermodynamical properties e.g. pressure which is taken to be constant throughout the box. From this assumption, one finds that quantum entropy is ln[ Average(nxnx + nyny+nznz)]. For a single quantum state, this seems to imply that entropy is ln[nxnx+nyny+nznz] ((1a)). In a previous note (2), we suggested that quantum entropy for a wavefunction (say an eigenstate) is Integral dx W*(x)W(x) ln[W*W] + Integral dp a*(p)a(p) ln[a*(p)a(p)] ((1b)). We found this is independent of L for a particle in a box with infinite walls so that displacing the wall adiabatically should mean that dS/dL = 0 which it does as L cancels from ((1)). We argue in this note that quantum pressure should be linked to the density humps for a sin(pave x)sin(pave x) wavefunction even though pave is constant at all x points. Thus it seems that one would not consider a constant pressure for sin(pave x)=wavefunction. Furthermore ((1b)) does not yield the same entropy as ((1a)). It is noted in (1) that ((1a)) is an exact expression for entropy and that the Maxwell-Boltzmann distribution entropy approximates it. In this note, we argue that a time reversal balanced scattering reaction leads to the Fermi-Dirac and Bose-Einstein distributions and these both lead to the Maxwell-Boltzmann distribution under certain conditions. (The Maxwell-Boltzmann distribution may also be obtained from time reversal balanced scattering with looser restrictions.)
Recently, there has been some interest in the literature in calculating temperature dependent quantum mechanical wavefunctions using the principle of maximum entropy. The idea seems to be to obtain a differential equation, similar to the Schrodinger equation, but with temperature dependent terms. It should be noted that this does not follow von Neumann statistical quantum mechanics. It should also be noted that other authors in the literature maximize or minimize other quantities related to information theory to calculate a wavefunction. The objective of this note is not to examine the maximization of entropy as applied to the calculation of a wavefunction, but to suggest a different form for entropy. Traditionally, authors seem to use the Shannon form -k W*(x)W(x) ln[ W*(x)W*(x)] as a spatial entropy and exchange W(x) for fp for the momentum one, with W(x) being the wavefunction. It is the spatial form which they seem to maximize in calculations. In this note, we consider a probability P(x intersection x) = W(x) fp sin(px) and use it in Shannon’’s entropy. Maximization of entropy then leads to an integral equation which, it seems, needs to be solved numerically, We can however, multiply such an equation by fp and sum over p to obtain results which can be compared with those of authors who use -k W*(x)W(x) ln[ W*(x)W*(x)] . NOTE: In later preprints, we have used W(x) fp exp(ipx) as probability and calculated Shannon's entropy. This has given an overall result for entropy including a term for Shannon's entropy for W(x), another term with Shannon's entropy for fp and a final term W(x) x d/dx W(x). This last term, linked to exp(ipx), is also equal to the covariance of x and d/dx ln(W(x)) which is used in the derivation of the Cramer-Rao inequality for Fisher information.
In (1), a variational approach is applied to Fisher’s information (with constraints) to obtain a differential equation in P(x)=probability to be at x. It is suggested the equation may be simplified using P(x)=W(x)W(x) where W(x) is real. The result is the nonrelativistic Schrodinger equation with mass=1. As a first observation, we note that Fisher’s information, which is a purely statistical quantity, should not necessarily yield a nonrelativistic equation which is an approximation of relativistic results. In particular, we argue that Fisher’s information is used to find <pp> not necessarily 1/2m <pp>, but this is a minor point. A more important issue, we argue, is the meaning of P(x) in a bound system. A classical bound system is sometimes said to have a P(x)=C/v(x) i.e. P(x)dx is proportional to dt, the amount of time a particle spends in each equally sized dx. In the high energy limit, the envelope of a quantum bound probability P(x)=W(x)W(x) (W(x)=wavefunction) is said to roughly match C/v(x), yet this density does not follow from the extremization of Fisher’s information (together with two general constraints which yield E and V(x)=potential). This begs the question: What is P(x)? If one considers a particle moving at constant speed (i.e constant momentum) then P(x)=constant because each dx point carries the same probability. Classically a particle placed in a box with infinite potential walls would have a P(x) =constant and p and -p motion would be separated in time. Thus if P(x) is to have meaning in a box, one must conclude that the probabilities associated with p and -p are not constant, but rather functions of x. This then leads to the notion of a complex new kind of probability (which maps to P(x)=constant) for a free particle i.e. W(x) =cos(px) + i sin(px) so that the modulus W*(x)W(x)=P(x)= 1. These ideas seem to go beyond the notion of Fisher information. What then is the role of Fisher information? We argue it is related to two very specific moments of the distribution cos(px) or rather W(x) = Sum over p a(p)cos(px) with a(p)=a(-p), namely and <pp>. In classical physics there exists p and pp at each x point (but one is the square of the other), but with distributions <p does not equal to <pp> in general. Both and <pp> are linked to physical concepts which exist in Newtonian mechanics, namely impulse and work. With the presence of cos(px) one has a combination of Newtonian mechanics and a statistical distribution. We argue that the role of Fisher’s information is to allow one to transform from to <pp>. In particular, -idW/dx appears in Fisher’s information (and is linked to ), but the specific role of a variation is to transform it into <pp> which appears in a conservation of energy equation. We argue that this is why Fisher’s information may be used to derive the Schrodinger equation.