Recently, there have been a number of articles in the literature describing Schrodinger type equations modified to account for temperature. In general, these are of the form of the regular Schrodinger equation with an extra potential term dependent on temperature and the wavefunction. It should be noted that temperature dependent Hartree Fock theory for a nucleus includes temperature induced changes to the potential felt by a single particle and so the single particle levels calculated are modified by temperature. Temperature dependent Hartree Fock, however, is not the same as the new Schrodinger type equations presented in the literature. In this note, we consider the case of a single particle in a box and consider dU=TdS, where U is the energy, T the temperature and S Shannon’s entropy in momentum space. We obtain a value for fp , the momentum distribution, by varying fp and use a Fourier transform to find the wavefunction. We then compare the result to predictions of equations in the literature.
Recently, there have been a number of articles in the literature describing Schrodinger type equations modified to account for temperature. In general, these are of the form of the regular Schrodinger equation with an extra potential term dependent on temperature and the wavefunction. It should be noted that temperature dependent Hartree Fock theory for a nucleus includes temperature induced changes to the potential felt by a single particle and so the single particle levels calculated are modified by temperature. Temperature dependent Hartree Fock, however, is not the same as the new Schrodinger type equations presented in the literature. In this note, we consider the case of a single particle in a box and consider dU=TdS, where U is the energy, T the temperature and S Shannon’s entropy in momentum space. We obtain a value for fp , the momentum distribution, by varying fp and use a Fourier transform to find the wavefunction. We then compare the result to predictions of equations in the literature.
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 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.
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 (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).
The Schrodinger equation, a partial differential equation in W(x,t) the wavefunction, was developed by Schrodinger from ideas of L. de Broglie and then given the interpretation of probability Probability(x,t)= W*(x,t)W(x,t) by M. Born. In this note we ask whether the form P(x)=W*(x)W(x) together with P(p)=a*(p)a(p) such that W(x)=Sum over p a(p)exp(ipx) follows from quantum thermodynamics. In particular we concentrate on a quantum adiabatic transformation and insist it be one of no entropy change (as in (1)). The quantum spatial density of a particle in a box with infinite potential depends on L as P(x)= 1/L density(x/L) (bound state). Thus Shannon’s entropy simply using P(x) is L dependent. One needs P(x)P(p) in addition to the first term changing by ln(1/L) and the second by ln(L) in order to remove L dependence during an adiabatic expansion. Similar considerations hold for the quantum simple harmonic oscillator (2). Thus the idea of a P(x) and P(p) which are linked (not independent) immediately follows form quantum thermodynamics as does the form of entropy S=Sx+Sp. As a result it seems that if one deals with probability, one must not only consider conditional probability, but also a transformation function f(px) which allows for inverse dependencies on p and x on a parameter. These ideas together with P(p/x)=P(x and p) / P(x) ((1a)) and P*(x/p) = P(x and p) / P(p) lead, we argue, to the form P(x)=W*(x)W(x) and P(p)=a*(p)a(p) where W(x)=Sum over p a(p) exp(ipx). We considered the approach of conditional probabilities in an earlier note (3), but did not give motivation for using them or the form exp(ipx). Rather we brought them in as assumptions, assuming that there should be both a P(p) and P(x). In this note, we argue that they seem to follow from the idea of quantum thermodynamics and a zero entropy change along an adiabatic path i.e. one which does not change the quantum bound state n.
In this note, we wish to examine how the variance of u(x)= [Sum over p p fp exp(ipx)] / [ Sum over p fp exp(ipx)] namely: P(x) Conditional Variance(p) changes as one moves from one point to another. If one tries to eliminate u(x) from the first order change, one obtains d/dx P(x) = 2 u(x) P(x) which is a quantum mechanical result. This also leads to the change in the variance depending only on the force and kinetic energy. d/dx P(x)=2 u(x) P(x) also dictates how Shannon’s spatial entropy changes from point to point. Thus, it seems that the time-independent Schrodinger equation may be thought of in terms of changes in entropy density. The ideas here are a little different from those of (1).