The Role of Gradient of the Logarithm of Density in the Quantum Mechanics, Classical Mechanics, the Euler Equation, Equilibrium Statistical Mechanics and Brownian Motion
It is interesting to note that d/dx [ln(d(x))] where d(x) is the spatial density, plays a central role in the time-independent Schrodinger equation, classical mechanics, the Euler equation, equilibrium statistical mechanics and Brownian motion and in all cases, even the quantum one, is related to force as -d/dx V(x), where V(x) is the potential. In this note, we try to examine the role of d/dx [ln(d(x))], and in particular try to see how quantum mechanics is related to classical and fluid mechanics and how it differs.
It is interesting to note that d/dx [ln(d(x))] where d(x) is the spatial density, plays a central role in the time-independent Schrodinger equation, classical mechanics, the Euler equation, equilibrium statistical mechanics and Brownian motion and in all cases, even the quantum one, is related to force as -d/dx V(x), where V(x) is the potential. In this note, we try to examine the role of d/dx [ln(d(x))], and in particular try to see how quantum mechanics is related to classical and fluid mechanics and how it differs.
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.
In the literature (1) it has been noted that (d/dx W)/W - (d/dx W*)/W* represents a classical velocity, where W is the wavefunction. For bound states, this expression is zero, yet (d/dxW)/W seems to play the role of density flux as d/dx (WW) = 2W d/dxW. When multiplied by E, the energy, it becomes and energy flux as there is constant energy at each point x. In this note, we wish to consider some examples related to this flux. In particular, for the case of V(x)=0 (no potential), the flux becomes a solution of the time independent Schrodinger equation. This and other considerations may suggest that periodic type functions (possibly related to photons or phonons) when multiplied by the wavefunction W, can lead to a new, higher energy wavefunction. We also consider the role of energy flux as it relates to discrete energy levels in quantum mechanics.
In a previous note, it was suggested one can use definitions of temperature and heat flow from the Boltzmann transport equation and calculate similar quantities for a bound state quantum wavefunction. It was also suggested that one could not use the “transport equations” such as the Euler equation for the quantum bound state system. In another note, however, it was shown that the quantum bound state contained important terms of the Euler equation, namely -d/dx ln(d(x)) and F(x)d(x) where d(x) is the density and F(x) the force. In this note, we consider the definition of pressure from the Boltzmann transport equation and calculate its gradient. We find the usual F(x)d(x) term in addition to an extra term. In setting F(x)=0, we find the extra term coincides with the gradient of pressure for a quantum particle in a box with no potential. Thus, it appears that a quantum particle, even without a potential, represents a pressure gradient which affects the form of the Schrodinger equation and causes quantum particles to differ from those of classical and classical statistical mechanics.
In a quantum bound state,spatial density is given by W(x)W(x), where W(x) is the wavefunction, average momentum by -i d ln(W) /dx and average kinetic energy at x by -1/2m d/dx d/dx W / W. One may note that: -1/2m d/dx d/dx W / W + V(x) = E is the time independent Schrodinger equation which may be solved for W(x) and E without any concept of spatial density. Furthermore, one may calculate and <p*p>, also without any concept of spatial density. This differs from the case of a classical fluid situation for which there is no W(x). We argue that kinematics of a bound state are contained in W(x) and that the Schrodinger equation is like a probability conservation hydrodynamic equation with spatial density replaced with W(x) and a V(x)W(x) term which we try to explain. In addition, the “conservation of momentum” hydrodynamic equation is essentially “d/dx” of the Schrodinger equation (if mass=m=.5). This strategy differs from the general approach used throughout the literature (for example (1) and many other cases), where spatial density is taken as the starting point of a quantum hydrodynamic formalism. Spatial density is utilized in a continuity equation of the form d/dt [W*W] + d/dx [W*W v] = 0 where v is a collective velocity and d/dt the partial derivative with respect to time. We argue one only needs the concept of a wavefunction as the normalization constant of the conditional probability P(p/x)= a(p)exp(ipx) / W(x) i.e. W(x)= sum over p a(p) exp(ipx). From this wavefunction, one may calculate and <p*p>. Thus we argue that hydrodynamic type equations should apply to W(x). We pointed out in (2), that W(x) in fact satisfies a momentum continuity hydrodynamic equation if mass=.5.
Quantum mechanical spatial probability densities may be obtained by solving the time independent Schrodinger equation. Statistical mechanical spatial densities may be obtained by maximizing Shannon´s entropy d(r)ln[d(r)] (where d(r) is the spatial density) subject to appropriate constraints. A classical spatial probability density may be obtained from 1/v(x) where v(x) is the velocity at position x obtained from kinetic energy + potential energy (x) =total energy. Recently, there have been attempts to link these various spatial probability densities. For example for the ground state wavefunction of a harmonic oscillator, the Schrodinger equation can be obtained from varying Shannon´s entropy with constraints because V(x) W*(x) W(x) (where W(x) is the wavefunction) can be written in the form of Shannon´s entropy times a constant [3]. Furthermore, Casal et al have maximized Tsalis´ entropy to obtain good approximations to the ground state wavefunctions of oscillators and Coulomb potentials. In [1], the classical density 1/v(x) has been calculated using energy levels from quantum mechanics, namely E=hbar w (n+.5). This density has in turn been used to calculate entropy and compared with results from wavefunctions of the Schrodinger equation with good results even for low n values even though for n=0, the wavefunction W(x)=1/C exp(-ax2) and 1/v(x) are not at all similar. The purpose of this note, is to attempt to examine the relationship between these different spatial densities.
In (1), the Schrodinger equation is derived using a hydrodynamical continuity of density approach. Thus, spatial density represents x probability. There, however, is also a flow velocity field in the problem v(x) and it is assumed this is irrotational. From this it is stated in (1) that v(x) is proportional to 1/m d/dx S(x) ((1)) where S(x) is some function and m is mass. In this note, we argue that a particle moving in one dimension with constant p (momentum) is automatically irrotational, but it should satisfy ((1)) from (1). In other words, a constant p particle hence constant velocity is associated with a function of x, S(x) even though its density in space is not. If one considers S(x)=ln(B(x)), then ((1)) is equivalent to: p = d/dx ln(B(x)). If one tries to associate B(x) with a probability, then ln(B(x)) is information = px. B(x), however, is a different kind of x probability from density(x) which should be constant for a free moving particle. In fact, using -id/dx instead of d/dx and exp(ipx) for B(x), one finds exp(ipx) as a second x probability. Its modulus yields the constant value of density(x). Thus the irrotational flow assumption leads to p= dS/dx. Incidentally, this form is equivalent to Fermat’s least time principle for light with p=p (x component). We argue that applying p=dS/dx associates an x probability distribution even to a constant p, so the assumption p=dS/dx is essentially a quantum mechanical statement, even if one does not derive the Schrodinger equation. The time independent Schrodinger equation may be obtained by using exp(ipx) as a relative probability and using energy conservation.