We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region.
We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region.
We calculate formal time derivatives of the mean values of the standard position, velocity, and mechanical momentum operators, i. e., the Ehrenfest theorem for a one-dimensional Dirac particle in the coordinate representation.We show that these derivatives contain boundary terms that essentially depend on the values taken there by characteristic bilinear densities.We do not automatically take the boundary terms to vanish (as is usually done); nevertheless, we relate the boundary terms to similar terms that must be zero if one requires the hermiticity of certain specific unbounded operators.Throughout the article, we thoroughly discuss and illustrate all these aspects, which include the relations to certain boundary conditions.To clarify, we call our approach formal because all operations involving operators (for example, some operators products) are performed without respecting the restrictions imposed by the sets of functions on which the self-adjoint operators can act.Moreover, the Dirac Hamiltonian that we consider in our calculations contains a potential that is the time component of a Lorentz two-vector; nevertheless, we also obtain and concisely discuss the Ehrenfest theorem for a Hamiltonian with the most general Lorentz potential in (1+1) dimensions.
In the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).
A process exists in the literature (1) of creating an information term based on Fisher’s information using spatial density P and adding it to a classical Lagrangian based on a velocity field i.e. velocity=dS/dx. Maximization of S and P is performed and a differential equation in terms of spatial density and S is obtained. Finally, it is noted that a change of variables: W= sqrt(p) exp(iS/hbar) may simplify the differential equation so that it becomes the Schrodinger equation with W as the wavefunction. In this note, we consider the wavefunction as appearing “naturally” as a normalization constant when using conditional probability and argue that W is the velocity field. Given that the Schrodinger equation includes a term of the form -1/2m d/dx d/dx W (in 1-D) i.e. the first derivative is related to the average velocity and the second to the ensemble average of p*p. This follows because unnormalized conditional probability f(p,x) is the Fourier basis function i.e. exp(ipx). The second derivative term - 1/2m d/dx d/dx W may be obtained from a Lagrangian containing (d/dxW) (d/dx W), thus formally one may use maximization to obtain the Schrodinger equation at a level using W directly. The maximization form, however, is identical to the “classical translational” form (2) of Fisher’s information. Thus, it seems that the Fourier basis exp(ipx) and the maximization of classical translational Fisher’s information are linked. In this note, we wish to investigate this in further detail. We also use translational properties together with conditional probability to show that f(p,x) where P(p/x)= a(p) f(p/x) / [Sum over p a(p) f(p/x)] is exp(ipx). In particular, we obtain an equation taken from the translation of the conservation of energy equation at point x to x+delta x. This links df(p,x)/dx with -dV/dx (the force). If the force is stochastic and yields “k” momentum hits, -dV/dx= Sum over k kVk(x). We argue that df(p,x)/dx also brings out this k term i.e. is an eigenfunction of k i.e. -idf(p,x)/dx =k f(p,x) yields exp(ipx) as a solution. This f(p,x) is automatically an orthonormal basis (multiplied by an appropriate constant) and helps to show why the “momentum operator” is linked to a “translation operator” i.e. d/dx. The bound state Schrodinger equation is considered in this note.
Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲, and in the other, a different quantity j̲̃ appears, which also has the units of a flux density. The quantum flux density j̲ is related to the probability density ρ via the continuity equation, and j̲̃ may as well be used to define a density ρ̃ that fulfills a continuity equation. Employing a model for the coupled dynamics of an electron and a proton, we document the properties of the densities and flux densities. It is shown that although the mean momentum derived from the two quantities is identical, the various functions exhibit a very different coordinate and time-dependence. In particular, it is found that the flux density j̲̃ directly monitors temporal changes in the probability density, and the density ρ̃ carries information about wave packet dispersion occurring in different spatial directions.
It was argued in a number of previous notes that for the nonrelativistic case, W(x,t) transforms under x->x+a, t-> t+b (with a and b infinitesimal) as exp(ipx - iEt), except with p replaced with = - i d/dx W(x,t) and E replaced with i d/dt W(x,t). It was suggested this was a driving condition of nonrelativistic quantum mechanics, and if this is the case, one might expect these ideas to carry over into relativistic quantum mechanics. For relativistic quantum mechanics, however, one has the Einstein energy-momentum relationship E*E= p*p + mo*mo (c=1) which leads to -d/dt d/dt W(x,t) and -d/dx d/dx W(x,t) being used. The momentum term is fine because it is related to < p*p conditional >, but the second derivative of time seems to be a little confusing as it represents < E*E conditional > and not <E conditional >. If <E conditional> has physical significance, as it seems to be the case in nonrelativistic quantum mechanics, one might expect an equation with this time flux i.e. i d/dt W(x,t) / W(x,t) = < E conditional > and not < E*E conditional >. Dirac wrote such an equation, but obtaining a single time derivative comes at a price. The Dirac equation, in one dimension, contains matrices, d/dx and d/dt in addition to a two vector wavefunction (U(x,t), V(x,t)) instead of W(x,t). Thus, the Dirac equation in one dimension leads to coupled equations with U(x,t) and V(x,t). The single derivative d/dx is deceptive because U(x,t) can be written in terms of d/dx V(x,t) and vice versa, so ultimately the Dirac equations contain d/dx d/dx which is needed to obtain < p*p conditional >. The question is: How can one obtain < E conditional >? This may be done by using a (d/dt + constant ) -1 operator i,e. almost an inverse time operator, in other words either (E+m)-1 or (E-m) -1 where E can be thought of as id/dt. Thus, the RHS of the Dirac equations is i d/dt U(x,t) or i d/dt V(x,t) which suggests an infinitesimal step forward in time, while the LHS side contains (id/dt +m)-1 or (-id/dt-m) -1 which seems to be representative of undoing such a step. It should be noted that one may take Einstein’s energy-momentum equation E*E = p*p + mo*mo, without any knowledge of quantum mechanics, and cast it as 2x2 matrix equation, linear in E, namely H (U,V) = E (U,V). The matrix H serves as an evolution operator and another matrix serves as a velocity operator. One may derive the Schrodinger zitterbewegung equation for the velocity matrix without any use of wavefunctions and derivatives which are key to quantum mechanics. Nevertheless, the idea of noncommuting matrices, vectors and zitterbewegung is present. There are several choices for the matrix H, but one leads to the one dimensional Dirac coupled equations Thus, the “spin” features may be obtained directly from E*E= p*p+mo*mo without any quantum mechanics. (1), (2). The purpose of this note is to examine some of these ideas.
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.