The continuity equation: d/dt partial d(x,t) + grad dot d(x,t)u(x,t) = 0 ((1)) where d(x,t) is density and u(x,t) a velocity field appears in the literature as a fundamental equation of “continuity”. For example, it follows from Boltzmann’s transport equation (and conservation of mass) and leads to a wave equation for sound when combined with a pressure equation (1). A continuity type equation also follows from Maxwell’s em equations and includes the Poynting momentum b El x B where b is a constant, El electric field and B magnetic field. This equation describes light. Finally the continuity equation is used in a number of derivations of the Schrodinger equation (e.g. (2)). In previous notes (3) we have argued that two flux/flow equations for free particles: d/dx partial A = p ((2a)) and d/dt partial A = -E ((2b)), with A=-Et+px being a solution, hold for both quantum particles with rest mass and without (e.g. photons/phonons). These equations are also involved with flux which begs the question: What is the relationship between the two? We investigate this in this note.
The continuity equation: d/dt partial d(x,t) + grad dot d(x,t)u(x,t) = 0 ((1)) where d(x,t) is density and u(x,t) a velocity field appears in the literature as a fundamental equation of “continuity”. For example, it follows from Boltzmann’s transport equation (and conservation of mass) and leads to a wave equation for sound when combined with a pressure equation (1). A continuity type equation also follows from Maxwell’s em equations and includes the Poynting momentum b El x B where b is a constant, El electric field and B magnetic field. This equation describes light. Finally the continuity equation is used in a number of derivations of the Schrodinger equation (e.g. (2)). In previous notes (3) we have argued that two flux/flow equations for free particles: d/dx partial A = p ((2a)) and d/dt partial A = -E ((2b)), with A=-Et+px being a solution, hold for both quantum particles with rest mass and without (e.g. photons/phonons). These equations are also involved with flux which begs the question: What is the relationship between the two? We investigate this in this note.
In the literature (e.g. (1)) the Schrodinger equation is sometimes derived using classical fluid equations which follow from Boltzmann’s transport equations. In particular a continuity equation and a conservation of momentum equation involving the potential dV(x)/dx are used. We have argued in Part I that the continuity equation leads to 0+0=0 for a free particle wavefunction exp(-iEt+ipx) because density is 1. Furthermore a bound quantum state also leads to 0+0=0 partly because d/dt partial density(x) =0. Following the analysis of (1) for which a new field S(x,t) is introduced through u(x,t) =velocity field = d/dx partial S(x,t) one may see that the solution S(x,t)= f(t) for any function f yields u(x,t)=0 for a bound state. In fact, for a bound state, the continuity equation contributes nothing to the derivation of the Schrodinger equation except for the result density(x) (no t present) which is known already. The Schrodinger equation for the bound state follows completely from the fluid transport equation containing dV/dx. For this to occur, however, one must use S(x,t)=-Et. It seems, however, that for a bound state with u(x,t) the natural approach is to have S(x,t) =0 Setting S(x,t)= -Et is equivalent to having exp(-iEt) appear in the wavefunction and we argue in this note that this is an independent assumption which does not follow from fluid mechanics. In particular, we link -Et for the bound state to A(x,t)=classical action = -Et+px. For a bound state there is an average energy value, but overall p average is 0. For the free particle quantum system we argued that two flow flux equations d/dx partial A = p ((1a)) and d/dt partial A = -E ((1b)) govern the system. These may be written separately as eigenvalue equations yielding: id/dt partial exp(iA) = E exp(iA) ((2a)) and -id/dx partial exp(iA) = p exp(iA) ((2b)). We show that exp(-iEt) for the bound state is a built-in assumption needed to make the fluid mechanical equation approach to the Schrodinger equation work. As a result, this assumption id/dt exp(iA)=E exp(iA) needs to be presented together with the two fluid equations, otherwise one does not derive the Schrodinger equation. In fact, one may use ((1a)) and ((1b)), the two flux/flow equations we propose for a free particle and create a momentum distribution P(x/p) = a(p) exp(ipx)/W(x) where W(x)=wavefunction = Sum over p a(p) exp(ipx). Using conservation of energy at each x point: [Sum over p a(p)pp/2m exp(ipx)] / W + V(x) = E leads to the Schrodinger equation without the use of fluid equations.
In Part II we noted that in the literature the Schrodinger equation is sometimes derived (e.g. (1)) from two fluid equations which follow from Boltzmann’s transport equation, namely the continuity equation (conservation of mass) and a conservation of momentum equation. We also suggested the following flow/flux equations for a free particle: d/dx partial A = p ((1a)) and d/dt partial A = -E ((1b)) or A=-Et+px a Lorentz invariant. For a quantum free particle one may convert these into eigenvalue equations (see Part II). For a bound state one has a system with no overall net momentum, but an overall energy. Thus one may expect the free particle quantum solution exp(-iEt) to still appear. In (1), u(x,t) = velocity field is defined as d/dx S(x,t) based on the idea of irrotational flow. We argued that for u(x,t)=0 S(x,t)=f(t) where t is any function which is ambiguous. Furthermore, in order to have the equations of (1) apply for a bound state S=-Et or in other words there is a principle that d/dt S = -E i.e ((1b)). In this note we examine both a free quantum particle and a bound state in terms of the fluid approach of (1). The result S=-Et remains, but in the case of the free particle ((1a)) emerges from dS/dx = u(x,t)=p. For the bound state overall p is 0, but S=-Et still holds. As a result there is a density from the fluid approach and a related stress potential which leads to a conservation of energy equation. The stress potential, however, is equal to the average kinetic energy at each x because E= W(x)+V(x) from (1). In (1) W(x) is calculated using assumptions linking sij = < (vi-ui)(vj-uj)> to derivatives of density. We calculate average kinetic energy directly using: P(p/x) = a(p)exp(ipx) / Wv(x) where Wv(x) =wavefunction = Sum over p a(p)exp(ipx). The two results are the same.
In Part I we argued that two flow equations for a free particle, namely: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1a)) may be applied to both particles with rest mass and those without. In fact we argue that there are at least three scenarios associated with ((1a)) and ((1b)). The first is classical mechanics for a particle with rest mass mo and constant velocity. Using ((1a)) and ((1b)) with p=E(v)v this scenario yields special relativity. A second solution is to write ((1a)) and ((1b)) as eigenvalue equations: -id/dx (partial) exp(iA) = p exp(iA) and id/dt (partial) exp(iA)= E exp(iA) where A=-Et+px. This yields a quantum mechanical solution for a particle with rest mo if one assumes p=Ev. In such a case one may use the special relativistic result EE = pp + momo and establish the Klein-Gordon equation. A third solution pertains to objects with zero rest mass. As argued in Part I, these do not follow p=E(v)v, but rather the classical wave result: v=E/p. In this note we examine massless particles, namely waves on a string, photons and sound, in more detail. For the zero mass case, dA=0 so there is motion in such a way as to keep A constant. If one considers periodic motion and uses ((1a)) and ((1b)), a solution which is real and periodic is: d/dt d/dt cos(A) = vv d/dx d/dx cos(A) ((2)) A second order partial differential equation is needed as one insists the solution be real. This differs from the quantum mechanical solution of exp(-iEt+ipx) which is complex for a reason. The modulus must be linked to a probability of 1 for a free particle. The zero mass solution is one of a smeared-out physical wave. We note that ((1a)) and ((1b)) deal strictly with E energy and p momentum, but cos(A)=cos(-Et+px) is associated with a period proportional to 1/E and a wavelength proportional to 1/p. This raises two questions. First, what physical entity is periodic? Secondly, how does the energy appear in the problem as energy and not just 2*3.14 frequency of the physical entity? We try to consider this issue for a wave on a string, photon and sound wave.
In (1), it is argued that the Schrodinger (nonrelativistic) quantum equation may be developed from steady state type classical equations linked to the continuity equation and Newton’s second law. Given the nonrelativistic treatment of (1), F=dp/dt = d/dt (mv) = m dv/dt partial + mv dv/dx. In (1) an equation (III.2) for many particles is obtained from the equation for a single particle: Md (dv/dt partial + v dot grad v) = dF + hbarhbar/4m grad dot (grad grad d - 1/d (grad d) (grad d) ). In (1), it is further assumed that v = grad (phi) for conservative forces, where phi is a function of x,t. One may then see that for d=1 case, for a single particle, that phi= -Et+px and that one may write a Schrodinger equation which is essentially a conservation of energy equation. Here, we consider starting with dp/dt=Force and using a special relativistic treatment. We note that for a conservative force, F = -grad V and suggest trying to write the LHS in terms of a gradient for a free particle so the gradient may be “pulled out” from both sides of dp/dt=F. From a physical point of view, F is a pull or push, but we argue that V (potential) is also physical and that given that p is almost like a force, that there should be meaning to p= grad phi(x,t). This is the basis premise. In the case, F=0, we show that one may obtain -EE + p dot p = constant via this approach which involves writing p = d/dx (-Et+px) (one dimension). We note that associating dp/dt = F with a Lorentz invariant -Et+px shows the close association of Newton’s second law with special relativity because dp/dt=Force is relativistic as long as p=mov/sqrt(1-vv/cc). We suggest that this approach associates operators id/dt partial and -i grad partial (hbar=1) with E and p and allows one to write a free particle equation (Klein-Gordon) in terms of exp(-iEt+ipx), which is a quantum form. We suggest that one way in which to see free particle quantum mechanics emerge (albeit in the simplest case-that of a free relativistic particle) is to try to consider dp/dt in terms of a gradient suggesting that there may be x, t features even for E,p constant. In Newtonian mechanics, a constant E and p immediately implies that derivatives are 0. This need not be the case if one considers p = d/dx (-Et+px). The legitimacy of considering p (which is closely linked to a force) is linked to the legitimacy of considering F = -grad V. In other words, V is a potential which is another way of considering the features of a force and p = d/dx (-Et+px) may mean that -Et+px (and exp(-iEt+ipx) may be another way of considering p,E at a deeper level than simply stating that one has a constant E,p for a free particle. We argue that one does not need to consider a continuity equation as done in (1) and that the full approach follows from analysis of a single free particle (constant E,p) in the special relativistic case, which then yields also the nonrelativistic one.
In previous notes we argued that two flow/flux equations d/dx (partial) A(x,t)=p and d/dt (partial) A(x,t) = E for free particles (where x/t is set to v after the derivatives are taken) may be used to describe classical mechanics, both relativistic and nonrelativistic quantum mechanics (for a free particle with rest mass) and lead to the form of special relativity for a particle with rest mass. In this note we consider (as in (1)) the situation for a particle with no rest mass. We argue that the two flow/flux equations still pertain, but that there is a different equation for momentum (which as a function of E(v) and v) is related to energy flow. For a particle with no rest mass, v=E/p while for one with mass p=E(v)v. How is this possible? We argue that v=E/p describes an object with energy which is entirely active in creating motion i.e. energy flow. In other words for pv=E, p→0 and v→0 means E→0 which does not allow for rest energy. This describes a classical wave and also the photon or phonon. Such an object cannot be at rest because its energy is equivalent to motion so to speak. It is possible to trap such a particle in a box so it bounces back and forth. In such a case, there is rest energy and a different notion of energy flow, namely p=E(v)v. Thus, we argue that there is no contradiction in having two equations for momentum, one pertaining to particles with rest mass and the other to those without.
In previous notes (1)(2), we argued one may write two flow/flux equations for a free particle using an unknown function A(x,t) i.e. d/dx (partial) A = E(v) v ((1a)) and d/dt (partial) A = -E(v) ((1b)) where E(v) is energy which is a function of rest mass and velocity v=constant. A(x,t), but after taking derivatives, x/t is set to v. As shown in (2), one may write A=tL(v) and find E(v)=mo/sqrt(1-vv) (c=1) (and p=Ev) i.e. the result of special relativity. Thus special relativity seems to follow directly from a flow/flux equation where motion v increases energy. Alternatively, one may obtain both relativistic/nonrelativistic quantum mechanics for a free particle, by converting ((1a)) and ((1b)) into differential equations i.e. id/dt exp(iA) = Eexp(iA) and -id/dx exp(iA) = pexp(iA) with A=-Et+px yielding the same result. In both cases, physical motion v has physical implications. For special relativity, energy changes from mo to mo/sqrt(1-vv) and there are effects on clocks and lengths. For quantum mechanics there is time and space resolution linked to dx→1/p and dt→1/E. In this note we try to investigate if there is a connection between the two. In particular we try to see why -Et+px within the free particle wavefunction exp(-iEt+px) is a Lorentz invariant.