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 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 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.
There have been various derivations of the Schrodinger equation in the literature (e.g. (1)) based on hydrodynamics (continuity and conservation of momentum equations) with the added assumption that a velocity field u(x,t) = d/dx S(x,t) where S(x,t) is a new field. Furthermore a mathematical transformation Wv= sqrt(d(x,t)) exp(i S(x,t)) is used leading to the new function Wv(x). In (2) we compared the bound state Schrodinger scenario to the hydrodynamic equations in (1). In (1) d u(x,t)/dt = -dV/dx -dW/dx ((1)). V(x) is the usual potential and W(x) is described as a fluid stress potential. We also compared (1) to a free quantum particle and found that S(x,t) must contain a -Et piece if the hydrodynamical approach is to match quantum mechanics. In (1), ((1)) is shown to be equivalent to: d/dt partial S + ½ u(x,t)u(x,t) + W(x) + V(x) = 0 ((2)) with W being expressed in terms of spatial derivatives of density. d/dt partial S yields -E and u(x,t)=0 for a bound state so ((2)) is of the form of a conservation of energy equation. If a free quantum particle is described spatially by exp(ipx) then one may create a distribution Sum over p a(p)exp(ipx) and calculate average kinetic energy [-1/2m Sum over p a(p)pp/2m exp(ipx)] / [Sum over p a(p)exp(ipx)] and set it equal to W(x) as done in (2). In this note, we wish to examine the idea of a free particle conditional probability exp(-iEt+ipx) and compare it to the Schrodinger bound state which itself may be considered a localized solution if one ignores the internal wavefunction and considers only the center of mass features exp(-i (masses +En) t + ipoverall x) where p(overall) =0. We argue that center of mass features are contained in the Schrodinger bound state solution if one integrates over x relative to cm, the x variable in the Schrodinger equation. The idea of time t only applies to the cm, there is no time in Sum over p a(p)exp(ipx). Furthermore p(overall) =0 only if one integrates W(x) (-id/dx W) over all x (relative to cm). Also the form -i d/dx Wavefunction(free particle p) = p Wavefunction(free particle) where Conditional Prob= exp(-iEt+ipx) carries over to the bound state with: -i d/dx exp(ln(Wv)) ={ -i d/dx ln(Wv)} exp(ln(Wv)) where Wv=Sum over p a(p)exp(ipx). In other words there is an imaginary momentum -id/dx ln(Wv(x)) at each x in the bound solution so u(x,t)=0 holds for the real part. As a result, the hydrodynamical approach to quantum mechanics is not as straightforward as it may appear on the surface. We try to investigate a number of ideas related to the treatment of quantum bound states using hydrodynamics.
In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.
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 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).
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.