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.
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 <pp>, 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 [WW] + d/dx [WW 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 <pp>. 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.
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 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.
Quantum mechanics is largely based on a wavefunction, which we consider to be a probability. The theory is, however, not solely probabilistic because if an interaction does not occur, x=vt holds as does the classical notion of impulse hit and kinetic energy = pp/2m (nonrelativistic case). The probability exp(-iEt+ipx) describes probabilistic interactions in time and space (we argue) and so one does not think of x and t as in x=vt. As a result, there is a probability for a quantum particle to interact with both slits of a two slit apparatus if they are about hbar/p apart. This leads us to the notion that W*(x)W(x) (W(x)=Sum over p a(p)exp(ipx)=wavefunction) is not spatial density, but rather an interaction profile. A particle can still exist at W(x)=0 points, it just has very little to no probability of interacting at these points on average, with the average considering both exp(ipx) and exp(-ipx) in a bound case scenario. Given that E=kinetic energy = pp/2m (nonrelativistic case only) and W*(x)W(x) not= spatial density, we suggest that one must be careful with interpreting d/dt partial W*(x)W(x) = constant grad dot Im (W* (-id/dx) W) (from (1)) as a continuity equation which represents hydrodynamic motion. We suggest that the quantum features are governed by exp(-iEt+ipx) and that any derivatives d/dx and d/dt are related to changes in phase and not hydrodynamic flow (in a classical sense). We note that for a free particle d/dt partial density = 0 = constant grad dot Im (W* (-id/dx) W). The same zero equation holds for a bound state (even though there is physical motion to the right and then left) , but if one creates a superposition of bound states (which is a probability statement), then the same equation is not 0. We note that for a classical bound state there exists a vrms(x), but this is an average quantity. It does not show any resolution pattern which is represented by W(x) and so one must be careful when interpreting it, we argue.
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 previous notes, we suggested that quantum bound states may be analysed in terms of the conditional probability P(p/x)= a(p)exp(ipx)/W(x) (where W(x) is the wavefunction) which allows for the calculation of average kinetic energy at each point x i.e. [Sum over p pp/2m P(p/x)]/ W(x). This added to V(x) yields E (average energy) at each x and yields the time-independent Schrodinger equation). We argued that probability is handled at the level of W(x)=Sum over p a(p) exp(ipx). Measurements (even internal) may knock a particle from one p1 state to another so another set of probabilities (P(x) and P(p)) arise, P(x) being based on: a(p1)exp(i p1x) exp(i p2x). Averages of f(p,x) follow from P(x) f(p,x) P(p/x). Thus, there are two sets of probabilities in quantum mechanics. Entropic dynamics (1) appears to begin with the probability P(x) and considers a transition probability P(x1 |x) with x1-x= an infinitesimal quantity. A product of transition probabilities is then taken to allow for finite x1-x differences. It is stated in (1) that there exists information about x but not p. The maximization of entropy density may be used to determine the form of P(x1 | x). A “drift velocity” is also included (which is later shown to be related to the phase of the wavefunction). Ultimately, the Schrodinger equation is also obtained. We argue one may use P(p/x) to also calculate P(x1 | x) and compare this result with that obtained through the maximization of entropy density. A type of drift velocity appears naturally in this approach. We argue that the form a(p)exp(ipx) already incorporates statistical ideas (i.e. maximum entropy) in that the modulus of exp(ipx) is identical for all x, endowing all x points with a “kind” of equivalent status. exp(ipx) is a function of p and x because one must calculate average kinetic energy at each x and this must change with x i.e. KEave(x) + V(x) = E. Thus, we think there are strong similarities between entropic dynamics and the approach of using P(p/x). There are also some differences in that the coefficient A of delta(x)delta(x) in P(x1 | x) = 1/Z exp{ A delta(x)delta(x) + B d Phi/dx delta(x)} is treated as independent of x in entropic dynamics, but is a specific function of x (related to d ln(W)/dx d ln(W)/dx + 1/W d/dx d/dx W) in the P(p/x) approach.
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).