In the literature (1), the Schrodinger equation: {p1p1/2m + p2p2/2m + V(x1-x2)} W(x1,x2,t) = id/dt partial W(x1,x2,t) may be transformed using P=p1+p2 and (p=m1p2-m2p1)/M and x=x1=x2 where M=m1+m2 ((1)). This leads to a bound Schrodinger equation in x and a “center of mass” equation” in P i..e Hcm =PP/2M which depends on m1 and m2, but not binding energy. . In a previous note, we argued that a free particle’s classical action A (relativistic or nonrelativistic) may be obtained by setting v=x/t and treating x and t as independent (2). This leads to dA/dx partial=p and dA/dt partial = -E. One may write eigenvalue equations: -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = E exp(iA). A solution is A=-Et+px with E,t,p and x independent. -Et+px is a Lorentz invariant and this form is used in both relativistic and nonrelativistic quantum mechanics (although E and p have different values). We argue that a free particle rest mass mo may actually involve internal quantum physics, yet the “system” is replaced with a parameter mo and exp(-iEt+px) used to describe it. One may try to apply this approach to a bound system, but exp(-iEt+px) is a statistical probability like object. Thus one needs a statistical theory. We have suggested using an ensemble of exp(-iEt+px) with E=En for all p to indicate a resonance. Then: { Sum over p pp/2m a(p)exp(ipx) } / {Sum over p a(p)exp(ipx) } + V(x) = En ((2)) This is the Schrodinger bound equation. We argue that W=Sum over p a(p)exp(ipx) should be real and this implies that Sum over p a(p) p exp(ipx) = 0 i.e. average momentum for this type of average is 0. As a result, one may create a 4-vector using { 0, mcc + En} and use the Lorentz boost approach to find the momentum and energy in a moving frame. One may then take the nonrelativistic limit to find kinetic energy i.e. the full center of mass kinetic energy. We argue that this is equivalent to considering exp(-iEt+ipx) for the overall bound system which is now translating in space. We argue that ((2)) is a different approach to ((1)) in that binding energy, which may be significant with respect to mcc, should be part of the overall energy of the translating bound system which is called the “center of mass” system in the literature.
In the literature (1), the Schrodinger equation: {p1p1/2m + p2p2/2m + V(x1-x2)} W(x1,x2,t) = id/dt partial W(x1,x2,t) may be transformed using P=p1+p2 and (p=m1p2-m2p1)/M and x=x1=x2 where M=m1+m2 ((1)). This leads to a bound Schrodinger equation in x and a “center of mass” equation” in P i..e Hcm =PP/2M which depends on m1 and m2, but not binding energy. . In a previous note, we argued that a free particle’s classical action A (relativistic or nonrelativistic) may be obtained by setting v=x/t and treating x and t as independent (2). This leads to dA/dx partial=p and dA/dt partial = -E. One may write eigenvalue equations: -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = E exp(iA). A solution is A=-Et+px with E,t,p and x independent. -Et+px is a Lorentz invariant and this form is used in both relativistic and nonrelativistic quantum mechanics (although E and p have different values). We argue that a free particle rest mass mo may actually involve internal quantum physics, yet the “system” is replaced with a parameter mo and exp(-iEt+px) used to describe it. One may try to apply this approach to a bound system, but exp(-iEt+px) is a statistical probability like object. Thus one needs a statistical theory. We have suggested using an ensemble of exp(-iEt+px) with E=En for all p to indicate a resonance. Then: { Sum over p pp/2m a(p)exp(ipx) } / {Sum over p a(p)exp(ipx) } + V(x) = En ((2)) This is the Schrodinger bound equation. We argue that W=Sum over p a(p)exp(ipx) should be real and this implies that Sum over p a(p) p exp(ipx) = 0 i.e. average momentum for this type of average is 0. As a result, one may create a 4-vector using { 0, mcc + En} and use the Lorentz boost approach to find the momentum and energy in a moving frame. One may then take the nonrelativistic limit to find kinetic energy i.e. the full center of mass kinetic energy. We argue that this is equivalent to considering exp(-iEt+ipx) for the overall bound system which is now translating in space. We argue that ((2)) is a different approach to ((1)) in that binding energy, which may be significant with respect to mcc, should be part of the overall energy of the translating bound system which is called the “center of mass” system in the literature.
In previous notes (1) we argued that the free particle action A (either nonrelativistic or relativistic) may have v (velocity) written as x/t and x and t varied independently to yield: dA/dx partial = p and dA/t partial = -E with A=-Et+px, the Lorentz invariant classical Action = t Lagrangian. We further argued that one may write eigenvalue equations i.e. -idexp(iA)/dx partial = p exp(iA) and idexp(iA)/dt partial = Eexp(iA) and call exp(iA) a type of probability. We noted that if a potential is present one may use P(p/x)= a(p)exp(ipx)/W(x) [W(x)=wavefunction = Sum over p a(p)exp(ipx)] and establish a conservation of energy equation KE(x) + V(x) = E such that there exists the same constant E at each x. We described this in two ways. First E represents binding energy so at each x one has the same binding energy i.e. each x point maps to the same E as part of the mass of an overall moving system. Secondly, we argued that E was a parameter indicating that equilibrium existed at each x point, much like temperature T. In this note we wish to consider a different idea suggested in (2), namely that if one has a quantum state: W(x)= Sum over n a(n,t) Wn(x) this may be written as: <i |H| j> a*(i,t)a(j,t) = <W H W>.. If Hamilton’s equations for complex variables are applied to these then: i d/dt partial a*(i,t) = d/da(i,t) partial { <W H W> } ((A)) (2) argues this is the time-dependent Schrodinger equation. In other words, probability weights of the Schrodinger equations (which are complex) obey Hamilton’s classical equation. In this note, we point out that the formalism dA/dx partial = p already introduces operators into classical physics, so it follows that H is an operator which may act on states. We consider the idea of ((A)) as applied to a free quantum particle moving. Next, we apply ((A)) to a quantum bound state and argue that a classical Hamilton’s equation applied to d/dt a(p) partial may describe equilibrium. Finally ((A)) may be applied to a sum of eigenstates, but this is already shown in (2).
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.
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.
In a previous note (1) we argued that both the relativistic A= -mo t sqrt(1-vv) and nonrelativistic A=t .5mvv classical free particle actions may be written with v=x/t and x and t varied independently to yield: dA/dx partial = p and dA/dt partial = -E. Thus A is equivalent to -Et + px for a free particle with x and t being treated as independent. We argue in this note that such an assumption is equivalent to a statistical picture for a free particle even at the classical level because x and t are treated as independent. Thus a constant p may occur at any x because no time is linked to x. Given that all space points are the same for constant p one expects a constant probability at all x points. Knowing p exactly means not knowing anything about which x one has. Secondly, one may create an ensemble of free particle p classical actions for a situation containing a potential V(x). Given that x and t are independent, there cannot be any notion of single p accelerating, although a vrms from an ensemble may. At a given x (x known exactly) one does not know which p is present. Thus the ideas of uncertainty of p and x follow immediately from this picture. If one considers a single particle and an ensemble of free particles in the x-t independent scenario, then each free particle action p is linked to A=-Et+px where E=pp/2m. This ensemble would arguably be stirred up by a potential V(x). If one wishes to consider an equilibrium scenario for which there is an overall En for the bound system one would expect an overall Ent for the system and for -pp/2m t pieces to disappear. Thus we postulate that pp/2m +{p part} G(p’s k’s) = En for all p where {p part}G(p’s, k’s) is an unknown function and V(x)=Sum over k Vk exp(ikx). Next one may consider the spatial situation for which V(x) is the classical potential for a single particle. The kinetic energy follows from an ensemble of free particles with the overall -Et removed. From classical considerations one would expect to have a KE(x). Thus one expects: KE(x) = {Sum over p a(p) f(px) } / {Sum over a(p)f(px) } with f(px), an orthonormal basis set, because the action contains px. (It is orthonormal because one wishes the identity of p to exist if not at a point, then over all of x.) We try to argue in this note that f(px)=exp(ipx), the momentum eigenstate which follows from A=-Et+px. We find G(p’s, k’s) which implies entanglement or mixing of a given p state with other p and k states from the potential V(x)=Sum over k Vk exp(ikx). Thus one expects an entropy linked to this momentum entanglement. Given the form exp(ipx), finding a set of ap’s which satisfy pp/2m + G(p’s, k’s)=En leads to a spatial distribution W(x)=Sum over p a(p)exp(ipx) so there should be an entropy contribution from this spatial part as well. In classical statistical mechanics one has both momentum and spatial contributions to entropy. We argue for S=Sp+Sx, but note that even though px has x, p symmetry as does the Heisenberg uncertainty principle var x var p >= hbar/2, Sp and Sx may not be symmetric in terms of important parameters such as n, energy level. In other words, Sp the entanglement entropy with V(x) may be the driver entropy sensitive to energy level changes while Sx the resulting spatial density may not be as sensitive. Furthermore the overall entropy Sp+Sx gives information about uncertainties in x and p as is already known in the literature where bound state Sp+Sx bounds are given.
We have argued in previous notes that both Newtonian mechanics and free particle quantum mechanics arise from special relativity. Thus, special relativity gives an expression for free particle energy and momentum and these are related to concepts of work. The Lorentz invariant -Et+px shows that x,t and x+hbar/p, t+hbar/E give the same solution to this potential-like term showing that there is stochasticity in the position and time delivery of interaction hits. In other words, even though the center-of-mass moves as x=vt, a momentum hit (proportional to rest mass mo) does not necessarily act at the center-of-mass. This gives the impression of shifts in probability, kinetic energy etc in the case of interference, i.e. interactions. For a single free particle, there is no interaction and -1/2m d/dx d/dx exp(ipx) = pp/2m exp(ipx). We argue that exp(ipx) is a probability and that in general one may add independent probabilities in an OR situation as in a 2-slit interference case. A bound state, however, is different because even though one adds weighted exp(ipx)s, they are part of a resonance or equilibrium and this is no longer an independent OR situation. We suggest that one uses a conditional probability when computing a Newtonian nonrelativistic conservation of energy equation to try to remove as much of the quantum features from this equation. As a result, one has: KEclassical(x) + V(x) = E = Sum over p pp/2m P(p/x) + V(x). One cannot bypass the fact that a quantum particle interacts differently from a classical one and that this involves a probabilistic granularity (wavelength), but classical notions of energy and energy conservation still exist. As a result, the bound state wavefunction Wn(x) = Sum over p an(p) exp(ipx) is not a usual OR sum, but is rather 1 over a normalization factor for an equilibrium picture. As a result, one cannot use Wn1(x)Wn2(x) = W(n1+n2) (x) as one would for independent probabilities. Furthermore, given that one has Wn(x) and Wn1(x)+Wn2(x) which are both of the form Sum over p a(p)exp(ipx), and both represent average energies, one needs a way to distinguish a resonance/equilibrium from a sum of resonance/equilibrium normalization factors. We suggest that the orthogonality of resonance Wn(x)’s provides this feature. The standard textbook explanation is based on Hermitian operators, but in the nonrelativistic case, we argue that one may obtain the result based on integration by parts based on the form of the p operator = -grad. Thus, it is ultimately the form exp(ipx) which leads to orthogonal bound state Wn(x)’s, together with the boundary condition that Wn(x)-->0 of x-> +infinite and -infinite.
We have argued in Parts 1 and 2, that the free particle quantum form exp(-iEt+ipx), which we call a complex probability, arises from special relativity in order to allow for conservation of energy and momentum (in an elastic collision). One may see from single particle bound states how exp(-iEt) plays the part of a binding energy. We have noted in Parts 1 and 2, that exp(-iEt+ipx) follows from the Lorentz invariant -Et+px, but that one does not consider the deterministic x=vt equation which applies to the center-of-mass, but rather treats t and x as independent variables which force the independence of E and p (two different interaction variables) and also force separate E and p conservation. The question becomes: How is it possible to have x,t play two such different roles in special relativity? Here we try to answer this question. We argue that special relativity is based on determinism, i.e. x=vt. One may create a Lorentz transformation for a particle with rest mass mo at x=0 with t, a running variable. As seen from a frame moving with -v, one has x’=vt’. In other words, a separate x’,t’ is associated with each v. In Newtonian mechanics, it is possible to have mov1 and mov2 in the same frame (i.e. a frame with a ruler and clock), but in special relativity, each mo in the rest frame maps to a different x1’,t’ and x2’,t2’. This, however, seems to lead to a contradiction. If one considers two particles p1,E1 and p2,E2 (one dimensional motion) which elastically collide, then before the collision, they may be thought of as a composite particle (for more than a single x,t point) with energy E1+E2 and momentum p1+p2. Given the validity of special relativity, we argue that one may associate a Lorentz invariant -(E1+E2)t + (p1+p2) x with this composite particle. Furthermore, any outcome Ei+Ej = E1+E2 and pi+pj = p1+p2 would satisfy this same invariant. At this point there is no contradiction. The contradiction arises if one considers a single t and x for a composite particle, while internally there exist t1’,x1’ and t2’, x2’. The composite particle, in terms of special relativity, is not compatible with each -E’t’+p’x. The energies and momenta, however, add for each particle (in conservation equations) and so we argue that one should have a relativistically compatible approach which allows one to link each particle to a composite particle and see conservation of E and p. We note that it is precisely x=vt for each particle which leads to x1’,t1’ and x2’,t2’ and so argue that if one gives up this notion of determinism for x and t and introduces independent x, t values (fluctuations), then one can indeed retain the -Et+px form for each particle and use it to create composite particles. This seems to give rise to an AND situation which does not exist for both p,x and E,t in the deterministic relativistic approach. In that approach, one can only think of p’s and E’s adding separately, without any linking of x’s and t’s. In the fluctuation approach, it is possible to have the same x and t fluctuation points for each particle. We argue that it is this kind of thinking which leads to the probabilistic form exp(-iEt+ipx) which still respects the Lorentz invariant form and at the same time allows for conservation of E,p with the same t,x fluctuation points allowed. x=vt does not allow one to combine x1’,t1’ and x2’,t2’ for a composite particle because these may coincide for only one x,t point and one wishes to describe a composite particle with an x,t which exists over more than one x,t. This is why we suggest the physical fluctuation scenario which exists together with x=vt. x=vt describes the deterministic motion of the center-of-mass, while exp(-iEt+ipx) describes probabilistic hits in x and energy in t. It is special relativity which forces one to associate time fluctuations with E and spatial ones with p. As a result, we argue that special relativity gives rise to both probabilistic and deterministic versions, both of which respect the form -Et+px. The deterministic version is linked with x=vt center-of-mass motion, while exp(-iEt+ipx) represents probabilistic appearances of E in t and p in x, i.e. there is a spatial probability for impulse hit delivery.