Speculation on Quantum Free Particle Probability and Spatial Equilibrium Part 3

Speculation on Quantum Free Particle Probability and Spatial Equilibrium Part 3 Francesco R. Ruggeri Hanwell, N.B. Nov.7, 2025 In Parts 1 and 2, we suggested that various physical scenarios such as -EE + p dot p cc = -momocccc and one dimensional reflection refraction probability conservation 1= P(reflect)+P(refract) are not involved with the direction of motion of a particle. If probability is involved (as in a wavefunction), these equations (even though they act on probability) do not show the direction of motion. In any physical test run, however, there is a specific direction of motion. As a result, we argued that there must exist equations linear in probability (exp(ipx) for a free particle) and id/dt partial, -id/dx partial (energy, momentum) which give rise to the physical non-directional cases. Here we retain the general idea of an equilibrium with no direction present, but argue that matters may be more complicated when different time events appear in space in a time-independent approach using exp(ipx)s. In particular, we analyze scattering and place 1-D reflection-refraction in the framework of scattering. We suggest that to create an equilibrium in both directions, one must consider both W=Aexp(ipx) + Bexp(-ipx) (for an incident p and reflected -p) and its flow -i W* d/dx W, together with the opposite direction scenario W*= A*exp(-ipx) +B* exp(ipx) and i(d/dx W*) W. Averaging yields p ( A*A - B*B), i.e. interference terms disappear in this scheme which directly considers motion in both directions i.e. A is linked with motion in both directions which does not match a physical run, but does match classical equilibrium (pressure). This result is not evident if one uses 1-D reflection-refraction at x=0 because one may then use a nd exp(ipx) and d/dx exp(ipx) set of equations to obtain directly 1=P(reflect) +P(refract), but interference terms creep in if one changes the junction to x1 as seen in Part 1. The averaging of both directions removes this issue. We see that exp(ipx) and exp(-ipx) represent equivalent kinetic energies in the time-independent Schrodinger equation. A math solution thus involves any identical energy directional derivative, i.e. exp(ipx) and exp(-ipx). There is no sense of time, so writing Aexp(ipx) + Bexp(-ipx) with A>B introduces a direction which is not in the Schrodinger equation, but is in a physical trial run. Given that the time-independent Schrodinger equation cannot discern time and probabilities add, a full solution should include various same-energy solutions which represent the particle at different times. There should then be a probability conservation linking these different time states, such as 1= P(reflect) + P(refract), but due to interference of W* d/dx W one has interference terms in x, but these may vanish if one considers motion in the opposite direction, i.e. Aexp(-ipx)+ Bexp(ipx), and classically averages expectations of -id/dx with each as one would do classically. Classical Equilibrium We suggested in Part 1, that exp(ipx) is a directional probability used in Newtonian elastic scattering. It is linked with the direction of p. Classical equilibrium, e.g. pressure is not linked with one direction. Pressure exists in all directions. We suggested in Part 1 that various non-directional physics equations exist such as: -EE + cc p dot p = -momocccc ((1a)) 1= P(reflect) + P(refract) ((1b) etc One may associate ((1a)) a directional derivative by using E→ id/dt partial and p= -i grad partial, but that does not change the fact that ((1a)) is essentially a non-directional equation. If one tries to use it (or its nonrelativistic limit equation, the Schrodinger equation) in a time independent manner, i.e. E W = (-1/2m grad dot grad +V(r) ) W ((2)) then ((2)) is an energy balance equation, which in terms of probability is an equilibrium which does not check direction (or time for that matter). As a result, if one solves ((2)) in 1-dimension (x) for V=0, one has: Aexp(ipx) + Bexp(-ipx) in a region of x ((3)) If A>B, then A seems to be an incident case and B, a reflected, but this is a directional statement (which physically appears in a trial run). ((2)), however, has an equally good mathematical solution: Aexp(-ipx) + Bexp(ipx) ((4)) If one considers a balance which holds whether a particle moves from the right to left or vice versa, as in 1-dimensional scattering, then one should consider an average of expectations of -id/dx of both ((3)) and ((4)). (1) calls this average, current, i.e. Current = ½ (1/m) { i (dW*/dx) W) - i W* d/dx W } ((5)) This leads to the expected: p ( A*A - B*B) which may be matched with a forward scattered piece C*C arsing from ((5)) applied to Cexp(ipx). To see this explicitly, one has: pA*A - p B*B + .5(A*B(-p)+ B*A (p) + A*B (p) + B*A(-p)) = pA*A - p B*B In other words, a solution of the Schrodinger equation mixes (adds) probabilities for equal energy exp(ipx)s and these may represent events at different times. The addition of these probabilities means that W*W and also W* (-id/dx) W include interference terms, but for time probability equations (which account for time events), there can be no interference. Thus, averaging in both directions removes the interference to show a probability linked with an event regardless of direction. In other words, P(reflect) or P(scatter) is irrelevant of the incident direction. To see these ideas appear in a more general manner, we consider general time-independent scattering, following (1) by using a Schrodinger in 3-space. For r-> infinite, V(r) has died down and one must have a completely general free particle solution: Lim r→ infinite Sum over l (Al jl(kr) + Bl nl(kr)) Pl(cos(theta) ((6)) Here jl is a Bessel function and nl, a Neumann. Pl(cos(thea)) is a Legendre polynomial. We consider elastic scattering, so any math solution with momentum k (i.e. kinetic energy = kk/2m) is fine. This means that an incident solution, say exp(ikx) should be mixed with a scattering solution: f(theta) exp(ikr)/r ((7)) In ((6). In fact, these are the only two solutions which may appear. Now, for r-> infinite jl(kr) = sin(kr-l*3.14/2) / (kr) ((8a)) R-infinite nl(kr) = cos(kr-l*3.14/2) / (kr) ((8b)) Also, lim r→infinite exp(ikx) = 1/(2ik) Sum over l i power l (2l+1) jl(kr) Pl(cos(theta) ((9)) Here the r→infinite limit ((8a)) may be used in ((9)). Thus, as explicitly shown in (1), one may take ((6)) and explicitly pull out and exp(ikz) piece and an f(theta) exp(ipr)/r one at r→ infinite, namely ((6)) at r-> infinite = lim r-inifnite exp(ikz) + exp(ikr)/(r) { Sum over l (2l+1) (exp(2 i delta(l) -1) / (2ik) Pl(cos(theta) ((10)) Here delta(l) is called a phase shift. It is a function of Al and Bl in ((6)), In particular, for scattering from an infinite spherical potential with radius, (1) shows that: delta(l) = arctan ( -Bl/Al) = aractan (jl(kro) /nl(kro)) ((11)) To make a long story short, W(r), the solution of the Schrodinger equation contains both exp(ikz) incident probability pieces added to elastically scattered pieces f(theta) exp(ikr)/r. One cannot simply calculate current as W* (-id/dx W) because this has cross-terms (as (1) notes). This is because a current balance is like an equilibrium with no direction. One needs to consider current as ((5)), but (1) uses tricks to bypass a long calculation and argues that J incident is based on using exp(ipz) directly in ((5)), i.e. p/m ((12a)) For the scattered current, he only uses exp(ikr)/r f(theta) to obtain: J scattered = e(r) / rr f*f p/m ((12b)) Here e(r) is a unit vector in the r-direction The interference terms are simply removed which means that ultimately ((5)) must remove them in a full calculation (without tricks used). If one considers motion exp(ipz)+f(theta) exp(ipr)/r and its expectation of -igrad averaged with exp(-ipz) +f(theta) exp(-ipr)/r and its -grad expectation, interference terms are removed and one sees the currents for the time separated events, i.e. the incident particle and reflected one. (One may then define: J scattered dot e(r) rr d(solid angle) / j incident = dsigma/d solid angle d(solid angle) ((13)) Thus, even though quantum mechanics uses directional probabilities exp(ipx) which allow equal energy different time probabilities to add, hence leading to interference of different time events, one may use a current (expectation of -i grad) and average over p and -p incident directions etc to remove interference. Even though interference appears in 2-slit interference, it does not appear in 1= P(reflect)+P(refract) or in f*f p/m e(r)/rr, the scattered current. This means that in time-independent calculations which do not consider direction (unlike 2-slit diffraction which is very much direction oriented and so shows interference), one must remove the quantum interference to find the classical probabilities of events separated in time, but having the same energy, e.g. exp(ipx) and exp(-ipx) or exp(ipz) and f(theta) exp(ipr)/r. The Case of Spatial Density The above arguments related to removing interference were directly related to a calculation of current, i.e. the expectation of momentum p, which is associated with a pressure balance. Pressure balance is not based on one direction and so W* dW/dx + (dW*/dx) W must both be considered. If, however, one calculates spatial density: W*W = spatial density ((14)) then it is not possible to remove interference. Thus, this interference must occur as it does in quantum bound states and 2-slit interference. As a result, interference is key for spatial density, but may be removed for current (i.e. momentum expectation linked to pressure balance). Conclusion In conclusion, exp(ip dot r) is a directional probability, but the time-independent Schrodinger equation is an energy based equation which does not discern time and accepts all equal energy solutions. If one considers 2-slit interference, then exp(ipx) and exp(-ipx) are two equal energy solutions, but one insists on a single directional result exp(ipx) as this is physical. An interaction with the slits occurs and one has an OR or probability sum exp(i p dot r1) + exp(i p dot r2), where r1 and r2 are space vectors from the center of each slit to the same point on a screen far away. Writing W*W leads to interference terms which are observable. One is not interested in any probability conserving equation for events which take place at different times in such a case. Also, one is interested here in W*W which shows interference, i.e. which is the same when averaged, i.e. .5(W*W+WW*) = W*W. In other situations, however, one is interested in an a priori probability which is linked to discerning time. In 1-dimensional reflection-refraction one knows about 1= P(reflect) + P(refract). This is a result is irrelevant of direction and separates time. Similarly, for elastic scattering from a potential V(r), one has an incident current and a scattered current. The problem is that solutions of the Schrodinger equation with the same energy represent events at different times and integers, e.g. Aexp(ipx) + Bexp(-ipx) (incident + reflected directional probability) and exp(ipz) + f(theta) exp(ipr)/r (incident and scattered probability). One does not want equations with interference here and so one must use a current which considers motion in both directions, i.e. a classical equilibrium as argued in Part 1. The form of this non-directional equilibrium current is given by Current = ½ (1/m) { i (dW*/dx) W) - i W* d/dx W } ((5)) which was not used in Part 1. If is the consideration of both directions which allows one to separate events in time from the interfering exp(ipx) pieces. We believe that this sheds some light on the interpretation of a solution of the time-independent Schrodinger equation as it adds all equal energy solutions even though these may represent events which occur at different times. We note that it is possible to extract classical probabilities of these events by removing interference. Thus, interference is not always observable in quantum calculations. The directional probability approach using exp(ipx)s shows interference if one uses W*dW/dx, but averaging with (dW*/dx) W removes it. References 1. Shankar, R. Principles of Quantum Mechanics (Plenum, 1988)