Quantum Elastic Scattering from a Spherical Infinite Potential and Directional Probability Equilibrium

Quantum Elastic Scattering from a Spherical Infinite Potential and Directional Probability Equilibrium Francesco R. Ruggeri Hanwell, N.B. Nov. 7, 2025 In a previous note, we asked the question: How does a single photon approaching an n1-n2 index of refraction junction in a one-dimensional problem know what its probability to reflect or refract is? We argued that this probability does not follow from some stochastic feature of the junction, but rather from a directional probability exp(ipx). We argued that exp(ipx) is a probability linked with conservation of momentum in Newtonian scattering and implies that there is a spatial uncertainty region of hbar/p. We argued that within a delta x region of the n1-n2 junction, the photon is able to establish a “balance” or equilibrium of the incident, reflected and refracted photon, because one does not know which it is during the time spent in dx. We argued that one must describe this by a balance of Aexp(ipx)+Bexp(-ipx) = Cexp(ip2x) at x=0 and a similar equation obtained taking d/dx. It is this balance or equilibrium which determines the weights of B and C relative to A, i.e. essentially solves the problem. If this is in fact the process which occurs, then we argue here that it should also apply to the scattering of a particle from a spherical infinite well potential at ro, for example. In other words, we consider all of the possible directional probability solutions outside of ro together as these participate in the balance or equilibrium at ro, the interaction region. In such a case, one has a general solution of the Schrodinger equation of: Al jl(pr) + Bl nl(pr) ((1)), where jl is a Bessel function, and nl, a Neumann. In order to have equilibrium between all possible exp(ipx) type directional probabilities, one must establish a constraint condition just as in the n1-n2 problem. The constraint condition for the infinite spherical potential at ro is that there is no probability for ro inside. Thus, we argue that the relevant weights in the problem (i.e. the solution of the problem) are established by the constraint condition at the interaction surface. This leads to 0 = Al jl(pro) + Bl nl(pro) ((2)). This yields a relative weight between Al and Bl. The question is then: How does this relate to anything physical? The fact is that even though ((1)) is a general math solution, with Bl now given in terms of Al through ((2)), one may now rewrite ((1)) in terms of a math sum of physical directional probabilities, namely exp(ipz) (incident particle) and f(theta) exp(ipr)/r (scattered particle). Thus, it is the use of directional probabilities (linked to momentum conservation) and constraint conditions which govern the weights of the allowable directional probabilities at an interaction point/region, i.e. ro in the spherical infinite potential case and x=0 in the n1-n2 junction reflection-refraction case. In particular, if one has an incident particle linked with exp(ipz) one wishes to know the weight f(theta) of exp(ipr)/r in order to find out what per cent of particles scatter into angle theta. This means finding a relative weight between the two directional probabilities and this occurs by a probabilistic sampling of all solutions at the interaction point (here ro). Then, only one choice of outcome occurs, but in keeping with the probabilistic balance, just as in the n1-n2 index of refraction junction reflection-refraction situation. We then try to generalize to the one-dimensional quantum bound state situation. Directional Probability Ultimately, as in Newtonian mechanics, one considers problems which involve a particle interacting in some manner, be it with a 2-slit apparatus (etc), an n1-n2 index of refraction junction or a potential V(x). In Newtonian mechanics, such interactions are deterministic with the particle being a point which is acted upon by a force. In previous notes, however, we suggested that one may introduce a probability directly into Newtonian 2-body elastic scattering. In particular, for an incident (e1,e2) energy and (p1,p2) momentum vectors, any outcome (ei,ej), (pi,pj) which conserves energy and momentum has equal product probability. This then is an interacting scenario with a probability. We argued that such a probability should be a complex number with unit modulus as there is no real value weight for a free particle. We noted that there are many problems with using exp(iE C1) and exp(i p C2), but that introducing Lorentz invariance of the probability and time reversal invariance (p→-p, x→-x) for exp(ipx) leads to exp(-iEt+ipx). Such a formalism suggests a characteristic uncertainty in x, called the wavelength bhar/p and gives rise to the non-Newtonian interaction with a 2-slit apparatus. The exp(ipx) form also has consequences on other interactions, we argue. Interaction Time in a Time-Independent Interaction Problem Given exp(-iEt+ipx), one may note that due to the uncertainty in t and x given by this probability, one is not following a particle in terms of x(t) as in Newtonian mechanics. In fact, exp(-iEt) may be used separately to analyze time uncertainty and exp(ipx), spatial uncertainty. In a 2-slit calculation one may use exp(ipx)s alone and this may be extended to reflection/refraction at an n1-n2 junction and from a potential V(r). There is a big issue, however, with using exp(Ipx)s as probabilities which add and by ignoring time. It means that one may add exp(ipx)s representing very different times. For example, in an n1-n2 index of refraction junction reflection-refraction problem, one writes: Aexp(ipx) + B exp(-ipx) (i.e. incident and reflected photon) ((1a)) and in an elastic scattering problem, one writes: exp(ipz) + f(theta) exp(ipr)/r ((1b)) incident and scattered particles An incident and a reflected particle exist at different times and even though probabilities add in OR situations, the physical situations do not coexist at the same time. One might argue that there is a steady state picture, but the same result must hold for a single particle and in that case, the times are different.We argue, however, that there is a very important “catch”, namely the interaction time. The key is to solve an interaction problem. This essentially means finding B in terms of A in ((1a)) and the weight f(theta) in ((1b)) given a weight of 1 for exp(ipz). There is a classical time-related probability in the problems ((1a)) ((1b)) which differs from the directional probability “exp(ipx)”. It is the exp(ipx) directional probability which may be used to find the classical probability in time in the following manner we argue. At an interaction point (math), there is x uncertainty due to exp(ipx) (and time uncertainty due to exp(-iEt) which we do not consider). We have argued in previous notes that this means that for reflection-refraction at an n1-n2 index of refraction junction, one may have either the incident, reflected or refracted particle in a tiny dx region as the interaction occurs. One may write a kind of probability balance or equilibrium based on some continuity equation(s) at this interaction point, we argue. In the n1-n2 problem one has: A exp(ipx) + B exp(-ipx) = C exp(ip2x) at x=0 ((2a)) Ap exp(ipx) -pB exp(-ipx) = Cp2 exp(ip2s) at x=0 ((2b)) It is these two continuity equations which define a kind of directional probability equilibrium during the interaction, we argue, and allow one to find B and C in terms of A, which is the goal of the problem. We argue that for an interaction with a potential, there must be a similar continuity or balance equation for the various exp(ipx)s involved at the interaction point or region. This is the approach to solving the probabilistic problem, we argue. Interaction with an Infinite Spherical Potential We consider a particle being sent along the z axis which interacts with a hard sphere, i.e. an infinite spherical potential with radius ro. The particle may pass through if it is above the sphere, or interact. The question is: What percentage of the particles interact and scatter into an angle theta? In order to solve this problem, one may suggest that there should be an equation allowing for a kind of balance/equilibrium between exp(ipz) and the scattered particle, described by: f(theta) exp(ipr)/r ((3)) The question is then to find this equation. We first note that in principle in an elastic problem, one may consider an energy balance equation: Kinetic energy + V(r) = Energy ((4)) Here, one uses kinetic energy = 1/2m (-grad dot grad) because one deals with exp(ipx) directional probabilities. Outside r=ro, the potential is 0 and so one must find the most general solution of Kinetic energy =E. The math solution is: Al jl(kr) + Bl nl(kr) for r>ro ((5)) where jl is a Bessel function and nl, a Neumann Al and Bl are completely unknown and so one seeks to find Bl in terms of Al. Above, we argued that physically one should have the directional derivatives exp(ipz) and f(theta)exp(ipr)/r. One should be able to write ((5)) in terms of these two probabilities, but first may use ((5)) to find a relation between Al and Bl at the interaction region, i.e. at r=ro ((6)) At this point, ((5)) must vanish because for an infinite well potential, then can be no probability inside r<ro. Thus: 0= Al jl(kro) + Bl nl(kro) ((7)) ((7)) provides a relation between Al and Bl which is the key to solving the problem. A constraint/ boundary condition (balance equation between the two particle directional probabilities at the interaction region) solves the problem. All that is left is to show how Al and Bl are linked with exp(ipz) and f(theta) exp(ipr)/r. To do so, we consider r→infinite, but note that the problem has been essentially solved at the interaction region r=ro. Considerations of the r-> infinite limit only lead to an association of Al, Bl with exp(ipz) and f(theta)exp(ipr)/r and f(theta) is linked to how many particles scatter into angle theta. (We use k and p interchangeably.) The analysis we use follows (1). For r→infinite: jl(kr) → sin(kr-l*3.14/2) / (kr) and nl(kr) → -cos(kr-l*3.14/2) /(kr) ((8)) and exp(ikz) = 1/(2ik)) Sum over l i power l (2l+1) {exp(ikrl*3.14/2) - exp(-i(kr-l*3.14/)) } Pl(cos(theta) ((9)) Here Pl(cos(theta)) is the Legendre polynomial. (8)) may be linked directly to exp(ikz) and the remaining pieces must be f(theta) exp(ikr)/r. In particular, (from (1)) for the hard-sphere case: ((5)) for r→ infinite = 1/kr (Al sin(kr-l*3.14/2) - Bl cos(kr-l*3.134/2) ) = 1/kr sqrt(Al*Al { Bl*Bl) sin(krl*3.14/2 + delta(l) } ((10)) (from (1)) where delta(l) = aractan (- Bl /Al) which is known using ((7)) Now using ((8)) and ((9)), (1) shows that in general: ((5)) at r→infinite = exp(ikz + Sum over l (2l+1) (exp(2i delta(l)-1)/(2ik) Pl(cos(theta)) exp(ikr)/r Thus, f(theta) is known if one knows delta(l) which is precisely what ((7)) gives. As a result, the elastic scattering problem is essentially solved by considering a balance/equilibrium equation at the interaction region, i.e. ((7)). This we argue is the essential physics linked with using the directional probabilities exp(ipx)s. Bound Quantum State We now try to generalize to the one-dimensional bound quantum state. In such a case, we know that a particle interacts at each x with a V(x). At any x, a given p may be described by exp(ipx) and if V(x) = Sum over k Vk exp(ikx), p may be given any impulse hit k. One should consider exp(ipx) and all other possible outcomes at x, namely all exp(ipx). The question is: What is the balance/equilibrium equation at x? We suggest that it is conservation of energy, i.e. {Sum over p a(p) pp/2m exp(ipx) } / {Sum over p a(p)exp(ipx)) + V(x) = E ((11)) There is then a second boundary condition which exists at x→ infinite and x-> -infinite namely that: Sum over p a(p)exp(ipx) tends to 0 ((12)) Again the directional probabilities exp(ipx) and some kind of balance or constraint equation at an interaction point x leads to ((11)) and a second constraint to a solution at x→ +/- infinite at which the interaction has stopped. We argue that this is consistent with the arguments made above. Conclusion In conclusion, we argue that there exists a directional probability in Newtonian 2-body scattering, namely exp(-iEt+ipx). This probability suggests an uncertainty in x involved in interactions involving p as seen experimentally in the interaction of a particle with both slits of a 2-slit apparatus, if the slits are separated by about hbar/p. The goal in many physical problems is to describe a probabilistic feature of an interaction. In a reflection-refraction at an n1-n2 index of refraction junction, the goal is to find the probability to reflect and refract. These are not directional probabilities, but are governed by them, we argue. In the case of scattering from a potential V(r), one has exp(ipz) + f(theta) exp(ipr)/r and wants to know f(theta) which is associated with the probability to scatter into an angle theta. Thus, there is probability not only in Newtonian 2-body scattering, but in other interaction problems as well. The key idea is that it is the directional probability form “exp(ipx)” which solves these problems. The question is then: How is this done? In previous notes, we argued that for reflection-refraction at an n1-n2 junction at x=0, there is actually uncertainty in position at x=0 due to exp(ipx). We thus suggested that a directional probability equilibrium or balance occurs in this dx because one does not know if one has an incident, reflected or refracted particle in dx. This is equivalent to two continuity equations at x=0, namely: Aexp(ipx) + Bexp(-ipx) = Cexp(ip2x) at x=0 and Apexp(ipx) -pBexp(-ipx) = Cp2exp(ip2x) at x=0 This, in turn, yields B and C in terms of A, which is essentially the solution of the problem. We argue here that this same idea applies to scattering from a potential V(r) and consider the case of a hard sphere, i.e. an infinite potential at r=ro. The idea is that one considers a balance/equilibrium of directional probabilities at the interaction region again, which is r=ro. In principle, the physical directional probabilities are exp(ipz) and f(theta) exp(ipr)/r. Mathematically, however, one may solve: -1/2m grad dot grad Function = E function. This leads to an r-dependence in a general solution of Al jl(kr) + Bl nl(kr). This solution must be 0 at r=ro. It is this equation (the balance of the directional probabilities at the interaction) which essentially solves the problem. One then may consider r→ infinite so that one may compare the form: exp(ikz) + f(theta) exp(ikr)/r with {Al jl(kr) + Bl nl(kr)} g(l) Pl(cos(theta) where g(l) is a known function in l. This leads one to find f(theta) which is directly linked to the probability to scatter in a direction theta. The problem, however, is essentially solved at the interaction point using a balance of directional probabilities there. We then extend these ideas to one-dimensional bound states. References 1. Shankar, R. Principles of Quantum Mechanics (Plenum, 1988)