Full text
Speculation on Quantum Free Particle Probability and Spatial Equilibrium Part 5 Francesco R. Ruggeri Hanwell, N.B. Nov.11, 2025 In this note, we try to understand why tunneling should be part of a quantum mechanical treatment of a one-dimensional bound state problem. We suggest that the ideas behind tunneling begin with the directional probability exp(ipx) used to describe conservation of momentum in Newtonian 2-body scattering. Given the notion of a wavelength in space, there must be a probability scheme in space which “physically” creates the wavelength, as we have mentioned before. This means one cannot have uniform probability in x and so some directional probability (linked to exp(ipx)) must be removed from one x region and moved to another. Thus, one cannot point to a particular x1 point, for which cos(px1)=0 and say that the particle has disappeared. Probability is simply shifted from one point to another and this leads to the idea of positive and negative values linked to this probability as one sees from the negative and positive values of sin(px) and cos(px). As mentioned in Part 4, one may create various averages of functions of p including the function 1 which is then linked to particle presence (i.e. spatial density although it does not directly equal this). Given the periodicity of exp(ipx), W(x) =Sum over a(p)exp(ipx) usually tends to be periodic as well, even if one considers a(p)=a(-p) and a real W(x). It is possible, however, to have smooth valued functions for {Sum over p f(p)exp(ipx) a(p)} / W(x). This might lead one to think that one has a classical scenario with no wavelength effects present, but we argue here that this is not the case. Furthermore, we argue that this type of argument seems to explain why tunneling is linked with a one-dimensional bound state solution. We noted in Part 4, that one may solve an equation like the time-independent Schrodinger one to obtain a linear combination solution of directional probabilities exp(ipx)s, i.e. W(x). In Part 4, we were concerned with what to do with W(x) and argued for imposing boundary conditionsconstraints -continuity on W(x). Here we consider the case of classical turning point, for which KEave(x)=0. For a classical particle, delta x being tiny is huge compared with the wavelength of exp(ipx). Thus, p→0 leads to wavelength = hbar/p, but for a tiny p, a large wavelength is not a problem as it is still not greater than delta x. Thus, one may have a p (momentum) become very small at a classical turning point (and even argue that it is 0.) In the quantum region, one cannot do this because a delta x near the classical turning point x1 is very small compared to hbar/p for p close to zero. One cannot have a single or a few small value p’s, but must rather have sizable p values. This means that the particle is in one of these p states or another and has sizable impulse and kinetic energy pp/2m. As a result, the potential barrier is not a problemit does not hold the particle back. What is zero is the average value of KE, i..e KEave(x) = { Sum over p a(p)pp/2m exp(ipx) } / W(x). As we have argued above, for a(p)=a(-p), exp(ipx)) becomes cos(px) and even with positive a(p), one has negative and positive directional probability values which allows sizable p, pp/2m’s to have an average of zero. As noted above, however, an average kinetic energy of 0 at a single x1 does not mean that it is zero at x1+delta. Just as with cos(px), removing values from some x region means that they appear elsewhere. KEave(x)=0 is not a constraint for x outside the classical turning point region. The quantum average KEave(x)=0 describes various probabilities for sizable momenta and pp/2m and so must deal with a particle moving in and out of the so-called bound region.
The question becomes: How does one make this appear like a bound state problem, i.e. one localized in space? This occurs by imposing the constraint W(x)=0 as x→ +/- infinite as noted in Part 4. It is this constraint which shows that even though one has sizable p values at the classical turning point (despite KEave(x)=0 but W(x) not=0), one must have the average W(x) drop to 0 as x→ +/- infinite. As a result, given the nature of the quantum calculation, it is physically expected that tunneling should occur for a one-dimensional bound problem because there is no constraint on a single p or a few p values becoming very small (or 0) at the classical turning point. Thus, they should keep moving past the classical turning point. This is the physical event which should occur (i.e. tunneling). Directional Probability In Parts 1 and 2, we suggested that there exists a directional probability to describe energy and momentum conservation in Newtonian 2-body scattering. We argued that for an initial (e1,e2) energy set and (p1,p2) momentum vector set, any (ei,ej) (pi,pj) vector set which conserves energy and momentum has the same probability. Furthermore, for energy and momentum, there should exist a probability for a single particle such that a product yields the (ei,ej) (pi,pj) case, i.e. and AND scenario. We showed that this leads to the unusual result: exp(-iEt+i p dot r) ((1)) If one considers a time-independent case and one dimension, then exp(ipx), a directional derivative, represents a probability to have a value of p. This means that one may use exp(ipx) to create an average: Sum over p f(p)exp(ipx), where f(p) is some function of p ((2)) An example might be f(p) = pp2/m a(p), where pp/2m is kinetic energy and a(p) a weight for exp(ipx). Then one may obtain average kinetic energy: KEave(x) = {Sum over p a(p)pp/2m exp(ipx) } / W(x) ((3)) where W(x)=Sum over p a(p)exp(ipx) One may, however, use f(p) = a(p) and then one has W(x) which is somehow linked to a probability for the particle to be at an x position. Note: W(x) is not P(x), it is only linked to this value. A feature of exp(ipx) is that is associated with a wavelength in space given by: Wavelength = hbar/p ((3)) A key idea is that one must be able to mark this length in space using probability and this must occur throughout space leading to both non-uniform probabilities in x and even positive and negative value of directional probability ii.e.
cos(px) and sin(px) both have positive and negative values ((4)) We argue that one must be careful at this point because a 0 value of cos(px) does not mean that the particle has disappeared, it only means that directional probability has been shifted from one x point to another. As a result, one cannot take cos(px) =0 or even W(x) =0 statements too literally, we argue. A directional probability, as argued in Part 4, is not a quantity (classical) probability. It may, however, be linked with a quantity probability in such a way that W(x)=0 has relevance, i.e. P(x) = W*(x)W(x) = spatial density ((5)) Then P(x)=0 and W(x)=0 are consistent and there is no particle at x=0, but in quantum mechanics, a particle with a fixed momentum is described by exp(ipx) and so it is the average which is 0 at a single point, but not zero at a point nearby. This even holds in the analysis of 2-slit interference. Thus, an average value created using exp(ipx)s may happen to be 0 at one point, but not at the next and so one cannot necessarily apply classical quantity arguments to it. To be specific, we consider KEave(x) ((3)). At a certain x1, W(x1) may not be zero, hence W*(x)W(x) is not zero, but for a(p)=a(-p), KEave(x) real may be 0. In classical physics, kinetic energy =0 means that the particle has a single momentum value of 0. For such a p=0, the directional derivative exp(ipx) suggests an infinite wavelength hbar/p. At first this seems like a paradox. We argue, however, that p is only very small (not exactly 0) at a classical turning point in a classical problem and that the wavelength is not then infinite, but fits in a delta x, which is considered to be a tiny value in the classical realm. Thus, ((3)) being 0 may be ascribed to very small p values and so there is essentially no possibility of even a single exp(ipx) having any momentum or energy, i.e. moving past the classical turning point. Such is not the case in the quantum realm. There, a tiny p means wavelength=hbar/p is very large compared with delta x regions which are considered small on this length scale. As a result, one cannot have “small” p values, i.e. the particle has not stopped moving at the turning point, only the average value of KE, KEave(x)=0. A particle which still has sizable possible p values may then easily move beyond the classical turning point. In fact, this is expected. There is no surprise that the particle moves beyond the classical turning point. The question now becomes: If that is the case, how can one possibly argue that a bound state, i.e, one localized in space, exists if particles may easily move beyond the classical turning points? In order to force localization, one imposes the constraint: W(x)=0 at x→ +/- infinite ((6)) It is ((6)) which essentially solves the problem as argued in Part 4. ((6)) means that W(x) is forced to move towards zero, meaning the particle has very little probability to be at x as x moves away from the classical turning. Now KEave(x) does not need to drop to 0 as the constraint ((6)) is only on W(x). Overall kinetic energy, however, is: Overall KE = KEave(x) W*(x)W(x) ((7))
Thus, it is irrelevant if KEave(x) is not small as x moves away from a boundary point as W*(x)W(x)->0 because W(x)-->0 ensures the product is 0 There is no particle and no overall KE as W(x)-->0. The constraint ((6)) then ensures the correct weighting for tunneling in and out, but the fact that tunneling occurs is completely expected in the quantum framework. Conclusion In conclusion, we argue that in classical mechanics, a particle stops at a turning point because its momentum and kinetic energy are actually zero at this point. In quantum mechanics, this is not the case as one has: KEave(x) = {Sum over p a(p)pp/2m exp(ipx)} / { Sum over p a(p)exp(ipx)}. In the classical limit, a tiny dx region relative to the system length is “huge” and so the idea of small p values, with wavelength hbar/p being so-called large, fit into such a huge dx. Such is not the case in the quantum realm, where a dx is tiny compared to hbar/p if p is very small. Thus, p values must be large and so regardless of the average KEave(x)=0, the actual p values are still sizable and there is no reason for the particle not to keep going outside the turning point (unless one has an infinite well potential). If this is the case, however, it seems that one would not have a localized system within the classical turning points and so this would not be a bound state. To allow for a bound state, one imposes the constraint W(x)=0 at x→ +/- infinite and so this creates quantitative tunneling probabilities, but in principle, KEave(x turning point) =0 does not mean that p values are small and there is no reason to expect that a particle stops at the classical turning point in the quantum scenario.
