Speculation on Quantum Free Particle Probability and Spatial Equilibrium Part 5
Ruggeri, Francesco R.
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 conditions- constraints -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 problem- it 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).
•Nov 11, 2025•0 citations