In previous notes, we argued one may formulate bound state quantum mechanics by assuming a stochastic potential which delivers “hits” of momentum k V(k) exp(ikx), while having an average form of V(x). A main constraint seems to be that the different k V(k) exp(ikx) hits may occur at any x point and that V(x) be of the form Sum over k b(kx) so that -d/dx b(kx) = -k b(kx). In other words, conservation of energy exists at the level of each hit as well as for the overall average. To force all of these constraints, a solution of b(kx)=V(k) exp(ikx) emerges which introduces periodicity and complex numbers. It should be noted that V(x)= Sum V(k) exp(ikx) so each term is actually a “piece of potential energy”. Furthermore, we argue that V(x) is decomposed for physical reasons i.e. V(x) is a low resolution result which incorporates all of the “pieces”. In a quantum mechanical situation, a “piece” V(k)exp(ikx) may act at t1 and V(k2)exp(i k2x) at t2. We argue that exp(ikx) should have physical meaning and leads to sin(kx) terms for the “k” and “-k” pieces. Thus, there is forward and backward motion which “cancel” i.e. have nothing to do with V(x). For a particle, however, in this region, there may be forward and backward motion, both of which have kinetic energy which should not be part of the average kinetic energy at x. The various physical (resolved) pieces of the potential lead to momentum distribution at each x point with P(p/x)= a(p) exp(ipx)/ W(x) (W(x)=wavefunction). In this case, one has a probability, not “pieces” of KE(x). At first, one may be alarmed by complex probabilities, but they are necessary to allow for cancellation of kinetic energy pieces due to the positive/negative values combined V(k)exp(ikx) pieces. Thus, even though KE(x) (average) matches classical results, the actual motion is more complicated. The potential seems to be the driver of the ensemble picture and should be linked to the microscopic physical nature of the potential. In this note, we wish to examine features and consequences of such a stochastic potential, including the idea of tunneling and infinite potential walls.
In previous notes, we argued one may formulate bound state quantum mechanics by assuming a stochastic potential which delivers “hits” of momentum k V(k) exp(ikx), while having an average form of V(x). A main constraint seems to be that the different k V(k) exp(ikx) hits may occur at any x point and that V(x) be of the form Sum over k b(kx) so that -d/dx b(kx) = -k b(kx). In other words, conservation of energy exists at the level of each hit as well as for the overall average. To force all of these constraints, a solution of b(kx)=V(k) exp(ikx) emerges which introduces periodicity and complex numbers. It should be noted that V(x)= Sum V(k) exp(ikx) so each term is actually a “piece of potential energy”. Furthermore, we argue that V(x) is decomposed for physical reasons i.e. V(x) is a low resolution result which incorporates all of the “pieces”. In a quantum mechanical situation, a “piece” V(k)exp(ikx) may act at t1 and V(k2)exp(i k2x) at t2. We argue that exp(ikx) should have physical meaning and leads to sin(kx) terms for the “k” and “-k” pieces. Thus, there is forward and backward motion which “cancel” i.e. have nothing to do with V(x). For a particle, however, in this region, there may be forward and backward motion, both of which have kinetic energy which should not be part of the average kinetic energy at x. The various physical (resolved) pieces of the potential lead to momentum distribution at each x point with P(p/x)= a(p) exp(ipx)/ W(x) (W(x)=wavefunction). In this case, one has a probability, not “pieces” of KE(x). At first, one may be alarmed by complex probabilities, but they are necessary to allow for cancellation of kinetic energy pieces due to the positive/negative values combined V(k)exp(ikx) pieces. Thus, even though KE(x) (average) matches classical results, the actual motion is more complicated. The potential seems to be the driver of the ensemble picture and should be linked to the microscopic physical nature of the potential. In this note, we wish to examine features and consequences of such a stochastic potential, including the idea of tunneling and infinite potential walls.
We argue that in a quantum bound state with one particle, the particle receives stochastic hits from a potential. As a result, if one examines a point x, the particle may be present at ti with momentum pi (in both the forward and backward directions) leading to a momentum distribution with a conditional probability P(p/x) = a(p) exp(ipx)/W(x) where W(x) is the wavefunction. Thus, one does not follow a stochastic particle in time, but rather describes it by a probability distribution at each x. One may next ask whether this conditional probability at x changes in time? This is not the actual time related to stochastic motion of the particle. To answer this question, we argue one must consider interactions which occur at x which are part of an energy equation. Even in classical physics, the energy equation at x indicates how a particle’s motion changes. Thus, we argue one does not simply take measurements at x of ti and pi, but that at each ti, a particle with pi is undergoing dynamic changes and these may be considered as well. If one describes the quantum picture by W(x) i.e. a momentum distribution, one may find a collective interaction given by -1/2m d/dx d/dx W + VW. The coupling between V(x)=Sum over k V(x)exp(ikx) and the particle conditional probabilities in W is apparent in the VW term. This energy term should describe whether the momentum distribution changes in time i.e should be proportional to dW/dt (partial derivative), but should also equal the average energy. Thus, a link exists between ensemble energy and time. In this note, we argue that these ideas may be applied to P(p/x) in a bound state resonance because the same E applies to each p.
The classical limit of quantum mechanics is often considered in terms of the time dependent Schrodinger equation. In this note we consider the bound case solution i.e. a time dependence of exp(-iEt) and as in previous notes argue that time has been removed from this statistical equilibrium picture as a the quantum particle tries to establish a kind of impulse potential field based on superpositions of exp(ipx) terms. Such a potential field is linked with a bound state and does not involve following the particle about in time as it undergoes stochastic impulse hits with exp(ikx) terms from V(x)=Sum over k Vk exp(ikx). On average a classical energy balance equation KE(x)+V(x) = En holds for every energy En, but the average is created by a spread of momenta i.e. W(x)=wavefunction= Sum over p a(p)exp(ipx). In this note we consider the case of a particle in an infinite potential well and note how the spatial resolution becomes very high at the same time that the momentum spread becomes very narrow in the case of high n (energy level). Thus as n becomes approaches infinite, it is not just the vrms(x) which is linked with x, but the statistical scenario of quantum mechanics approaches a situation in which x is being resolved to a very high level and p is almost constant. Furthermore quantum entropy becomes a constant (independent of n) as n→ infinite suggesting a theory which is nonstatistical in this limit.This would represent a classical picture except for the fact that time is still removed from the quantum bound state i.e. forward and backward exp(ipx), exp(-ipx) still interfere. Given that x and p are well resolved locally for high n it seems one may follow the high n quantum particle in space and time and manually remove the forward and backward portions to create a theory of a particle moving in one direction under V(x) i.e. classical physics. This bypasses the idea of a wavepacket whose x-spread increases in time and allows one to base classical physics on a bound state solution of a quantum particle in an infinite well.
Newton’s second law dp/dt = F(x)=-dV/dx may be written in terms of x alone i.e. dp/dx v(x) = -dV/dx or 1/(2m) d/dx pp = -dV/dx leading to the idea of kinetic energy pp/2m and potential energy V(x). In (1) we argued that there is velocity, rest mass mo, and momentum in quantum mechanics. A free particle, however, does not undergo smooth acceleration as in classical mechanics, and we argued there exists a momentum space wave exp(ipx) accompanying the moving particle. In such a scenario, force is considered to be probabilistic momentum hits of the form Vk exp(ikx) leading to a free particle momentum distribution W(x)= Sum over p a(p) exp(ipx). Both of these expressions involve interference due to momentum waves which do not track a particle in time. A mass flux mov or energy flux p=Ev relativistically seems to be the main consideration. Nevertheless, summing Vkexp(ikx) over k to obtain V(x) yields a function varying in x alone. This is a kind of average based on interference and Sum k Vk exp(ikx) is an average linked to momentum hits. We speculate that average momentum hits i.e. -d/dx V(x) are linked to momentum flux changes. In other words, even though an impulse is based on p and does not distinguish v i.e. p=m1v1=m2v2, momentum flux is velocity dependent and appears for an average momentum hit i.e. -dV/dx. In such a case (pave+dpave)(vave + dv ave) = 2 pave dv(ave) . This expression, however, contains both forward and backward motion which is part of quantum mechanics. Moving to an average picture seems to introduce time and direction i.e one uses pave d v(ave). This is already a change from a quantum to a classical scenario albeit for averages (which have not been specifically defined). Thus we consider pave dv(ave). If this is the case, then: d/dx pave 1/m space = -dV/dx or 1/(2m) pave pave = -d/dx V(x). Instead of treating pave pave as two separate averages one considers pave=prms or pavepave=<pp>. Thus the quantum bound picture (at any energy level) becomes a classical type of scenario which includes time and velocity. Note that for a two slit interference pattern, one does not need velocity and time because the interaction with the potentials from both slits is based on exp(ipx). One does not even need the form of the potential. Interference is the same for p=m1v1=m2v2 (even though the process takes place during different time intervals, the pattern is not affected). In other words, we argue that Newton’s second law might follow from a picture of momentum hits in which a flux i.e. p=mov is treated as a quantity which in an average scenario leads to a second flux i.e. an average momentum flux which reintroduces direction, time and velocity.. This is somewhat like a hydrodynamics scenario.
In quantum mechanics, one often speaks of wave-like behaviour. In a previous note, however, we argued that wavelike behaviour is based on “averaging” of complex, periodic probabilities and so represents an ensemble average taken at different times. If one does not want an average over different times, but rather follows a quantum particle in time, it seems it undergoes a constrained random walk. This random walk differs from usual statistical ones in that the stochastic hits from a potential must on average yield V(x) at each x. We argue this is the source of the complex, periodic terms exp(ipx) which are part of a Fourier series of V(x). The potential influences the momentum distribution of a particle causing it to also exhibit complex, periodic probabilities of the form exp(ipx). Thus, if one considers “or” probabilistic situations for p and -p, one may obtain real periodic functions such as sin(px) and cos(px), but we argue these are time averages. A particle is either in a state p or -p (or some other momentum state) at x at a particular time. Thus, the wave-like behaviour follows from averaging over time exp(ipx) terms, but at each instant it seems a particle follows a constrained random walk type motion possibly changing its direction of motion with each stochastic hit from a potential. Overall, however, there must be a periodicity in the bound system, but we have argued this (based on exp(iEt)) is related to changes of a time averaged ensemble. We argue that this periodicity is also found in the constrained random walk picture if one considers a series of xi points (xi) with associated momenta (pi) and uses P(xi)P(pi/xi) as the probability at each point. Then the average kinetic energy for a constrained random walk added to V(x)W(x)W(x) yields EW(x)W(x). We also argue that W(x)W(x) is related to the idea of constrained random walk. The idea of a constrained random walk in quantum mechanics has no doubt been suggested in the literature, but the main focus of quantum mechanics does not seem to be with such a random walk, but rather with averages (taken at different times) which related to a wave picture that exists only as an ensemble average.
Quantum mechanics utilizes a wavefunction W(x)=Sum over p a(p)exp(ipx) (in time independent cases) with exp(ipx) being what we call a free particle complex probability. In such a case, W(x) is an OR scenario of these various probabilities. We have argued in previous notes that exp(ipx) is the probability for a p impulse hit at x, even if the center-of-mass of the particle is not at x. This means that impulse hits may occur probabilistically in space as in the 2-slit example which requires adding two probabilities as an OR case for each slit. The point then seems to be that W(x) (or even its momentum frozen form W*(x)W(x) gives a profile in space of the number of interactions (various p impulse hits) at each x. Classically, the kinetic energy (nonrelativistic) is pp/2m. In a classical bound problem this kinetic energy is considered to exist (in calculations) at the center-of-mass, but forces are also said to be rendered at this point so pp/2m, the center-of-mass and the force hit point all coincide. In quantum mechanics, a p impulse hit and the center-of-mass do not coincide. If p is said to hit at a probabilistic x, then it seems that pp/2m should also be seen to exist at this same hit point. A physical interaction point verifies in a sense the location of pp/2m in a probabilistic manner, i.e. a(p) exp(ipx)/W(x) is the probability used to calculate KEaverage and this is based on impulse hit locations. By normalizing using W(x), one removes the relative number of average hits at each x and obtains a pure energy value at x. (This may then be used in a KEave(x)+V(x)=E equation.) On average W(x) has zero points, which means that if one considers W*(x)W(x) to represent the motion-frozen-out spatial distribution of hits, then no hits are rendered at certain x points (on average). One might argue then that the particle cannot exist at x, but the center-of-mass can exist anywhere. One might next argue that there is no kinetic energy at x, but that cannot be true because 1/2m d/dx d/dx W has the same 0 points as W and the two divide. Thus, there is average kinetic energy at all bound points and this represents an average based on impulse hits (exp(ipx)). Classically, if one has many particles in a potential, one may write density(x)V(x) to find the potential energy of the particles. This implies that the particles interact with V(x), e.g. neutrinos usually do not interact. Density(x) in a classical sense means that the center-of-mass and force hit point are both at x. In quantum mechanics, exp(ipx) represents the probability for a force hit point, W(x) represents an average and W*(x)W(x) represents a motion-frozen average hit point. W*(x)W(x)V(x) seems to represent the average potential energy of the particle at x because W*(x)W(x) is linked with interaction. Even if a particle is not present, however, there is energy V(x) in space. It seems that if W*(x)W(x)=0 at some x, then there is no average particle interaction at this point, but there is still average kinetic energy Sum over p a(p)pp/2m exp(ipx) / W(x) based on the probability for average probability hits. The key idea is that one normalizes probability by dividing by W(x) in the average kinetic energy calculation. W*(x)W(x) is normalized not at x, but integrating over dx and setting this value to 1. One has relative hit at x values represented by W*(x)W(x). For exp(-ipx)exp(ipx)=1, each dx has the same average hit value (with one not being able to distinguish details in hbar/p, i.e. this is a motion frozen result. The question then becomes: Why would one write: W*(x)W(x) KEave(x)? One might argue that this term occurs if one takes the quantum mechanical equation: KEave(x) +V(x) = En and multiplies by W*(x)W(x). One may ask: Does V(x)W*(x)W(x) have physical meaning? V(x) is the potential energy at x. The average interaction probability with V(x) at x should be W(x)V(x) and not W*(x)W(x)V(x). The latter is a classical mechanical idealization. If W*(x)W(x) is 3 at x1 and 2 at x2, then it is as if the particle interacts 1.5 times more at x1 and so if V(x1)=V(x2), then this approach assumes 1.5 the interaction energy at x1. KEave(x), however, is simply free particle energy (average) at x and one does not need to take this average value and multiply it by the relative number of interactions at x. W(x) has zero points meaning that there is no interaction at such a point on average. KEave(x) on the other hand is the average energy at x based on the particle kinetic energy presence pp/2m being at the point of an impulse hit. This does not mean that an interaction is occurring with V(x) at this point in this formalism. Even a free particle has an exp(ipx) probability when it is not interacting. The sum Sum over p a(p)exp(ipx) means that all of the interactions linked with x have occurred and one is making an OR calculation. KEave(x) is free particle kinetic energy at x and V(x) is potential energy (not interaction potential energy with the particle). Multiplying by W*(x)W(x) is a classical minded approach to creating a classical equation: density(x) (KEave(x)+V(x)) = E density(x) ((1)) Here KEave(x) would represent the average kinetic energy of a single particle and density(x) would be the average number of particles at x. density(x)KE(x) has a clear meaning. W*(x)W(x) KEave(x) on the other hand is different. KEave(x) is actually the classical kinetic energy value at x and so it is not clear why one would multiply it by W*(x)W(x). W*(x)W(x) represents the probability for an impulse hit at x in a p-frozen out picture. The notion of exp(ipx) (i.e. probability for a p hit at x) is already buried within the quantum calculation of KEave(x) (which yields the classical value). In order to find the physical meaning of W*(x)W(x) multiplied by KEave(x) or V(x) it seems that one has broken dx into equally sized segments which correspond to different detection hits of the particle. This is then somehow equated with the presence of the particle per unit time in dx which is then somehow rendered as the average energy at x multiplied by the detected presence at x through a steady rate of measurement/time. Thus, multiplying by W*(x)W(x) is not multiplying by a physical density.
In a previous note (1), we argued that a potential V(x) may be written as Sum over k V(k) exp(ikx). Thus, -dV/dx is only an average force which may in fact be zero, yet -d/dx V(k) exp(ikx) is not zero leading to possible forward and backward motion/acceleration (due to each V(k) exp(ikx) which act at a different times). This gives rise to a momentum distribution P(p/x) =a(p) exp(ikx)/W(x) (W(x)=wavefunction) at each x. In this note, we wish to link these ideas to the Bohm Aharonov effect by considering two recent (2018, 2020) examples from the literature. In the first (2), the magnetic potential A(r) is written in a Fourier series in the same manner as V(k)exp(ikx). Thus, although grad A = B (magnetic field) = 0, grad A(k) exp(ikx) should not be zero and so there may be hits from “virtual photons”. (2) applies perturbation theory to find a correction to energy E and then considers exp(i Integral over t delta E dt) as in adiabatic theory. The second approach (3) applies the usual Hamiltonian containing the magnetic potential A, but imagines a little box at various xi positions such that A(xi) is a constant at each xi. In such a case, one may show directly that W(x)= W(x=0) exp( ie/hbar Integral xo to x1 A(x) dx) which yields the Ehrenberg–Siday–Aharonov–Bohm effect. In this approach, one essentially has a different constant potential at each point and hence a different energy. In a previous note, we argued that even though average kinetic energy may remain the same, a change in constant potential using Sum over k V(k) exp(ikx) gives rise to different physical effects which try to argue are related to the Bohm Aharonov effect. We also try to consider a more general case.