Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.
Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
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 Part 1, we suggested that exp(ipx) represents a spatial probability for the rendering of momentum hits. A single particle has a center-of-mass which undergoes x=vt, but momentum hits are not rendered from this point, but are rather delivered in x according to the complex probability exp(ipx) which is linked with the resolution delta x = hbar/p. This accounts for the fact that a particle can probabilistically interact with each slit of a 2-slit apparatus if they are about hbar/apart. If one is only interested in energy, pp/2m, the classical result holds for both exp(-ipx) and exp(ipx). In a classical bound state, one considers only energy p(x)p(x)/2m + V(x)=E because each p(x) exists at x in dx->0. One cannot distinguish any probabilistic hit distribution. exp(ipx) and exp(-ipx) differ even though they are linked with the same kinetic energy pp/2m (nonrelativistic case). We argued in Part 1, that the Newtonian conservation of energy equation actually includes a force profile linked with V(x). The catch is that for V(x) one has a force at each x, dx->0. Such is not the case for exp(ipx). We argued in Part 1 that one must consider not only pp/2m, but the associated spatial probability distribution for hits (i.e. energy together with force). Here, we note that this idea gives rise to an equilibrium calculation. This is somewhat startling because classically, and even quantum mechanically for higher energy levels, a particle moves to the right part of the time and then to the left for other times. As a result, a calculation which respects time would not average over exp(ipx) and exp(-ipx). Only an equilibrium calculation in time would do so. We suggest that one must consider the Newtonian conservation of energy equation as applying to both p(x) and -p(x) and so from an energy point of view one may consider it to be an average. The catch is that each term p(x) and -p(x) yields the same kinetic energy profile in the Newtonian case and the same V(x). For a quantum particle the kinetic energy with probabilistic force distribution is not the same for p and -p. As a result, to use V(x) which applies to forward and backwards motion, one is forced to consider the average and this leads to an equilibrium scenario. We show that the average kinetic with probabilistic force hit profile in space is not the same for +p and -p. Measurements, however, take time and may be averages of much data and so the notion of an average or equilibrium seems to make sense. We note, however, that an average is very different from the Newtonian forwards-backwards motion picture. Only in the high energy limit does the energy pp/2m with probabilistic force profile become the same for both p and -p. Thus, the main point of this note is to stress the notion of a single particle bound state as an average and to point out the consequences of this.
In Part 1, we noted that W*(x)W(x) represents the average number of interaction hits at x for an equilibrium (p, -p) scenario. This follows from exp(ipx) representing the probability for a p hit at x. If -p is also present with the same weight, then exp(ipx)+exp(-ipx) is proportional to cos(px), showing that some x points have low or no probability for a p hit. We argued, however, that these hit points are not at the center-of-mass so the center-of-mass could be at a 0 point. More importantly, average kinetic energy exists at these low or no hit points. In this note, we wish to examine the average kinetic energy more closely. It is built upon both classical Newtonian (nonrelativistic) ideas and the notion of a probabilistic x point for p hits, given by the probability exp(ipx). We note that both x and V(x) are continuous, but suggest that a probabilistic picture is compatible with non-continuous impulse hits. In particular, lengths are of the order of 10 power -10 m (atomic) or 10 power -15 m (nuclear) and particle speeds, on the order of 10 power 7 m/s. This suggests that V(x) and exp(ipx) are associated with ensembles. In the classical case with very slow speeds and large lengths compared to the above numbers, a tiny dx seems to represent an ensemble. Each dx represents an average result p(x)p(x)/2m + V(x) = E. In the quantum case, one needs to use an average based on exp(ipx) (as argued in Part 1) because pp/2m is seen to exist and change at an interaction point. Thus, {Sum over p a(p)pp/2m exp(ipx) }/{Sum over p a(p)exp(ipx)} represents an average KE at an x interaction point which may be linked to the V(x) at this same point. One must have kinetic energy at an interaction point for it to be compatible with V(x)
In Newtonian mechanics, force and energy are clearly separated. For example, pp/2m = Integral F dx = Integral dp/dt dx = m/2 vv = pp/2m. The equation pp/2m + V(x) = E is considered to be strictly an energy equation. In special relativity, mocc is considered to be strictly an energy. We have suggested in previous notes that mo is proportional to inertia which is a force. Thus, mo and mocc are thought of in two very different manners simultaneouly, i.e. as an energy and as a force. We propose that it might be beneficial to think of both mo and V(x) in terms of energy and force distribution simultaneously. In particular, we consider that V(x) is not only potential energy at x, but is also linked to force or interaction at x through -d/dx V(x). The reason this duality of energy-force may be important is that it is directly linked to quantum mechanics, we argue. In general, it is argued that classical mechanics is a high energy level limit of quantum mechanics, We argue, however, that for a single particle bound state, the energy linked with a momentum p (at any energy level) is pp/2m (nonrelativistic) which is deterministic (and also follows from special relativity) and has nothing to do with quantum mechanics. The difference between classical mechanics and quantum, is that there exists an interaction (impulse hit) probability of exp(ipx) which means that instead of a hit occurring at the center-of-mass point, it may occur within an hbar/p range. Thus, one may have a particle interact and move through either of two slits if they are separated by about hbar/p. If p, however, is changed in an interaction, p1->p2, then the change in energy p2p2/2m - p1p1/2m (one dimension) is classical-deterministic. It is, however, the duality of V(x) with force that force one to consider not only energy pp/2m (deterministic), but also its associated force distribution given by exp(ipx) / W(x), where W(x) = Sum over p a(p)exp(ipx). As a result, one creates the average kinetic energy term: {Sum over p a(p) pp/2m exp(ipx) } / W(x). One may call exp(ipx)/W(x) a probability and so formally one has KEave(x), but physically exp(ipx) represents a force distribution. The balance to V(x) is a combination of a deterministic energy pp/2m with a force distribution exp(ipx)/W(x). This, we argue, is a physical statement and helps to describe why V(x) may be expanded as a series of virtual particles (e.g. photons etc). It is not just energy, but force that must be considered together. It is only in classical physics, pp/2m + V(x) = E that one solely considers the energy aspects, i.e. pp/2m and V(x) only as energy, but the more general picture, we argue, is to consider not only energy, but force distribution together. This is the quantum picture and it incorporates not only probability, but deterministic Newtonian mechanics (or special relativity) through the use of pp/2m.
Quantum mechanics is largely based on a wavefunction, which we consider to be a probability. The theory is, however, not solely probabilistic because if an interaction does not occur, x=vt holds as does the classical notion of impulse hit and kinetic energy = pp/2m (nonrelativistic case). The probability exp(-iEt+ipx) describes probabilistic interactions in time and space (we argue) and so one does not think of x and t as in x=vt. As a result, there is a probability for a quantum particle to interact with both slits of a two slit apparatus if they are about hbar/p apart. This leads us to the notion that W*(x)W(x) (W(x)=Sum over p a(p)exp(ipx)=wavefunction) is not spatial density, but rather an interaction profile. A particle can still exist at W(x)=0 points, it just has very little to no probability of interacting at these points on average, with the average considering both exp(ipx) and exp(-ipx) in a bound case scenario. Given that E=kinetic energy = pp/2m (nonrelativistic case only) and W*(x)W(x) not= spatial density, we suggest that one must be careful with interpreting d/dt partial W*(x)W(x) = constant grad dot Im (W* (-id/dx) W) (from (1)) as a continuity equation which represents hydrodynamic motion. We suggest that the quantum features are governed by exp(-iEt+ipx) and that any derivatives d/dx and d/dt are related to changes in phase and not hydrodynamic flow (in a classical sense). We note that for a free particle d/dt partial density = 0 = constant grad dot Im (W* (-id/dx) W). The same zero equation holds for a bound state (even though there is physical motion to the right and then left) , but if one creates a superposition of bound states (which is a probability statement), then the same equation is not 0. We note that for a classical bound state there exists a vrms(x), but this is an average quantity. It does not show any resolution pattern which is represented by W(x) and so one must be careful when interpreting it, we argue.