In (1), a formula for average dwell time in a step-potential damped region (from x=0 to x=ininite) is given by the Buttiker equation: time= m/(hbar k) Integral (0, infinite) W*(x)W(x) dx, where W(x), the wavefunction carries no units and is of the form: 2ik/(ik-k1) exp(-x/k1). Here k is the wavenumber in the V=0 region. We argue that in quantum mechanics one deals with average operator values: Integral dx W*(x) operator W(x) to yield an observable result. We call this observable result the measurement or interaction reality because it is based on the factor W*(x)W(x). We suggest that this factor is not the probability for a particle to be present within dx at x, but rather the probability for it to interact within dx at x. In other words, quantum observables based on using the factor W*(x)W(x) represents the information a measurement would reveal. The idea then seems to be that quantum mechanics only represents measurement/interaction reality. Mathematically, however, there are other situations. For a bound state, one has: KEave(x) + V(x) = En. KEave= -1/2m d/dx dW/dx / W and this value represents the classical kinetic energy at x. It is only when one multiplies KEave by W*(x)W(x) that one brings measurement/interaction reality into the picture. Furthermore, using W(x) means using a non-W*(x)W(x) weighted length scheme even though x is a quantum observable. If one has a bound state in an infinite well potential, the lowest bound state (x=0, x=L) is W(x)= C sin(3.14*x/L). Integrating from 0 to L or from 0 to L/2, yields L and L/2 respectively. Integrating from 0 to L/4, however, one does not yield L/4, neither does one obtain L/4 for integration from L/4 to L/2. Even though the metric in x-space for x which is fed into sin(3.14*x/L) is 1, i.e. the usual uniform classical x space, W*(x)W(x)dx is not the usual classical space, it has a different metric, an interaction or measurement metric. The point is that one seems to have two different length systems, i.e. two different metrics. Given that one may ask: What is the interaction probability at x?, one seems to have the usual classical metric in space because x is measured according to this metric. This metric could be established by a very high energy bound state in an infinite potential well. For the lowest bound state system, however, measurement indicates that there are few interactions near x=0 and x=L and so because measurement is the only way of establishing reality in quantum mechanics, it is suggested that the particle is mostly absent from these low to zero wavelength value points. We argue, however, that a wavefunction is only a measure of interaction and does not mean that the particle is not present at these zero wavelength value points. This is why the average kinetic energy is the same at all x points from 0 to L, we argue. We suggest that there can exist a reality other than the measurement based one for a quantum system. Just because one does not measure a particle at W(x)=0 does not mean that it cannot exist at this point. This suggests that using Integral W*(x)W(x) dx to calculate a dwelling time in a tunneling region is a measurement based length and so one should be aware that this kind of reality is being used in such a calculation.
In (1), a formula for average dwell time in a step-potential damped region (from x=0 to x=ininite) is given by the Buttiker equation: time= m/(hbar k) Integral (0, infinite) W*(x)W(x) dx, where W(x), the wavefunction carries no units and is of the form: 2ik/(ik-k1) exp(-x/k1). Here k is the wavenumber in the V=0 region. We argue that in quantum mechanics one deals with average operator values: Integral dx W*(x) operator W(x) to yield an observable result. We call this observable result the measurement or interaction reality because it is based on the factor W*(x)W(x). We suggest that this factor is not the probability for a particle to be present within dx at x, but rather the probability for it to interact within dx at x. In other words, quantum observables based on using the factor W*(x)W(x) represents the information a measurement would reveal. The idea then seems to be that quantum mechanics only represents measurement/interaction reality. Mathematically, however, there are other situations. For a bound state, one has: KEave(x) + V(x) = En. KEave= -1/2m d/dx dW/dx / W and this value represents the classical kinetic energy at x. It is only when one multiplies KEave by W*(x)W(x) that one brings measurement/interaction reality into the picture. Furthermore, using W(x) means using a non-W*(x)W(x) weighted length scheme even though x is a quantum observable. If one has a bound state in an infinite well potential, the lowest bound state (x=0, x=L) is W(x)= C sin(3.14x/L). Integrating from 0 to L or from 0 to L/2, yields L and L/2 respectively. Integrating from 0 to L/4, however, one does not yield L/4, neither does one obtain L/4 for integration from L/4 to L/2. Even though the metric in x-space for x which is fed into sin(3.14x/L) is 1, i.e. the usual uniform classical x space, W*(x)W(x)dx is not the usual classical space, it has a different metric, an interaction or measurement metric. The point is that one seems to have two different length systems, i.e. two different metrics. Given that one may ask: What is the interaction probability at x?, one seems to have the usual classical metric in space because x is measured according to this metric. This metric could be established by a very high energy bound state in an infinite potential well. For the lowest bound state system, however, measurement indicates that there are few interactions near x=0 and x=L and so because measurement is the only way of establishing reality in quantum mechanics, it is suggested that the particle is mostly absent from these low to zero wavelength value points. We argue, however, that a wavefunction is only a measure of interaction and does not mean that the particle is not present at these zero wavelength value points. This is why the average kinetic energy is the same at all x points from 0 to L, we argue. We suggest that there can exist a reality other than the measurement based one for a quantum system. Just because one does not measure a particle at W(x)=0 does not mean that it cannot exist at this point. This suggests that using Integral W*(x)W(x) dx to calculate a dwelling time in a tunneling region is a measurement based length and so one should be aware that this kind of reality is being used in such a calculation.
In Part 1, we identified W*(x)W(x) (where W(x) is the spatial part of a wavefunction) as the probability to interact. In the classical limit, if one draws a curve roughly touching the peaks of W*(x)W(x), then this probability is sometimes identified in the literature as proportional to the amount of time spent in dx, i.e. dt= dx/v(x). In the classical limit, however, dt(x) is not only the amount of time interacting in dx at x, but also the amount of time spent passing through dx at x. Such is not the case in quantum mechanics. For a bound state at any energy level, there is a KEave(x) = -1/2m d/dx dW/dx / W = E-V(x) = classical KE(x). This means there exists a vrms(x) so for a fixed dx, there is a quantum average passage time through dx, which may be more related to an ensemble average passage rather than any physical passage time. In such a case, we suggest that W*(x)W(x)dx is not only the probability to interact, but may be considered proportional to an ensemble average interaction time at dx. The expression V(x) W*(x)W(x) means that one weights the potential energy by the average amount of interaction time spent in dx at x. In other words, the quantum observable reality discussed in Part 1 seems to be equivalent to the interaction experience of the particle. It is not merely the case that there is a potential energy at x, there is also a certain interaction time spent in dx at this x (even in the classical case) and this is part of the potential energy experience of the particle. Thus, the potential energy V(x) is weighted by W*(x)W(x) to describe this experience. In the classical case, this dt interaction time is also physical passage time, but in quantum mechanics it is not (as will be shown). As a result, the quantum average Integral dx W*(x) operator W(x) = Integral dx W*(x)W(x) (operator W ) / W(x) represents both a reality based on external measurement or the particle’s experience interacting. Both are interaction realities. KE(x) = -1/2m d/dx dW/dx /W is also a physical expression which should have as much meaning as V(x) or E, but it is not based on the W*(x)W(x) weighted view of the particle. In Part 1, we focused on external measurements, but here we concentrate on the interaction of the particle with V(x) and suggest that this also gives a kind of physical meaning to <W V W> = Integral dx W* V W.
It is sometimes stated in the literature (1) that quantum mechanics is based on conditional probabilities. If this is the case, one would expect such conditional probabilities to appear in a single particle bound state. A free quantum particle is associated with a probability to interact of exp(ipx) (linked to 2-slit interference, 1-D reflection-refraction at an n1-n2 junction etc), but the kinetic energy and momentum are pp/2m (nonrelativistic) and p at all x even though each x has a different phase according to exp(ipx). The momentum is p at x, but the probability for an impulse hit at x is not the same even though its modulus is. When different probabilities exp(ipx) add in an OR situation, this leads to a spatial pattern which no longer has a modulus of 1, i.e. translational invariance is broken. Such a pattern also occurs for a single particle bound state W(x)=Sum over p a(p)exp(ipx). W(x) can have 0 points, but the average kinetic energy -1/2m d/dx dW/dx / W is not 0 at such a point. One has, on average (a statistical calculation), no average interaction at such an x point. One might ask why this should have anything to do with average kinetic energy? If one writes a conservation law: KEave(x) + V(x) = 0, then one considers kinetic energy interacting with potential energy. If there is no interaction at some x point for a given p (not W(x) on average), then it should not be part of the conservation of energy at this point. One is only interested in kinetic energy which interacts with potential energy at each x point. Thus, a(p) exp(ipx) + a(-p) exp(-ipx) (with a(p)=a(-p)) may have a 0 point at some x and so it does not have an average interaction at such a point, i.e. pp/2m (a(p)exp(ipx) + a(-p)exp(-ipx)) = 0. As a result, in a bound state, one considers kinetic energy per p, -p multiplied by the average probability (p, -p). Different a(p)exp(ipx) + a(-p)exp(-ipx) (a(p)=a(-p)) weights in this average picture indicate the extent to which the particle interacts. This idea holds for a single p,-p pair, but one must normalize over all such pairs, leading to the idea of a conditional probability. Given that hbar/p must be “created physically” in space so to speak, one must have interaction probability which creates this physical situation even in cos(px)+ i sin(px). This leads to the idea of interaction probability being 0, positive, but <1, or even negative and it is the interaction which allows one to consider conservation of energy with a potential V(x) in the first place. As a result, one cannot include a full pp/2m if there is negative, 0 or only particle interaction. Thus, pp/2m is multiplied by 2 a(p)cos(px) /W(x) for a(p)=a(-p).
We have argued in previous notes that both Newtonian mechanics and free particle quantum mechanics arise from special relativity. Thus, special relativity gives an expression for free particle energy and momentum and these are related to concepts of work. The Lorentz invariant -Et+px shows that x,t and x+hbar/p, t+hbar/E give the same solution to this potential-like term showing that there is stochasticity in the position and time delivery of interaction hits. In other words, even though the center-of-mass moves as x=vt, a momentum hit (proportional to rest mass mo) does not necessarily act at the center-of-mass. This gives the impression of shifts in probability, kinetic energy etc in the case of interference, i.e. interactions. For a single free particle, there is no interaction and -1/2m d/dx d/dx exp(ipx) = pp/2m exp(ipx). We argue that exp(ipx) is a probability and that in general one may add independent probabilities in an OR situation as in a 2-slit interference case. A bound state, however, is different because even though one adds weighted exp(ipx)s, they are part of a resonance or equilibrium and this is no longer an independent OR situation. We suggest that one uses a conditional probability when computing a Newtonian nonrelativistic conservation of energy equation to try to remove as much of the quantum features from this equation. As a result, one has: KEclassical(x) + V(x) = E = Sum over p pp/2m P(p/x) + V(x). One cannot bypass the fact that a quantum particle interacts differently from a classical one and that this involves a probabilistic granularity (wavelength), but classical notions of energy and energy conservation still exist. As a result, the bound state wavefunction Wn(x) = Sum over p an(p) exp(ipx) is not a usual OR sum, but is rather 1 over a normalization factor for an equilibrium picture. As a result, one cannot use Wn1(x)Wn2(x) = W(n1+n2) (x) as one would for independent probabilities. Furthermore, given that one has Wn(x) and Wn1(x)+Wn2(x) which are both of the form Sum over p a(p)exp(ipx), and both represent average energies, one needs a way to distinguish a resonance/equilibrium from a sum of resonance/equilibrium normalization factors. We suggest that the orthogonality of resonance Wn(x)’s provides this feature. The standard textbook explanation is based on Hermitian operators, but in the nonrelativistic case, we argue that one may obtain the result based on integration by parts based on the form of the p operator = -grad. Thus, it is ultimately the form exp(ipx) which leads to orthogonal bound state Wn(x)’s, together with the boundary condition that Wn(x)-->0 of x-> +infinite and -infinite.
We try to consider free particle and single bound state quantum mechanics from the viewpoint of classical physics. We first assume that a free particle moves as x=vt. Secondly, we assume that dx and dt cannot be 0 as in Newtonian mechanics, but must be finite. This begs the question: How does one measure dx and dt? If one can measure a particle at a given x and t, then dx→0, dt→0. Thus, finite dx, dt means that a given p=constant and pp/2m=kinetic energy (nonrelativistic) interact with a force over a finite dx. In other words, one does not know where the interaction occurs within dx. As long as the free particle moves without interacting, then x=vt and there is no interest in any dx or interactions. For example, one may measure very large x and t ranges. Even if there are dx and dt effects at the start and end points, these are small compared to the large x and t values. As a result, one has a F(x,p) probability within dx, which repeats itself for the next dx (i.e. is periodic), where F is a probability for a p impulse hit at x within dx. At the same time, the particle moves with x=vt it seems within dx. In general, one might then think that F(x,p) and x=vt are decoupled. In previous notes, we have argued that one way to couple F(x,p) and x=vt is through special relativity, i.e. A = -Et+px. X and t values used would presumably satisfy x=vt, but for any x,t one has x+hbar/p and t+hbar/E as solutions yielding the same A value and so might consider dx= hbar/p and dt=hbar/E. The question we ask here is whether one may arrive at this conclusion without using special relativity. The answer seems to be that one can if one suggests that F(x,p) must contain information making it consistent with x=vt, i.e. with every x and t point having the same weight. Clearly, however, each x,t do not have the same weight within dx and dt, but are rather governed by F(x,p). This means, however, that F(x,-p) also represents a probability distribution and so if one has F(x,p)F(x,-p) there is no momentum at a given x point. In such a case, the distribution within dx must disappear and one must have a uniform distribution. This holds even if F(x,p) has the same dx value for any p. Thus, F(x,p)F(x,-p) = constant and must be periodic in x. This implies that F(x,p) = 1/F(x,-p) for constant=1. Given that F(x,p) =0 at the initial and final points of dx, then F(x,-p) = infinite at these points and that is not allowable. Thus, one is forced to use complex variables and try exp(i g(p) x). If g(p) is a function of p (i.e. not a constant), then conservation of momentum implies that exp(i g(p1) x) exp(i g(p2) x) = exp(i g(p1+p2) x), suggesting exp(ipx). What, however, if g(p) = constant? In such a case, one has issues creating an average kinetic energy: ( Sum over p a(p) exp(i constant x) pp/2m) / (Sum over p a(p) exp(i constant x) ((1)) This leads to a kinetic energy which does not depend on x and cannot be used with V(x) in a conservation of energy equation. Thus, we choose exp(ipx). Note: there is no reason to choose a(p,x) because the physics of finite dx and dt is restricted to exp(i constant x). Finally, we try to explain why the same dx=hbar/p and dt=hbar/E together with x=vt follow from special relativity.
In classical physics, a particle traveling to the right is the mirror image of one traveling to the left. In terms of a spatial density (the time spent in each constant dx), the value is the same, as is the energy. A similar notion of such a spatial invariance is in the classical conservation of energy equation for a bound particle: p(x)p(x)/2m + V(x) = E. Quantum mechanics seems to be different because exp(ipx) and exp(-ipx) are distinctly different spatial probabilities, i.e. they are dynamic. It is not just that case that a particle moves to the right for exp(ipx) and to the left for exp(-ipx). The impulse hit probabilities (represented by exp(ipx) and exp(-ipx)) are distinctly different. If one wishes to create an OR situation and write: W(x) = Sum over p a(p)exp(ipx) and use this in a conservation of energy equation, then simply including only positive or negative p values in W(x) leads to a complex average kinetic energy -1/2m d/dx dW/dx / W which cannot be used with V(x) and E. Furthermore, W_ (only negative p) and W+ (only positive p) values do not yield the same average kinetic energy. Fortunately, given that this is a statistical average for a localized object (somewhat analogous to a gas), one may combine W+ + W- and obtain a real kinetic energy. We have argued in previous notes that there exists a quantum-classical duality. For example, for a free particle one has two independent notions. One is x=vt which describes the center-of-mass of the particle. We take this to represent a state description which treats v and -v as mirror images. exp(ipx), on the other hand, represents probabilistic impulse hits and does not define the state by itself, we argue, only its interactions. Interactions are physical and define what happens in an interacting system. One may, however, measure a system, i.e interact with exp(ipx), and this should define a measurement state ( a second state ensemble average). For example, exp(ipx)exp(-ipx) = 1 which is equivalent to the constant dt in each dx associated with x=vt. Thus, the two states coincide, but exp(ipx)exp(-ipx) means that one may place the 2-slit apparatus at any x point and is not about the center-of-mass, as x=vt is. In some cases, the two states are quite different. W*(x)W(x) represents an interaction state, but vrms(x) from .5mvrms(x)vrms(x) = -1/2m d/dx dW/dx / W represents a center-of-mass state description. (Note: in these cases a state means an average state, i.e. an average over many ensemble entries.) If one takes W*(x)W(x) and integrates over x, then interference terms vanish and one obtains a classical counting of distinct p value states with their weight a*(p)a(p).This, we argue, leads to the notion of orthogonality of probabilities. In classical probability, there is no such notion, but there is the notion of AND (multiply) probabilities. Given exp(ipx) and exp(-ipx), if one multiplies one removes p, i.e. the interactions and only sees the state left. W*(x)W(x) yields a set of peaks which represent a pattern of probabilistic hits in space which define a state which one would measure in space. The interference pattern, however, must be one such that if one integrates over x, one only obtains a classical sum of probabilities a*(p)a(p)’s as mentioned above. This leads to the notion of orthogonal exp(ipx)’s (which follows mathematically anyway). If one considers different bound En states, then these two are orthogonal because W*n1(x)Wn2(x) mean n1 and n2 are associated with different energies and integrating over x cannot show a result which looks like a state result, where W*n(x)Wn(x) shows a state result. In order to achieve this orthogonality (linked directly to the notion of a state) one must have that Wn(2) usually has 0 points and is positive and negative with the possible exception of a ground state. In other words, if Wn1*(x)Wn2(x) integrated over x is not 0, one violates conservation of energy. As a result, we argue an average probability for a bound state must statistically include +p and -p contributions and must create a function which has positive and negative peaks (except possibly for the ground state), i.e. W(x) must have 0 points. This W(x) is linked to a measurement state W*(x)W(x), but this is strictly considering exp(ipx)s. We argue that there are two pieces of information for a free particle, x=vt and exp(ipx). There must also be an average state linked to x=vt and this given by KEave(x) = .5m vrms(x)vrms(x) + V(x) = E which shows that the particle may exist and have average kinetic energy even at W(x)=0 points. When measuring a system, one interacts with the exp(ipx)s and so the W*(x)W(x) state is the one which is traditionally presented, but whether one measures the center-of-mass motion or not, one also has the notion of the x=vt, i.e. vrms(x) state (average) as well, we argue.
Given a single particle bound wavefunction, W(x)=Sum over p a(p)exp(ipx), it is traditionally stated that W(x)W(x) is the probability to find a particle at x. We have argued that W(x)W(x) is actually a resolution profile or a probability for interaction and does not indicate whether a particle is at x or not. (We discuss this briefly in the note.) It is linked to an observer trying to find the particle, on average at x, not if the particle is present there and interacting with V(x). If one normalizes Integral dx W(x)W(x)=1, then it follows that Sum over p a(p)a(p) = 1. Thus, a(p)a(p) has the appearance of a probability as well and one might argue that it is the probability to have a momentum p. The catch is that with W(x)W(x) one considers spatial resolution/interaction without knowing p and with a(p)a(p), one knows nothing about x. Given exp(ipx), the probability for a free particle impulse hit at x, one knows both p and x and may use this probability to calculate KEave(x) = {Sum over p a(p)pp/2m exp(ipx)} / W(x). We note that this probability does not have 0 points when W(x)=0. It shows the particle does not avoid W(x)=0 points in internal interactions with V(x), we argue. a(p)exp(ipx) / W(x) is a third probability. Given that it holds at a fixed x and ranges over p (it is sometimes called P(p/x)) it should be linked to a(p)a(p), but it considers spatial issues. We suggest that it should be combined with W(x)W(x) as this demonstrates resolution or measuring interaction hits and is linked to an external observer. The product probability, however, imposes the 0 points of W(x). This simply means, we argue, that a person making measurements does not interact with the particle at these W(x)=0 points, not that the particle does not exist there. Using this probability, as we have shown before, yields an average of the Shannon’s entropies created using W(x)W(x) and a(p)a(p) as probabilities. Here we try to understand the relationship between these three probabilities more closely without using the usual P(x)=W(x)W(x), P(p) = a(p)a(p) and P(p/x) = a(p)exp(ipx)/W.