We have argued in previous notes that it is special relativity which gives rise to the free particle probability exp(-iEt+ipx). This probability is directly associated with physical behaviour because the somewhat random variables are x and t and while p and E are exact. One has probability in t and x for the interaction variables E,p. This suggests a kind of resonance feature, we argue.This idea of resonance seems to carry over into a single bound state and helps to explain why such a quantum system is associated with more than simply a statistical calculation. Using exp(-ipx) as a probability for a time-independent problem (i.e considering impulse hits in x only), one may mathematically express the average kinetic energy as: {Sum over p a(p)pp/2m exp(ipx) }/W(x), where W(x)=Sum over p a(p)exp(ipx). One may then use: KEave(x) + V(x) = En. As an average, one has different p values creating KEave(x), with only the average KE plus V(x) equalling En. The notion of conservation of energy, however, must hold and this begs the question: How is the energy for each p obtained such that energy is conserved. We suggest that the only source is V(x) and so if one moves from a p1 to a p2, at some x, then if p2>p1, the extra kinetic energy must be borrowed from the potential energy. This is not simply an average calculation, energy conservation must be imposed. This suggests that a physical resonance is established, which allows for high p values using borrowed V(x) energy. As a result, a quantum particle is not stopped at a classical turning point, but it receives its high p and passes this point using borrowed energy. The energy belongs to the potential V(x) and the particle is not free to move off with the borrowed energy. We suggest that tunneling occurs with borrowed potential energy and that the simplest condition W(x)->infinite shows that a particle which escapes the classical turning point cannot go too far with its borrowed energy. The catch is that if another potential is placed near the classical turning point of the first, it can lend energy to the particle so that the borrowed energy from the first potential is returned and the particle is pulled into the second system, as seen in experiments. Thus, borrowed potential which is part of a resonance mechanism allows a quantum particle to bypass the classical turning point restriction and one can swap borrowed energy from one system to another to actually have a particle tunnel out of one bound system into another system.
We have argued in previous notes that it is special relativity which gives rise to the free particle probability exp(-iEt+ipx). This probability is directly associated with physical behaviour because the somewhat random variables are x and t and while p and E are exact. One has probability in t and x for the interaction variables E,p. This suggests a kind of resonance feature, we argue.This idea of resonance seems to carry over into a single bound state and helps to explain why such a quantum system is associated with more than simply a statistical calculation. Using exp(-ipx) as a probability for a time-independent problem (i.e considering impulse hits in x only), one may mathematically express the average kinetic energy as: {Sum over p a(p)pp/2m exp(ipx) }/W(x), where W(x)=Sum over p a(p)exp(ipx). One may then use: KEave(x) + V(x) = En. As an average, one has different p values creating KEave(x), with only the average KE plus V(x) equalling En. The notion of conservation of energy, however, must hold and this begs the question: How is the energy for each p obtained such that energy is conserved. We suggest that the only source is V(x) and so if one moves from a p1 to a p2, at some x, then if p2>p1, the extra kinetic energy must be borrowed from the potential energy. This is not simply an average calculation, energy conservation must be imposed. This suggests that a physical resonance is established, which allows for high p values using borrowed V(x) energy. As a result, a quantum particle is not stopped at a classical turning point, but it receives its high p and passes this point using borrowed energy. The energy belongs to the potential V(x) and the particle is not free to move off with the borrowed energy. We suggest that tunneling occurs with borrowed potential energy and that the simplest condition W(x)->infinite shows that a particle which escapes the classical turning point cannot go too far with its borrowed energy. The catch is that if another potential is placed near the classical turning point of the first, it can lend energy to the particle so that the borrowed energy from the first potential is returned and the particle is pulled into the second system, as seen in experiments. Thus, borrowed potential which is part of a resonance mechanism allows a quantum particle to bypass the classical turning point restriction and one can swap borrowed energy from one system to another to actually have a particle tunnel out of one bound system into another system.
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).
We have suggested in previous notes that there is a tendency to consider quantum mechanics as a kind of hybrid probabilistic-deterministic theory because it is supposed to ultimately “merge’ with classical mechanics for high energy levels (tiny wavelengths). Newtonian mechanics is deterministic and so somehow it seems quantum mechanics should also be seen as deterministic, hence the sense of surprise at tunneling when KEave(x)=0, even though vrms(x)=0 does not mean that actual values creating the probability are 0. Here, we wish to argue that exp(-iEt+ipx) (or the exp(ipx) part for spatial interactions) is really a momentum probability scheme which may possibly bypass many aspects of the actual physics to describe a problem in a simpler form in terms of a momentum conserving probability. A key idea is that exp(ipx) is actually the form to use when one has randomness as we will argue for reflection-refraction from an n1-n2 index of refraction junction. We argue that this idea is already known for a particle in an infinite well for which one has sin(pave x), i.e. 1/2i (exp(i pave x) - exp(- i pave x)). In other words, one has the form exp(i pave x), even though it is linked with an average p. Given that one is not considering deterministic physics, there is nothing wrong with exp(i pave x) being composed of Sum over p a(p) exp(ipx), where exp(ipx) holds in all space, not just within the infinite well potential. exp(ipx) is a momentum conserving probability, but on average momentum must also be conserved by a free statistical entity within the well. Similarly, W(x) for a single particle bound state consists of many exp(ipx)s, but represents a fixed E ave and on average this Eave must be conserved in time, hence the time dependent exp(-i Eave t). As a result, one may see that the quantum mechanical exp(-iEt+ipx) is an attempt to describe a system in a simple manner in terms of momentum conserving probability. To see this in more detail, we consider (1) which describes reflection and refraction of a photon at an n1-n2 junction in depth. In a previous note, we have already pointed out that in n2, c/n2 simply means that the photon is interacting on average with atoms/molecules in n2, but has c in between such interactions. It is only on average that one has c/n2 and even E and p2 = p n2. Nevertheless, one may use exp(ip2 x) as a probability which describes average properties of the system, just as it does in the examples in the above paragraph. The question then becomes: Do the microscopic details of the photon physics (the actual interactions) dictate the probabilities to reflect/refract or does the simplified exp(ipx) set linked to momentum conservation do so? Based on (1), it seems that reflection is completely independent physically from “refraction”, the latter requiring re-radiation and affecting all glass molecules. As a result, it appears that it is difficult to suggest that it is the microscopic interactions which dictate probability because reflection and refraction are microscopically so different. On the other hand, reflection-refraction is not a .5-.5 probability random process as experimental results show. The question then becomes: If some kind of systematic probability driver is not supplied by the microscopic interaction, what provides it? We argued in previous notes that one may introduce the probability exp(ipx) into Newtonian scattering to account for an initial p1,p2 (momentum vector set) having equal probability to produce any pi, pj (outcome set) as long as momentum is conserved. We suggest here that it is exp(ipx) that one must use for random interaction if one insists on conserving momentum. Thus, exp(ipx) bypasses microscopic details of interaction and provides an “ any outcome pair of momenta” is equally likely if momentum is conserved. This is a major simplification of the problem, but the question is: What constraints does it impose on physical results? For a free particle, it suggests that it may interact probabilistically with both slits of 2-slit apparatus if the slits are about hbar/p apart. For randomness, one would expect equal probability for each slit and this is the case as long as one also considers exp(ipx) for each slit to a the same point on a screen. In the case of reflection-refraction at an n1-n2 index of refraction junction, we try to see how the simplification exp(ipx) can be consistent with the microscopic details of the interactions and at the same time dictate the probabilities to reflect and refract. We try to consider why an exp(ipx) scheme predicts how nature behaves while a random one with no momentum conservation considerations does not. In particular, we argue that exp(ipx) represent randomness in a two body scattering problem in which momentum is conserved and so this same exp(ipx) must be carried over into problems which involve interactions instead of simply suggesting that one has .5 probability to reflect and .5 to refract because the two processes are independent and hence random. They are random, but with conserved momentum and hence an exp(ipx) scheme is required, we argue.
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 exp(-iEt+ipx) is a modulus 1 (representing a uniform distribution under reduced information) complex probability. It represents the spatial probability for hits in space by the inherent force. Given that this complex probability may interfere, it has an effect on average kinetic energy: -1/2m d/dx d/dx W/W where W= Sum over p a(p)exp(ipx). One has various free particle interaction probabilities as the particle is interacting with a potential. Both the numerator and denominator of the average kinetic energy are related to the interaction spatial density, but by dividing the two one obtains the classical kinetic energy. Thus, the classical picture is present alongside the quantum one. The interference of exp(ipx)s in a time independent problem can give rise to interaction points with zero or very low probability. This, however, does not mean that the velocity is 0 at these points. For example, a particle in an infinite well has KEave(x)= -1/2m d/dx d/dx sin(pave x) / sin(pave x) = pave pave/2m. This result matches x=vrms t which is the motion of the particle because it is not interacting with a potential. Nevertheless, the interaction spatial density is W*W = C sin(pave x) sin(pave x).We have argued that exp(-iEt+ipx) is a probability which conserves energy and momentum in a two body elastic collision. Energy conservation should hold for a 1-dimensional single particle bound state as one creates an energy resonance when creating a bound state. This is important because we argue that transitions from one energy level to another should be linked with energy considerations, in particular < W1 V W2>. We first note that the Schrodinger equation, which is a conservation of nonrelativistic energy, establishes an energy resonance. V(x) is the classical energy which may be considered to be a number at x, one has E on the RHS of the equation and average kinetic energy: {-1/2m Sum over p a(p)pp/2m exp(ipx)} / {Sum over p a(p)exp(ipx)) on the LHS. Thus, it requires two pieces of information {-1/2m Sum over p a(p)pp/2m exp(ipx)} and V(x)W(x) to determine if one has a resonance energy state. In principle, if one only had V(x) W(x), this is a math function which may be written as Sum over i an Wn(x), where n is the energy level. We argue that given V(x)W(x), one should first calculate -1/2m d/dx dW/dx / W to see if one has an energy resonance, because this is a steady solution (if no V perturbation exists). Otherwise, V(x)W(x) may be seen as a sum of states an Wn(x) and should not depend on the energy of W(x). Given that V(x)W(x) represents an interaction between W(x) and V(x), Sum over n an Wn(x) represents the weighted bound state amplitudes and < W1 V(x) W2 > * <W2 V(x) W1> should be proportional to the transition probability and not depend on any energy of W1 or W2. One may create units of probability/per sec by dividing by 1/hbar and multiplying by density(E) which has units of 1/E. Thus, we argue that it is the single particle bound state resonance based on conservation of energy which gives rise to Fermi’s Golden rule. We note that (1) derives Fermi’s Golden Rule in a somewhat different manner. We suggest that -Et+px, a Lorentz invariant, is akin to a potential for a particle with rest mass and so exp(-iEt+ipx) is linked with a potential. This also holds for an electrostatic potential, while a photon has cos(-Et+px) apply directly to the inherent force and not a potential. A single particle Schrodinger equation, however, deals with a particle with rest mass mo and so it is the potential energy which is the key idea for quantum interactions. We suggest that quantum mechanics is inherently a potential based theory (with the exception of the photon) and this is why Fermi’s Golden Rule is based on <W1 V W2>. In other words, in quantum mechanics, force follows from a potential and not the other way around as in Newtonian mechanics. In quantum mechanics, the rest mass of a particle represents an inherent force which is linked with a probabilistic hit pattern in space given by exp(-iEt+ipx). Thus, the inherent force gives rise to a probabilistic potential which in turn gives rise to a term proportional to force -grad (-iEt+ipx) = p. This seems to explain why quantum mechanics equations deal with potential instead of force. The probabilistic wavefunction is linked with a potential related to the inherent force for particles with rest mass.
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.