Two particle product type wavefunctions may be created so as to be symmetric (bosons) or antisymmetric (fermions) on then interchange of the identical particles. As a result, spatial density at a point does not change upon interchange. This approach, however, appears to be mathematical in nature. In Part I of this note, we argued that exp(ipx) free particle momentum wavefunctions within a bound wavefunction W(x) interfere and that full wavefunctions Wn(x) in a linear combination also do the same. As a result, probability interference is ultimately linked to the physical mechanism associated with the periodic exp(ipx) wave behind the Pauli rule that fermions may not occupy the same quantum state. In this note, we try to extend this idea and argue that one does not simply exchange identical particles in the wavefunction as a mathematical procedure, but that actual exchange may occur in the probabilistic quantum mechanical case. We note that a time-independent treatment of scattering from a target already includes the idea of “negative” or out of phase interference within a wavefunction W(x)= exp(ikz) + f(theta, phi) exp(ikr)/r so as to remove some incoming flux as outgoing scattered flux now exists. We suggest that the “minus” sign used in the antisymmetric fermion wavefunction behaves in a similar manner. In (1), we argued that W(x)W(x) in a bound state includes all possible a(p1)exp(i p1 x) a(p2)exp(i p2 x) combinations and represented a product probability. In other words, a particle at x with p1 P(p1/x) may be knocked into p2 at x. Similarly, we argue that cross terms in a two particle fermion wavefunction represent flux W*b(x1)Wa(x1)W*a(x2)Wb(x2) flowing from out of states as an interchange occurs. Main terms such as W*a(x1)Wa(x1)W*b(x2)Wb(x2) represent the presence of particles in the a, b states. If a=b, the presence and flow probabilities cancel locally and there is a vanishing wavefunction i.e. the two fermions may not occupy the same state. We argue that one should be able to describe the Pauli exclusion rule in this manner and note that Feynman (2) has used probability arguments in boson scattering to argue the boson enhancement factor. In other words, the mathematical argument of a symmetric wavefunction is not used, but a more physical argument is given.
Two particle product type wavefunctions may be created so as to be symmetric (bosons) or antisymmetric (fermions) on then interchange of the identical particles. As a result, spatial density at a point does not change upon interchange. This approach, however, appears to be mathematical in nature. In Part I of this note, we argued that exp(ipx) free particle momentum wavefunctions within a bound wavefunction W(x) interfere and that full wavefunctions Wn(x) in a linear combination also do the same. As a result, probability interference is ultimately linked to the physical mechanism associated with the periodic exp(ipx) wave behind the Pauli rule that fermions may not occupy the same quantum state. In this note, we try to extend this idea and argue that one does not simply exchange identical particles in the wavefunction as a mathematical procedure, but that actual exchange may occur in the probabilistic quantum mechanical case. We note that a time-independent treatment of scattering from a target already includes the idea of “negative” or out of phase interference within a wavefunction W(x)= exp(ikz) + f(theta, phi) exp(ikr)/r so as to remove some incoming flux as outgoing scattered flux now exists. We suggest that the “minus” sign used in the antisymmetric fermion wavefunction behaves in a similar manner. In (1), we argued that W(x)W(x) in a bound state includes all possible a(p1)exp(i p1 x) a(p2)exp(i p2 x) combinations and represented a product probability. In other words, a particle at x with p1 P(p1/x) may be knocked into p2 at x. Similarly, we argue that cross terms in a two particle fermion wavefunction represent flux Wb(x1)Wa(x1)Wa(x2)Wb(x2) flowing from out of states as an interchange occurs. Main terms such as Wa(x1)Wa(x1)Wb(x2)Wb(x2) represent the presence of particles in the a, b states. If a=b, the presence and flow probabilities cancel locally and there is a vanishing wavefunction i.e. the two fermions may not occupy the same state. We argue that one should be able to describe the Pauli exclusion rule in this manner and note that Feynman (2) has used probability arguments in boson scattering to argue the boson enhancement factor. In other words, the mathematical argument of a symmetric wavefunction is not used, but a more physical argument is given.
Interference is an important observed feature in quantum mechanics. It is often seen in terms of peaks and troughs in two slit interference or in the profile of bound state wavefunction (e.g. harmonic oscillator). In this note, we argue that interference also plays a central role in determining the two types of quantum particles which exist i.e. fermions and bosons. Whereas a wavefunction is subject to a global constraint (its integral over x is 1), a local spatial probability constraint is imposed (at each x) in the case of at least two particles built from a separable wavefunction Wa(x1)Wb(x2). Here there are two indistinguishable particles, two allowable energy levels a and b and one wishes to know the probability of finding a particle at x1 and one at x2 regardless of the energy level. Let us call this P(x1,x2). The modulus squared of the wavefunction yields probability P, so a constructed wavefunction may have the property that the interchange of x1 and x2 yields a phase factor. This argument is presented often although (1) suggests it should be modified, yet the conclusion that there are only fermions and bosons remains. In this note, we consider the existence of two types of interference as the physical basis of this result. From the Schrodinger equation or from P(p/x)= a(p)exp(ipx) / W one may see that W→-W and a(p)-->-a(p) is still a solution. In quantum mechanics not only do exp(ipx) terms interfere, but eigenfunctions Wn may also interfere. Given that a bound eigenfunction may be represented by Wn or -Wn, there exists two kinds of interference. Even in the case in which there is only one particle, one may already create two possibilities: Wa(x)+Wb(x) and Wa(x)-Wb(x). If one sets a=b, the second becomes 0 at each x. This we argue already points to the idea of fermions even though only one particle is considered. This result is due strictly to alternate phase (1, -1) interference). The idea of alternate phase interference Wa-Wb also seems to be linked to a difference in wavefunction i.e. a “kind” of wavefunction representing the change in moving from energy level a to b. Thus we try to argue that there is interference beyond the usual exp(ipx) namely an alternate phase interference which applies to eigenfunctions Wn(x) and leads to the existence of fermions. Bosons occur with no alternate phase interference. These two types of particles emerge from the constraint that P(x1,x2)=P(x2,x1) i.e a local spatial probability constraint.
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.
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.
Interference between exp(ipx) momentum eigenstates in W(x)=wavefunction= Sum over p a(p)exp(ipx) or bound eigenstates in W(x,t)=Sum an(En)exp(-iEnt) Wn(x) are well known in quantum mechanics. The resulting interference (seen in W*(x,t)W(x,t)) seems to be the effect of an OR probability condition i.e. adding complex probabilities with both the real and imaginary terms being periodic and containing both positive and negative values. (Here the wavefunction is treated as a kind of probability (i.e. a conditional probability.) In this note, we wish to argue that interference does not always occur. In particular, a free particle has a wavefunction of exp(-iE(p)t) exp(ipx where E=pp/2m, but only exp(ipx) terms interfere in W(x)=Sum a(p)exp(ipx). exp(-iE(p)t) terms do not interfere, rather a new time cycling resonance is created described by an overall exp(-iEt) where E represents overall energy (kinetic and potential). In other words it seems that a new kind of energy is created namely potential energy which must be considered together with kinetic energy. A particle in a bound state, treated as a free particle for statistical purposes, maintains a kind of identity and energy flow independent of the potential. Its energy identity, however, described by exp(-iE(p)t) where E(p)=pp/2m is transformed as it now interacts with exp(ikx) terms of V(x) to contribute to an overall new cycling resonance of exp(-iEt). Thus, exp(iE(p)t) terms do not interfere. For a linear combination of bound states (e.g. in a kind of heat reservoir) W(x,t)= Sum over n an(En)exp(-iEnt) Wn(x) and a new cycling resonance is not created, rather the reservoir kicks one n state to another and exp(-iEnt) terms interfere. Interference gives rise to the humps or troughs found in the original exp(ipx) or exp(-iEnt), but no longer yields a modulus of 1 at each x or t point. Thus interference shows the nonlocality in the problem. exp(ipx) on the other hand contains both locality and nonlocality. There is a wavelength proportional to 1/p and cos(px) and sin(px) are both nonlocal, but the modulus of exp(ipx) is 1 which is a local property. Quantum mechanics allows both for interference which yields nonlocality and noninterference, e.g. exp(-iEnt) which contains both locality and nonlocality. We investigate this idea in this note.
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).
In Part III we considered the quantum free particle wavefunction exp(ipx) as consisting of two parts cos(px) and sin(px) treated as two different dimensions. We also argued that each, is a probability distribution in space and so associated with an internal force. One may notice that exp(ipx) does not contain time, but still predicts the direction of motion. Given that one cannot consider x(t) within a wavelength hbar/p even though on average the particle moves as x=p/m t, there is no real sense of time within a wavelength. Extending this idea, one repeats cos(px) humps. Given that the particle is moving, one must somehow juxtapose two spatial probability densities. The derivative of the distribution is proportional to average momentum and so one may see the notion of action-reaction occurring between cos(px) and sin(px) and within cos(px) and sin(px) accounting for the sign changes with x. We argue that d/dx cos(px) regions which change sign imply that the impulse direction changes. Furthermore, cos(px) has regions near the endpoints of the wavelength which have very low values (close to zero) which is fine if one has two juxtapositioned distributions. If, however, one creates a single particle bound state, exp(ipx) and exp(-ipx) for a(p)=a(-p) (a specific parity) give the appearance of a cancelation leaving the impression of 2cos(px) i.e a single “static” distribution. This occurs because there is no internal time in the picture and so p and -p probabilities add. The so-called canceled pieces have not really disappeared. as will be argued, they simply do not contribute to an average impulse which is what experiments apparently detect. Thus “negative” quantum probability is linked to impulse in the opposite direction, we argue. Thus interference is really linked to the notion of probabilistic impulses i.e. p exp(ipx) even though exp(ipx) is used as a kind of probability. exp(ipx) contains internal action-reaction impulse hits which have experimental consequences as we argue is shown in a reflection/refraction problem, but are also linked with impulse hits associated with p. Thus there are two sets of forces to deal with. This, we argue, explains why one writes the energy conservation equation AA/c1=BB/c1+CC/c2 for reflection/refraction where A,B,C represent the incident, reflected and refracted rays and AA simply shows that intensity/flux is positive as two equations: A+B=C and Ap - pB = p2C with E=pc. The two equations,, needed to solve for B,C in terms of A,, describe the internal force in exp(ipx) and the probabilistic impulse p exp(ipx). For a single particle bound state, one considers a linear combination of free particle states indicating that impulse hits represent the underlying physics. Given the apparent spatial invariance of a free particle exp(ipx), one may wonder how a kinetic energy profile appears. We argue that different p values yield different peak points for maximum impulse hits and these are out of synch leading to a spatial distribution of KE(x) or d/dx KE(x) which is equivalent to -dV/dx =Force where V(x) is the potential.