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.
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.
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.
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.
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.
Quantum mechanics is based on two different probability schemes, namely W(x) =wavefunction and W*(x)W*(x)=P(x), with one being linked to the other. Why should there be two probability schemes? In (0) we argued that there are two spatial behaviour schemes because an impulse p (momentum) is automatically associated with two spatial distributions cos(px) and sin(px) in exp(ipx), yet a particle with constant p moves as x=p/m t. Thus if one is only concerned with motion, each x point has the same weight, and this is physically measurable as a photon or electron travels a certain distance in a certain time in free space. On the other hand, an impulse interaction is associated with exp(ipx) is a completely different spatial picture, and this too is measurable as in a 2-slit experiment. Thus it seems the two schemes are measured differently. We argue that this is an important issue because if there are two x-probability schemes there should be different entropies associated with each. Ttraditionally (1), one sees two quantum entropies: - W*(x)W(x) ln {W*(x)W(x)} and - a*(p)a(p) ln{ a*(p)a(p) } where W(x) = Sum over p a(p)exp(ipx). These are associated with the x=p/m t scheme or x-dependence based on motion or average motion. W*(x)W(x) represents spatial density which should be measured presumably by some sort of scattering, but we ask whether some information is lost in this approach? In particular, we note that a wavefunction may be obtained for a bound state. In the high energy limit, the envelope over the density W*(x)W(x) is said (2) to match C/v(x) from classical physics where .5mv(x)v(x) + V(x) = E. C/v(x), however, is proportional to delta t (with delta x being constant) and has nothing to do with the strength of an impulse hit. In the classical case, there is a single v(x) and so a single p(x) in each delta x thus no impulse entropy within a delta x i.e. at a point x. In quantum mechanics, however, we argue that a constant p automatically has an x probability distribution exp(ipx) associated with it and there might be an entropy linked to this.This entropy is associated with a probability to receive a p impulse at x and is so not linked to interference while the W*(x)W(x) is. In (3) we introduced the impulse or wavelength entropy (for a free particle |cos(px)|/C ln{| cos(px)|/C},, but here we wish to investigate its properties more closely.
For quantum bound states, classical style probabilities P(p)=a(p)a(p) and P(x)=W(x)W(x) [where W(x)= the wavefunction = Sum over p a(p) exp(ipx)] exist. These are real and normalized to one, but unusually, they separate p and x completely. For example, P(p) is the probability to find a momentum p anywhere in the system - no information about x is known. This seems to suggest that the P(p/x) is not defined in a classical sense. (Similar arguments hold for P(x).) We have argued in a previous note (1), that due to a lack of time reversal balance, there is no clear classical notion of P(p/x) in quantum mechanics while in classical statistical mechanics, there is a time reversal balance of two-body collisions and so P(p/x) is well-defined. In particular, a momentum p collides with a pj to form another momentum pk. There is ambiguity if p is changing, but if at the same time a pk is changing (colliding) back into a p at x, then the ambiguity is removed. In quantum mechanics, the ambiguity is not removed at x, but one may discern one p state from another throughout the entire x space. This suggests orthonormal probabilities e.g. say exp(ip1 x) and exp(i p2 x). These exp(i p x) probabilities comprise the wavefunction which is linked to P(p/x) = a(p) exp(ipx)/W(x), but although Sum over p P(p/x)=1, P(p/x) is complex with the real and complex portions taking on positive and negative values. Nevertheless, it is W(x) which is the solution of the Schrodinger (and other) equations. It is known that solutions Wn(x) of the Schrodinger equation have discrete En (energy) values. It is possible to have [H, Operator]=0 and possibly to have H written in terms of this operator. In such a case, Wn(x) might be both an eigenfunction of H and the Operator. Alternatively, a linear combination of Operator eigenfunctions may be used. If En is discrete, one may expect the operator eigenfunctions to also be discrete, leading to quantization of the Operator eigenvalues. In some cases, one may have two operators which commute with H, e.g. LL and Lz (related to angular momentum). Lz eigenfunctions might not be linked to H so one might ask why Lz should be quantized. Another H, however, may distinguish between Lz eigenfunctions and so they also have discrete eigenvalues. In this note, we try to consider why one should have orthonormal eigenfunctions of these commuting operators which make up the wavefunction and try to interpret their physical meaning.
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.