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Free Particle Quantum Mechanics, Delta Functions and Energy-Momentum Conservation Francesco R. Ruggeri Hanwell, N.B. Nov. 28, 2025 In previous notes, we introduced a complex probability into Newtonian two-body elastic scattering to ensure that for a given (e1,e2) and (p1, p2) (momentum vectors), any (ei, ej) (pi, pj) have the same probability to occur. This ultimately led to the Lorentz invariant probability exp(-iEt+i p dot r). Here we begin with the delta functions delta( pinitial - pfinal) and delta( Einital - Final) and see that one may write these as Integral da C exp(-i pfinal + i p initial) and a similar expression for E. Here a is simply a math variable. (Note: This is only one math form for a delta function, but this particular form will prove useful from physical considerations later.) The delta function may be used in an expression Integral delta(pinitial - pfinal) dpfinal and only pfinal values which satisfy conservation of momentum yield a nonzero integral. Furthermore, this integral may be multiplied by various functions and still maintain conservation of momentum. At this point, the above is simply a calculation formalism. The question we ask is whether one may make the argument that a particle itself is responsible for conservation of momentum in an interaction. We suggest this may be the case based on Newton’s notion of action-reaction. It is not simply that one applies a force to a particle and it changes. The particle itself resists and this is the idea of inertia as well as action-reaction. The particle itself is strongly involved in a reaction and we suggest that a particle carries a property of ensuring conservation of momentum and energy itself. It is not something that is imposed by the interaction, otherwise there would be no need for the notion of action-reaction. The particle reacts and the manner in which it reacts leads to conservation of momentum and energy. We suggest that there seem to be grounds for considering a particle to actually be the driver of energy and momentum conservation and that this actually extends to further features, namely probabilistic results in interactions which do not appear in Newtonian mechanics. This property might then be taken as exp(i p a) and exp(i E b), where a and b are math variables used to establish the delta function. One, however, cannot really establish a property of a particle if math running variables a and b are present. The entire expression should consist of particle state variables. This seems to suggest the consideration of Lorentz invariance because E and p change when the particle is seen from a frame moving with -v. One may create a state based exp(-iEt+ipx) energy-momentum conserving property of a particle which is Lorentz invariant. We next note that p and E are linked to interactions, but ones that differ. In particular, p is linked with an impulse which has nothing to do with velocity , i.e. E. On the other hand, E = .5movv in the nonrelativistic limit and is dependent on v, i.e. a m1v1= m2v2 have the same impulse, but different kinetic energies. Thus, we suggest that exp(-iEt) and exp(ipx) be used separately depending on the type of interaction one considers. In a quantum single particle bound state, one is interested in impulse hits with the potential we argue and so does not need to consider exp(-iEt) as one cannot measure the particle moving through the potential in a classical manner. We argue that a constraint (momentum and energy conservation) may actually be due to properties of the particle and that this ultimately follows from Newton’s action-reaction. In (1), we
argued that exp(-iEt+ipx) follows from Lorentz invariance which is based on the Newtonian idea of work increasing energy, i.e. 1/sqrt(1-vv/cc) and A= -Et+px have minus signs for this very Newtonian reason. We argued that dx = hbar/p and dt=hbar/E follow as fluctuations which leave A unchanged due to the presence of the minus sign.Here we again argue that a Newtonian idea is responsible for exp(-iEt+ipx), namely action-reaction. We show that the two approaches are identical. Conservation of Momentum and Energy and the Delta Function In Newtonian mechanics, conservation of energy and momentum are presumed to be laws which must be imposed on an interaction. In other words, one may consider mathematically a delta function: delta( pinitial - pfinal) in Integral dpfinal delta(pinitial - pfinal) function(p’s) ((1)) A similar expression holds for E conservation. A possible way to write a delta function is: C Integral da exp(i p a) ((2)) and a similar expression for E. Here a is a math parameter and the above is nothing but a math formalism. Furthermore, ((2)) is not the only way in which to express a delta function and so there is bias in presenting it instead of some other expression. The form ((2)) will prove useful later because of its link to Lorentz invariance. Action-Reaction and Conservation of Momentum/Energy In the above section, we described momentum and energy conservation as if it is a law which must be imposed in a calculation. This begs the question: Why does it not appear naturally through consideration of forces? After all, a reaction involves forces and they should act in such a way as to conserve momentum and energy. One should not need to introduce delta functions. Momentum and energy, however, are properties of the particles interacting and not of the interaction force, it seems. As a result, the particles themselves must somehow impose the presence of their momenta and energies in a reaction and we suggest that this has already been described by Newton through his idea of action-reaction. Inertia and action-reaction allow a particle to impose its energy and momentum properties and so to speak ensure conservation of momentum and energy. In other words, we argue for a property of a particle, based on its E and p, which allows it to conserve energy and momentum when it interacts with other particles. If one were to use the biased expression ((2)) for a delta function, one has: exp(i p a) and exp(i E b) ((3)) Here a and b are math parameters, but one cannot introduce math parameters (which are running variables) into a function which is supposed to be a property of the particle. A and b should be properties of the particle. This suggests a Lorentz invariant form for ((3)) i.e.
exp(-iEt+ipx) ((4)) Thus, a fundamental idea is that a particle carries a property which is linked to dynamical-interactional features. This property seems to act like a complex probability because conservation is obtained by considering products of ((4)) for the initial particles and a product with a minus phase for the final. If this property ((4)) governs interactions in terms of momentum and energy conservation, one may notice that a second feature is present in ((4)) namely that of x and t units: delta t = hbar/E and delta x = bhar/p ((5)) (Here we insert hbar without explanation. It is linked with photon features as discussed in (1).) As a result, there seems to be uncertainty in x and t and if ((4)) describes momentum and energy conserving properties, then it seems that there is an uncertainty in x and t property present as well which should apply to interactions just as much as conservation of momentum and energy. This feature in fact appears in 2-slit interference and 1-D reflection-refraction from an n1-n2 index of refraction junction and is responsible for the ultimate probabilistic nature of these results. Thus, a momentum and energy constraint (in the form of a delta function) leads to an expression which may assign a property to a particle (based on E, p) which ultimately enforces this conservation in a reaction. It is the reaction of the particle in an interaction which ensures that it is the particle which is the driver of conservation of energy and momentum. This leads to a particle associated function of E, p which can reproduce conservation of momentum/energy, i.e. the delta function. If one insists on Lorentz invariance, then one has ((4)). We note that ((4)) represents two different kinds of interactions (as we have pointed out in previous notes). P is linked to an impulse hit and is independent of v in the sense that m1v1=m2v2 yields the same impulse. E (.5movv in the nonrelativistic case) is completely dependent on v and describes motion within a potential. Results of (1) In (1), we argued that x=vt does not fully describe a free particle. Special relativity for mo at x=0, t leads to: x’= g(v) v t and t’ = g(v) t ((6)) We argued that one cannot solve for g(v) correctly unless one considers Newton’s that work leads to E increasing and that ((6)) must also apply to mo, p=0. This we argued leads to: -Et+px = A and g(v) = 1/sqrt(1-vv/cc) ((7)) We then noted that A is unchanged for: t+ hbar/E and x+ hbar/p ((8)).
This suggests a probability in space and time. We argued for a Lorentz form which preserves ((8)) and this yields ((4)) with on a priori consideration of momentum and energy conservation. This conservation, however, automatically follows from ((4)). Thus, both the approach of this note and (1) are ultimately linked to Newtonian work-force ideas and use the notion of Lorentz invariance. The approach here is based directly on conservation of energy and momentum. If one considers -(E1+E2) delta t + (p1+p2 delta x = A ((9)) as being linked to probabilities, then Prob(p1)Prob(p2) should equal Prob(p1+p2) and so ((9)) is consistent with exp(ipx), i.e with conservation of momentum. A similar argument holds for E. Thus, the approach of (1) and this note are essentially the same, we argue. Conclusion In conclusion, we argue that in Newtonian mechanics, one imposes conservation of energy and momentum a priori, mathematically through a delta function. We show that one way to write a delta function is C Integral da exp(ip a) where a is a math parameter. This ,however, is simply one math form which yields the delta function and does not necessarily have any physical relevance as many other forms exist. The point we make is that Newtonian action-reaction and inertia seem to suggest that it is the particle itself which is the driver of conservation of energy and momentum and not the interaction force. We propose that there should exist a physical function of p and E which is a property of the particle and leads to conservation. One cannot have exp(ip a) exp(i E b), where a and b are running math variables, only physical variables may appear. This, we argue, leads to the notion of a Lorentz invariant exp(-iEt+ipx). This form does not only yield conservation laws, but introduces uncertainty regions hbar/p and hbar/E which are responsible for a probability which shows these regions (i.e. exp(-iEt+ipx) and ultimately shows that it is the particle itself which is not only the driver of momentum and energy conservation, but of the probability found in 2-slit interference and 1-D reflection-refraction at an n1-n2 index of refraction junction. References 1. Ruggeri, Francesco R. x=vt is Not the Full SpaceTime Description of a Free Particle (preprint, zenodo, 2025)
