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Quantum Free Particle exp(-iEt+ip dot r) and State Probability Part 2 Francesco R. Ruggeri Hanwell, N.B. Nov. 17, 2025 In this note, we point out that although P(x)dx=dx/L for an arbitrary length L makes sense for a particle at rest, moving particles are characterized by four variables: x,t, E and p and so there should exist a probability P(x,t,E,p) which is Lorentz invariant, i.e. P(-Et+p dot r). Furthermore, for any given E,p, t values, P(E,t,p, x) should be “equivalent” for all x. This is not possible for a real-valued function (as we have noted before), but is possible for a two vector, because one may shift some “probability value” from one component to the second, hence conserving the overall quantity (through a constant modulus for example). One might suggest that this continues to occur for all x values, but that would imply an infinite initial value for the first value which is not physical. An alternative is to have two periodic functions. Thus, it is the a priori requirement of an P(E,p,x,t), Lorentz invariance and a sense of equivalence for all x, given constant t,p,E values which leads to the conclusion of a periodicity in space, something that is completely absent from P(x)dx=dx/L and so may initially have the sense of being unphysical. The three conditions listed above, however, are completely physical and it seems one must consider their consequence as being physical as well. This consequence is periodic probability in space which may take on positive and negative values in an OR (add situation). In particular, if one considers a single function which is at first losing value along x, but then stops losing value and gains value, it must have a slope of zero at such a change. If the gains and losses match, then one may always place a y-axis in-between to have positive and negative values of probability for each function of the two-vector. In an OR or add case, however, different p values at the same x, may lead to positive and negative values being added at this x, i.e. interference, but this follows directly from the requirement of a two vector which must somehow remain equivalent for all x. It is the shifting of the value of the two vector functions in x, leading to an uncertain region in x which also leads to interference and the shifting of P(x,t,p,E) probability in x in an OR situation, i.e. interference. Thus, one might consider interference unusual, but it is no more unusual than the shifting of probability, i.e. the creation of a wavelength for a free particle. One effect leads to the other, we argue. The Notion of State Probability A key idea of Part 1 was to introduce the notion of a state probability P(E,p,x,t). In classical physics, one sometimes uses: P(x)dx = dx/L where L is an arbitrary length ((1)) ((1)), however, really holds for a particle at rest. In such a case, the only variable characterizing the particle is essentially position. A moving particle, however, is characterized by four variables: E,p,x,t with v=pc/E if the particle has a rest mass ((2))
If the particle does not have a rest mass, then v=c. As a result, as we argued in Part 1, one should a priori consider the existence of a probability: P(x,t,E,p) ((3)) For some reason this is not considered in Newtonian mechanics, although ((1)) is common knowledge. If P(x) exists for a particle at rest, there is no reason, we argue, not to consider the existence of ((3)). One does not even have to consider two body Newtonian elastic scattering and assign equal probability to any (ei,ej) energy (pi,pj) momentum vector outcome set which conserves energy and momentum. ((3)) should exist a priori. Furthermore, given the well-known idea that physics equations have the same form in different frames moving at constant speed, a person writing ((3)) in any rest frame should have the same mathematical form for probability, using the state variables x,t,E, p of his/her frame. Thus: P(-Et+p dot r) or P(-Et+px) in one dimension should hold ((4)) The final requirement is that given fixed t,E and p values, P(-Et+px) should be equivalent for any x value. This cannot occur for a real function, but in principle it may occur for a two dimensional vector: { f(-Et+px), g(-Et+px) } where f and g are unknown ((5)) The basic idea is that f() may decrease in value with x and g() increase in such a way that a certain value (e.g. the modulus) remains constant. One might suggest that f() always decreases and g() always increases, but this implies an infinite initial value for f(). As a result, it seems that one has no choice but so suggest that: f(-Et+px), g(-Et+px) oscillate such that the modulus remains constant ((6)) This way some probability is transferred from f() to g() as x progresses and then transferred back over another range of x. As a result, three conditions: (A) The existence of P(E,p,x,t) (B) the Lorentz invariance of the math form of P(E,p,x,t), i.e. P(-Et+px0 (C) The equivalence in some form of P(-Et+px) for any x given fixed E,p,t lead to a two vector with two real functions which are periodic and have constant modulus. This leads to cos(-Et+px) and sin(-Et+px) as solutions and one may immediately see that each has positive and negative values. Thus, the modulus of the probability is never negative, but each function of the two-vector of probability take on positive and negative values of probability. This
means that the notion of periodicity in both x and t is directly associated with positive and negative probabilities. It seems, however, that one usually focuses on the idea of: wavelength=hbar/p and period = hbar/E ((7)) These are the physical properties which seem to be stressed in the literature, but we stress that they do not exist without the notion of positive and negative probability and the shifting of probability from one x to another etc. In other words, we suggest that the basic probabilistic behaviour of interference is present in the periodicity of cos(-Et+px) and sin(-Et+px) which results from the completely physical conditions (A)-(C). The Notion of Interference In the above section, we postulate a probability P(E,p,t,x) based on all four state variables of a moving particle. Given Lorentz invariance of the formula for P and an equivalence for all x , for fixed t,E,p, we concluded that one must have a two vector with functions which take on both positive and negative values. These are linked with the idea of a shifting of probability between the two functions which first involves the first function sending probability to the second and then later the second sending material back. When a function receives material it increases in value, but when it loses material, it decreases and so one may see how a negative -positive probability arises. Given cos(-Et+px) and sin(-Et+px), one has a shift of probability from one to the other as x changes. If one holds x fixed (as well as E and t) and changes p, then one may have various cos(-Et+px) with different p’s supply positive and negative probability values at a fixed x. This is called interference, but seems to follow from the nature of cos(-Et+px) and sin(-Et+px). Interference is simply a different way to shift probability in such a way that overall it is conserved. Thus, as unusual as interference when compared to the ideas of classical physics, it follows directly from three physical conditions (A)-(C). The Notion of Conservation of Energy and Momentum Given that the postulate P(x,t,E,p) is a probability, AND (multiply) situations must be allowed and these may be interpreted as representing two body elastic scattering, i.e. P(E1,t,p1,x) P(E2,t,p2,x) = P(E3,t,p3,x) P(E4,t,p4,t) one-dimension ((8)) One must have E1+E2 = E3+E4 and p1+p2 = p3+p4 ((9)) and a check using exp(-iEt)exp(ipx) = P(-Et+px) shows this is indeed the case. Thus, the ideas of conservation of momentum and energy, which already appear in Newtonian mechanics, also follow from (A)-(C).
Conclusion In conclusion, we argue that P(x)dx=dx/L (for a length L) is often used in classical physics. Technically, we argue that this result applies to a particle at rest. A moving particle is physically described by x,t,p and E with v=pc/E for a particle with rest mass and v=c, otherwise. We argue that one may a priori expect the existence of a state probability P(x,t,E,p). Moreover, its mathematical form must be Lorentz invariant so one has P(-Et+px). Finally, given a fixed E,p,t one must have an equivalence for all x values. We call these conditions (A),(B),(C). This cannot be achieved with a single real valued function, but it can for a two vector ( f(-Et+px) , g(-Et+px)). One may imagine f() loses probability by transferring it to g. If this were to go on forever, f() would initially need to be infinite. This cannot be, so one must adopt the notion of periodicity. f() gives some probability to g() for some x range and then it is given back. This leads to two ideas, the notion of repeated x intervals, called wavelengths=hbar/p, and the idea of probability in f() and g() being both positive and negative at different x values. We suggest that the former idea is usually only considered. The positive and negative values of probability suggest transfers of probability and one may extend this idea to a sum (OR situation) of cos(-Et+px) values for fixed E,t,x and varying p. In such a case, instead of shifting probability between cos() and sin(), one shifts it between various cos() (and separately between sin()s). Thus, we argue that the basic probabilistic idea of interference is already found arising as a consequence of three physical conditions (A),(B),(C).
