Full text
The Significance of Time Reversal Invariance of the Quantum Free exp(i p dot r) Part 2 Francesco R. Ruggeri Hanwell, N.B. Oct. 17, 2025 In Part 1, we suggested that a free particle probability, originating in the desire to have equal product probabilities for any (ei,ej), (pi, pj) momentum vectors for an initial (e1,e2), (p1,p2) ,ultimately leads to the complex unit modulus Lorentz invariant exp(-iEt+i p dot r) free particle probability. (Details of the derivation of this function are given in Part 1.) We also noted that this complex probability has three interesting properties. First, it is a dynamic probability in the sense that exp( -i p dot r) and exp( i p dot r) do not have the same value. Thus, this probability can detect the direction of motion in space. The unit modulus indicates that all free particles are on the same footing (just as the classical P(x) = dx/L does), but exp(i p dot r) shows direction through positive and negative values of sin(px) and cos(px) for p along the x direction. This we argue is important for the idea of interference. Second, there is a characteristic length present in space, namely the wavelength hbar/|p|. These two properties are linked to each other. In Part 1, we argued that there is another important property, namely invariance under time reversal, i.e. p → -p and x→ -x leaves exp(i p x) invariant. We argued that the same exp(ipx) represents a movie being run forward or backwards. In the forward case, the starting point is linked by t(initial), but in the running of the movie backwards, this point becomes t(final). We argued that there is no time flow when using the probability exp(i p dot r). This allows one to add exp(i p dot r)’s for events at different times as in one dimensional reflection-refraction, discussed in Part 1. It might seem like a paradox at first to argue that exp(i p dot r) indicates direction of motion and then to state there is no time flow when using exp(i p dot r). We suggest that there is no paradox because exp(- i p dot r) and exp(i p dot r) are different due to the different momenta, i.e. p and -p. Even with time reversal invariance and no sense of time, there is still the sense of momentum in one direction or the other. exp(i p dot r), however, goes further, because p and -p are linked with sin(px) and sin(-px) = -sin(px) and so there can be interference. In other words, the presence of a negative momentum may lead to the diminishing of an “initial” probability term. This is good because if one has events in time, like an initial photon moving towards an n1-n2 junction (index of refraction), then a reflected or refracted photon may be created. If there is no time in the problem (i.e. one uses exp(i p dot r)’s), then one has Aexp(i p x for the incident photon and B exp(-i p x) for the reflected on the same side of the equation, i.e. in the same range of x values, while C exp(i p2 x) exists to the left of the n1-n2 junction. The term B exp(-i p x) may be linked to the diminishing of Aexp(ipx) allowing some probability to appear as C exp(i p2 x). In the case of a single particle elastically scattering from V(x), with exp(ipx) + f(theta)/r exp(ipr), it is the backscattering portion of the second term which interferes with exp(ip x) to diminish some probability to account for forward terms of exp(ipr)/r. The difference in values of exp(ipx) and exp(-ipx) is crucial for interference, we argue, but there is still a scenario of no time flow, so one may have exp(ipx)s representing various different times as argued in Part 1.
Properties of exp(-iEt+ip dot r) In Part 1, we tried to provide a derivation of exp(-iEt+i p dot r). We also noted its key properties: (A) exp(i p dot r) not= exp(-i p dot r) (B) There exists a characteristic length, called the wavelength = hbar/|p| (C) For p→ -p, x→ -x exp(ipx) remains invariant. Point C implies that exp(ipx) represents both a film running forwards and backwards. In other words, if an event is linked to t(initial) in the forward run, it is t(final) in the backwards one. Thus, the same point is characterized by both t(initial) and t(final). We conclude that there is no time flow if one uses the probability exp(i p x) (or exp(i p dot r). This is a key idea because it allows one to combine exp(ip dot r)s for events occurring at clearly different times in the same equation as long as one uses exp(i p dot r) probabilities. Two examples discussed in Part 1 are: A single particle scattering from a potential V(x) exp(ipx) + f(theta) exp(ipr)/r ((1a)) One dimensional reflection-refraction: Aexp(ipx) + Bexp(-ipx) = C exp(i p2 x) at x=0 and Apexp(ipx)-pBexp(-ipx) = Cp2 exp(ipx) at x=0 ((1b)) Because one uses exp(ipx)s, events existing at different times appear in the same equation, but this does not mean that one can ignore the relative values of p. For example, an incident photon must have a momentum that is minus that of a reflected one. The fact that exp(ipx) and exp(ipx) are different is key because it means that there can be a diminishing of exp(ipx) due to: sin(-px) = -sin(px) or even B being negative in ((1b)) One may use exp(i p dot r) probability which means no time present, to have an “initial beam” exp(ipx) in ((1a)) be diminished by back-scattering terms in f(theta) exp(ipr)/r to account for the forward scattering terms of f(theta) expI(ipr)/r because probability must be conserved. Why The Modulus of exp(-ipx)exp(ipx)? In quantum mechanics, one argues that W*(x)W(x) or exp(-ipx)exp(ipx) represents the classical probability. We note that for a particle with constant p, one may write classically: P(x) dx=dx/L where L is an arbitrary length ((2))
P(x) does not indicate the value of p or v nor does it show its direction nor does it separate events in time.Time is completely removed from ((2)). This is like exp(i p x), but exp(ipx) indicates differences for p and -p and allows for interference. The key point of time removal, however, allows one to link exp(ipx)/sqrt(L) to 1/L in a consistent manner. We argue here that it is because both P(x)=1/L and exp(ipx) are time reversal invariant, i.e. show no signs of the presence of time that they may be associated. It is not simply a math trick that: exp(-ipx)/sqrt(L) exp(ipx)/sqrt(L) = 1/L ((3)) ((3)) allows one to remove the first two properties (A) and (B) from exp(ipx) while retaining the third, C, which is time reversal invariance, i.e. removal of time. This, we argue, is why one may create sums of various exp(i p dot r)s linked to different times and then take the modulus to obtain a classical type probability. The classical probability itself is not linked to any specific time, only to x. Conclusion In conclusion, in Part 1, we derived a complex free particle probability exp(-iEt+i p dot r) and argued that it had three important properties. First, we noted that it differentiates between direction of motion, i.e. exp(ip dot r) not= exp(-ip dot r). Secondly, there is a characteristic length wavelength = hbar/|p|. These two features are closely related because sin(-px) = - sin(px) meaning that it is phase shifted or reflected about the x axis. This suggests the possibility of cancelation and addition of probability in x. In Part 1, we strongly stressed the third feature, time reversal invariance of exp(ipx). This probability remains the same under p→ -p and x→-x. This is the same as stating that it has the same value for a movie run forwards and backwards. An initial t in the forward movie is the final t in the backwards movie. In other words, there is no flow of time when one uses exp(ipx)s. This allows one to write equations with exp(ipx)s representing events that occur at clearly different times. At first there might seem to be a paradox between exp(ipx) not= exp(-ipx) and time reversal invariance, but we argue in this note, that exp(ipx) and exp(-ipx) show different relative momenta which must exist even if time is removed. In fact, we argue that it is the possibility of cancellation/addition of probability in such a case (i.e. interference) which allows an incident exp(ipx) to be diminished to account for probability appearing in another place. As an example, in the case of a single particle scattering elastically from V(x) , a potential, one writes: exp(ipx) + f(theta)exp(ipr)/r. Classically, the incident particle should not exist at the same time as the scattered one and so one would not write probabilities for the two in the same equation. With exp(ipx)s, however, one may do so. In this case, new probability appears as scattered material, but this must ultimately come from the incident probability exp(ipx), so backscattered terms of f(theta) exp(ipr)/r must account for interference which achieves this. (We note that this scattering problem is well-known in the literature and it has been shown that this is the case.) We suggest that the time reversal invariance of the probability exp(ipx) (i..e no time present), in addition to exp(ipx) not= exp(-ipx) allows one to create equations which show exp(ipx)
probability conservation. We then suggest that this probability may be converted to a classical probability through W*(x)W(x), where W(x) = sum of the various exp(ipx) in x. We argue that this holds because for a constant p particle, P(x)dx=dx/L classically. This result removes all reference to time, just as exp(ipx) does. exp(ipx), however, has other features like a wavelength and exp(-ipx) not=exp(ipx) and these must be removed to obtain the classical result, through exp(-ipx)exp(ipx), but the time reversal invariance feature (no time) remains, we argue.
