Free Particle Quantum Probabilities and Classical Conditional Probability Relations

  In previous notes, we argued that one may introduce a unit modulus complex probability into Newtonian mechanics to account for the uncertainty in the outcomes of elastic two body scattering. Such scattering must conserve energy and momentum and so for an initial (e1,e2) and (p1,p2) vectors, one may propose that any set (ei,ej) and (pi, pj) has the same product probability as the original. At first, one might think that this leads to exp(iE) and exp(ip), but enforcing Lorentz invariance and time reversal invariance (i.e.  p→-p  x–.-x) one obtains exp(-iEt+ipx) (or exp(-iEt+i p dot r). The question then becomes: How is this probability used? First, one may note that it is physically relevant because it may be applied to the analysis of 2-slit interference and one dimensional reflection-refraction at an n1-n2 index of refraction junction. 

  On the other hand, if one considers a large number of particles with various momentum which move in a box without interacting, then one has P(x)=1/sqrt(L) and P(p) = 1/B, where L is some length and B, some momentum range. This implies there is no special preference for any momentum. At first it seems that exp(-iEt+i p dot r) has nothing to do with the non-colliding particles in a box, because there are no interactions and exp(-iEt+i p dot r) was specifically constructed with interactions in mind. Nevertheless, one may observe that exp(-ipx)exp(ipx) = 1 which is linked to P(x).

      We suggest that even though a priori, the non-colliding particles in a box example does not seem to be directly linked to exp(-iEt+ipx), there is another Newtonian case which involves interactions, namely those with a potential V(x). Such interactions must conserve momentum and energy of the particle-V(x) system and so exp(-iEt)exp(ipx) should apply to such an analysis. In Newtonian mechanics, however, one does not use exp(ipx) to calculate motion with a V(x) present. As a result, one must account for this. It seems that if one uses wavelength = hbar/p, classical values of p lead to a wavelength which is so tiny compared to the system length that one may use the Newtonian approach dp/dt = -dV/dx. This does not mean, however, that exp(ipx) is irrelevant. It manifests itself in 2-slit interference and 1D reflection-refraction (n1-n2 junction) and it is possible that there are systems for which the wavelength is of the order of the system length. In such a case, one should be able to analyze the system using exp(ipx)s. This would imply a probabilistic OR situation with free particle probabilities exp(ipx) and various impulse hits leading to: { Sum over i a(p) pp/2m exp(ipx) } / W(x) +V(x) = E, where W(x)=Sum over p a(p)exp(ipx). 

   Given the presence of p and x, one might suggest that P(p/x) = a(p)exp(ipx)/W(x) as done in (1). We note that exp(ipx) is symmetric in p and x and so a conditional probability treatment should deal with a system with uncertainty in both p and x. A given p in exp(ipx) has no uncertainty in p and so it is not until one considers the V(x) interaction that uncertainty in p arises.

 In (1), using conditional probability relations,  we showed that such an assumption leads to P(x) = W*(x)W(x) and P(p) = a*(p)a(p). Here we argue that the classical relations of conditional probability do not appear until one obtains a situation with both uncertainty in p and x as exp(ipx) is symmetric in both of these variables. We also example the case of a(p1)=1, a(p)=0 for p not= p1 (1 dimension). We show that this then leads to P(x)=P(p)=P(p/x)=P(x/p) = 1 which describes the non-colliding particles in a box. Thus, there is a roundabout way to link the exp(ipx) formalism to the noncolliding particles in a box scenario.