Intrinsic Quantum Square Root Flux Related Probability of Mutually Exclusive Events

Classical probability often deals with mutually exclusive events such as a coin landing heads or tails or a die landing on a certain number. There is no information as to which event occurs and so one assigns equal probabilities to each possible outcome i.e. a coin is associated with the probability ½ and a die with ⅙. 

    Mutually exclusive events also appear in the quantum world. For example, a single photon moving in one dimension along the x axis in a medium with index of refraction n1=1 may strike an interface along the y axis with index n2>n1. The single photon either reflects or refracts. If one knew nothing  about one probability or another, one might assume the probability to reflect to be ½ and that to refract ½, but there is no symmetry making these mutually exclusive events seem identical as in the coin or die case. One might next suggest that the stochasticity is in the mechanism which causes reflection or refraction, which may be very difficult to ascertain.

  For a photon, however, E=pc (E=energy, p momentum and c, speed of light). The incident, reflected and refracted photons all have the same energy, so from the point of view of energy alone, nothing happens to the photon. It is momentum and velocity which change. Thus an impulse seems to occur at the interface such that energy remains unchanged. Instead of suggesting the reflection/refraction mechanism is stochastic, the photon itself may be the driver of the two events, reflection and refraction. If the photon is the probability driver, there is a certain determinism to the problem, but the physical outcomes are probabilistic. The probabilities for the mutually exclusive events add to 1 and so are part of the same equation. If the photon contains an intrinsic built-in type of probability, one would also have an equation containing these built-in probabilities. This intrinsic probability, however, would be one which exists in time and space, it seems. Thus, one would need to match built-in probabilities at a certain x and t. If the built-in probability is a function of p,x,t,E, then weights could be assigned to the reflected and refracted functions with the incident having a weight of 1. 

    A final issue is to link a weighted built-in probability for say the refracted photon to its physical probability. Given that the built-in probability is associated with time and space, it is not really linked with a probability for an event, but rather with a flux. One may, however, easily associate a mutually exclusive event, say reflection or refraction with their steady state picture flux divided by c/n, the speed. Such a flux does not depend on x,t, but the built-in one does as it is a driver. To link the two, one may suggest that the built-in probability is a unit complex vector in x,t,p,E (Lorentz invariant).

   As a result, somewhat deterministic (continuity and links with steady state scenario) approaches may be applied to a built-in flux related type of probability which includes all mutually exclusive events for a single photon and acts as a driver, in order to calculate the mutually exclusive probabilities of reflection/refraction of a single photon. In other words, a built-in type of probability related to flux creates or determines the observable probability of physical events. In the reflection-refraction case, the built-in probability does not directly manifest itself, but in other cases, such as two slit interference it does. The mutual exclusiveness for the reflection-refration example, however, occurs because the refracted photon is confined to a mutually exclusive region of space. If this restriction does not exist, as in an n-slit example, the exp(-iEt+ipx) driver probabilities may interfere.