The Effects of Averaging within Probability in Quantum Mechanics

   In classical probability, an average is a basis measure of a set of  data. We suggest that the set of data is like a vector and that the average reduces this vector to a scalar and thereby loses information. As a result, one often uses the average together with a standard deviation. An average, however, is not the same as a probability, but it does take a vector and reduce it to a scalar.

    In quantum mechanics, we suggest that the wavefunction,  W, is a vector probability and that the so-called probability of presence or spatial density W*W is a scalar, i.e. a modulus. In other words, we argue that a kind of averaging has already occurred in creating W*W which loses much critical information. W*W is a kind of average probability which only corresponds to an averaging of measurements and does not allow one to solve momentum related physical problems such as 2-slit scattering and one dimensional reflection-refraction. In particular, we note that exp(ipx) is a 2-vector and exp(ipx)exp(-ipx)=1 a scalar with little information.  exp(ipx) represents a spatial resolution of hbar/p which is completely lost in 1. Thus, we argue that W*W is not really a probability of presence. 

   Averaging may even occur in W. For example, for a bound state, time averaging exists because one uses exp(ipx) and exp(-ipx). This averaging reveals the resolution peaks/troughs, but  a 2-vector is reduced to a single component. Physically, the two vector has not disappeared, it is only averaged down to one. Similarly, classical physics arises by using an envelope function (1) which touches peaks of W*W and so we argue that this is a kind of blurring process which removes information.

    In (1), the notion of W*W in nonrelativsitic quantum mechanics has been extended by using a nonrelativistic limit of a Dirac 4-vector, i.e. has been reduced to two 2-spinors  (W,  sigma dot p /2m W) ((1)) where sigma are the Pauli matrices. This leads to W*W + 1/(4mm)  gradW* dot grad W as the “probability of presence”, but once again a vector probability, this time a double vector because for a free particle one has exp(ip dot r) linked to each component and so W*W leads to a double loss of information when taking Wtotal *  dot Wtotal.  In particular, for a free particle, the vector form ((1)) retains the resolution hbar/p even though for exp(-ipx) the bottom component is phase shifted. Thus, we argue that the vector W is a better representation of probability of presence and that W* dot W is again a kind of averaging which loses information in the problem.