Joint Conditional Probability and Bound Quantum Density

In a number of previous notes, we have argued that the bound state quantum problem may be considered in terms of a conditional probability P(p/x) i.e. Sum over p p*p/2m P(p/x) + V(x) = E with P(p/x)= a(p)f(p,x)/ W(x) (where W(x), a normalization constant, is identical to the wavefunction). In other words, one considers a momentum distribution at x (with the different times pulled out). This momentum distribution may be thought of as being created by a stochastic potential which on average is V(x). Thus, within a little region delta x, in which one potential hit occurs, a particle has a joint conditional probability to move from momentum p1 to p2. Because the potential is stochastic, it is not clear where in the region delta x this occurs. Thus, we suggest a joint conditional probability of Sum on p1, p2 f(p1,x)f(p2,x). This is equivalent to W(x)W(x) the quantum density or P(x). From this, one may calculate P(p)= Sum over x P(p/x) P(x) leading to the idea that f(p,x) is an orthonormal basis. Thus, one moves from probability to linear algebra. We suggest that the stochastic force may be given by: -Sum over k k Vk(x), where -d/dx Vk(x) =-k Vk(x) or at least Vk(x)=Vk(kx). If one considers the bound state Schrodinger equation: Sum over p P(p/x) p*p/2m + V(x) W(x) = E W(x) then V(x)W(x)= Sum over p Vk(x) a(p-k)f(p-k,x). Vk(x), however, is associated with a hit of momentum=k, so one would expect that Vk(x)f(p-k,x) = f(p,x). This together with Vk(x)=Vk(kx) may lead to V(p,x) = b(p) f(p,x) where f(p,x)=exp(ipx)