Zeros of Quantum Spatial Probability Density

Quantum spatial probability is given by W*(x)W(x) where W(x)=Sum over p a(p)exp(ipx) is the wavefunction. If one knows about a particle’s location through W*(x)W(x) then momentum is uncertain. On the other hand, a*(p)a(p) is the probability for a particle to have a specific p free particle momentum. In that case, there is no knowledge of the location of the particle.

     In (1) it is argued that for an infinite potential one dimensional box, the wavefunction is sin(n3.14/L) and so W(x)W(x) may have zero values within the box. It is stated that this is unphysical as there is no reason a particle cannot be at such an x point. (The resolution in (1) is to consider a linear superposition of energy eigenfunctions based on certain rules.)

    In this note, we consider that a(p) values suggest there are probabilities for a particle to have a constant p momentum. This momentum means the particle may move through all points x because even in the quantum free particle case exp(ipx), a two dimensional dynamical probability implies: exp(ipx)exp(-ipx)=1. Thus a measurement of a p value (i.e. a constant value) even with no known corresponding x value suggests the particle may be anywhere in the system, even at zero points of the wavefunction. The bound wavefunction is real and so is an average which removes the dynamism of the system i.e. the two A particle does not exist in all p states at once. The zeros of W(x) appear only because exp(ipx) and exp(-ipx) interfere revealing one dimension of the two dimensional probability. Thus we argue that W(xa)=0 does not mean that a particle cannot be at xa.