Momentum-Position Uncertainty Principle

The momentum-position uncertainty principle delta p delta x >= hbar/2 is often derived in texts using Heisenberg’s heuristic approach or a calculation based on the commutator of p and x together with Schwartz’s inequality (or other operators, although such an approach does not hold for E and t). Delta x and delta p are square roots of variances of x and p. For example, Integral dx xx W(x)W(x) is used in the one dimensional calculation of a bound wavefunction W(x), where W(x)W(x) = P(x). From the point of view of statistics, this appears completely normal, but we argue quantum mechanics is based on a wavefunction W(x)=Sum over p a(p)exp(ipx). x*x may be expanded as a Fourier series and interacts with W(x) like a potential - in fact like the oscillator potential. Thus, we argue one may obtain the x,p uncertainty principle by using ideas from the harmonic oscillator ground state. Interestingly, the oscillator ground state has momentum and spatial wavefunctions of the same form as the classical statistical mechanical density and has a similar x dependence as which usually does not occur