Quadratic-norm proof of the Heisenberg uncertainty relation
= Quadratic-norm proof of the Heisenberg uncertainty relation
For an admissible <quantum state> with zero <position> and <momentum> means, positivity of the squared <norm> $\lVert(p-isx)\psi\rVert^2$ for every real $s$ gives
$$
(\Delta p)^2+s^2(\Delta x)^2-s\hbar\geq0.
$$
The <discriminant> is nonpositive, hence $\Delta x\,\Delta p\geq\hbar/2$. Subtracting the <expectation values> from the <position operator> and <momentum operator> extends this outline to nonzero means.