= Equality of two quadratic position-momentum orderings
{title2=$\widehat x\widehat p^2\widehat x=\widehat p\widehat x^2\widehat p$}
On a common invariant domain for the operator products, let the <position operator> and <momentum operator> obey $[\widehat x,\widehat p]=i\hbar I$. Then $\widehat x\widehat p=\widehat p\widehat x+i\hbar I$, so both displayed expressions equal $(\widehat p\widehat x)^2+i\hbar\widehat p\widehat x$. In position representation both act on a smooth function $f$ as $-\hbar^2(x^2f''+2xf')$. This particular equality does not imply that arbitrary classical reorderings remain equivalent in <quantum mechanics>.
Back to article page
