Quantum operator ordering
= Quantum operator ordering
Products of noncommuting <linear operators> depend on their order. For position and <momentum>, $xp-px=i\hbar$; substituting classical commuting expressions into a product can therefore lose an additive quantum term. For example, expansion of the oscillator product $a^\dagger a$ includes a <commutator> contribution $-1/2$, leading to the correct <zero-point energy> in $H=\hbar\omega(a^\dagger a+1/2)$.