 Quantum operator ordering

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