152 CHAPTER9. THEHARMONICOSCILLATOR
+
√
n<φn|a†|φn− 1 >+√
n+ 1 <φn|a†|φn+1>
=√
n(n−1)<φn|φn− 2 >+(n+1)<φn|φn>+n<φn|φn>+√
(n+1)(n+2)<φn|φn+2>
= 0 +(n+1)+n+ 0
= 2 n+ 1 (9.71)sothepositionuncertaintyis
∆x=√
̄h
2 mω(2n+1) (9.72)Noticethattheexplicitformofthewavefunctionφn(x)wasnotneededtocompute
∆x;thecalculationiscompletelyalgebraic,andrequiredonlytherelations(9.68).It
iseasytoseethatanyexpressionoftheform
<φm|F[x,p]|φn> (9.73)whereF[x,p]is any polynomial inthe xandp operators, can alsobe computed
algebraicallybythesamemethod.