Work out p̂H (t) for the system in Example 6.7 and comment on the correspondence with the classical equation of motion.

Example 6.7 A particle of mass m moves in one dimension in a harmonic-oscillator potential: 1 -zmwx. 2m Find the position operator in the Heisenberg picture at time t. Solution: Consider the action of xH on a stationary state n. (Introducing Vn allows us to replace the operator-i1/h with the number e-Ent/, since e-it/he-iExt/h) Writing in terms of raising and lowering operators we have (using Equations 2.62, 2.67, and 2.70) = XH (1) n (x) = * (t) k (1) n (x) 2mw Thus 35 = h - e E/ i /h [ + 1 &n+1(x) + n n1(x)] =2mo e-i Ent/hei/h (+ +_) n(x) 2mw (+ +) e-/(x) 2mw h V2mw +ne En-1/n - e-i E1/k [ + leit 1 n-1(x)] - [eicx + + e-ico h 2 mw [n + 1e +1(x) + neiwt #n-1(x)]- *H (1) = H (0) cos(@t) + -]. En+11/h. XH (t)= Or, using Equation 2.48 to express in terms of and p, 1 ma Vn+1(x) - PH (0) sin(wt). (6.73) (6.74) As in Example 6.6 we see that the Heisenberg-picture operator satisfies the classical equation of motion for a mass on a spring.