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

Transcribed Image Text:

Example 6.7 A particle of mass m moves in one dimension in a harmonic-oscillator potential: 1 -zmw²x². 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-iĤ1/h with the number e-¡Ent/ħ, 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) = Ü* (t) kÛ (1) ¥n (x) ħ 2mw Thus 35 = ħ h - e¬¡ E₁¹/ħ ¿i ¹/h [√₂ + 1 &n+1(x) + √n ¥n−1(x)] =√2mo ħ · e-i Ent/heiĤ¹/h (â+ +â_) n(x) 2mw (â+ +â) e-¹₁/₁(x) 2mw h V2mw +√ne En-1/n - e-i E₁1/k [√₁ + leit 1 √n-1(x)] - [eicx â + + e-ico₁ à h 2 mw [√n + 1e²² √²+1(x) + √ne¯iwt #n-1(x)]- *H (1) = H (0) cos(@t) + à-]. En+11/h. XH (t)=₁ Or, using Equation 2.48 to express à 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.