Question

14. Consider the Hamiltonian describing a one-dimensional oscillator in an external electric field + mo???(1) - Ex(1) Calculate the equation of motion for the operators p(t) and x(i) using (6-67) and the commutation relation [p(), x(0)= Show that the equation of motion is just the classical equation of motion. Solve for p(t) and X(t) in terms of p(O) and x(0) Show that [x(11), X(t))] 0 fort, + 12

Answer 1

- Here, the given Hamiltonian is- H = pt + Ź mw2x²(t) * - c{XH 0 and it is given that the commutation relation - [PH, *)] wie-it - 0. He know;  = [4, Â] + de  - where, à may or may not be time-dependent and Therefore. for. equation of X(t) we.com write dumat) - [H, 2(t)] + X). —D: wow.[û, x()] = [ 17+ { mw2x+1).- eemt), ] - stan [ p*6). 46).] + mw? [ 2+0), 4(+)] 1. e4 [xlt), alt] - stron {pey[ Plt), a4)] + . [pe), môj po} - stone {pct)a-it + plto-it} - mx - zit pt). - - ita pel). putting this in above equation we get de tot - À PCD + everything ot

Similarly. coupe it [H. Pw] + 2) 478) - it [H. PO ] — @ [it, PH)] = [p?t) + + mw²x²4 h el nos, pH)] now. azt [P20.04]+ + mw? [x24, PG)] + bez [xl).PⓇ] = £mw?{ [20), P (A)] + [xG), P4) J xes} +-03 (20), PH] = {mw? { 26t). it + it. xDFes.it since. [86), aG)]=-it im was at ciest. - its tes + mw? n(hy) — 0. defe i to Les + mw2 n(t)). evento = - tex + m w 2 x(H) )- és – mw2x69. — Therefore i or apet e s- mw2 213 On substituting a l m dll ) = ee. mwtxt) m d ² xt) & mw²att. = es or dt a

lect (es. 13 - de + wax = leve e —@: dap - tw2p=0 sot of equ ® is or int o pt For to and t At t-o. . 12 A e Be o physible sol is PG Be-i t P = P (od there fore. to = P(o) = P e ... P solt of eat 6 lee-w2) = t (22/m-w2 ) 2 t o at = e em .co + De toetusest ate and t of the to a physible solution for is. *(t) = Be (et/m - wa)?t whies wă= ex/m - ☺ At t=0, n(t) = x (0) D = acol ur a (t) = De *!!) - 16 e - (wo? -w2y x (to) and X(+2) are two different retalization of the position of the particle. for two different-time (titt). so, [a (ti), x(+₂) ] to. they do not commute