Consider a simple one-directional quantum harmonic oscillator with Lagrangian
1 . 2 1 L = -2MX – -2MW2X2.
a) Rewrite the trajectory as the classical trajectory (the solution of the Euler-Lagrange equations-xei (0) plus the deviation from it (y (t)) . So that the trajectory can be calculated as (x(t) = xci(t) + y(t)). Describe in words the result obtained. b) Given that the amplitude K(xf, tf, xi, t1) for the particle to travel from xi at time ti to xf at time xf in the path integral formalism (the propagator) is given by the expression:
77/W imw 1/2 \ K(X21 TIX1) 0) = (27riri sin wT x exp 2h sin wT[(xT + 4) cos wT — 2x x 2] Generalize this expression for an arbitrary number of dimensions