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Consider a simple one-directional quantum harmonic oscillator with Lagrangian L =1/2 mx2 —1/2mw2x2

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Consider a simple one-directional quantum harmonic oscillator with Lagrangian

L =1/2 mx² —1/2mw2x².

a) Rewrite the trajectory as the classical trajectory (the solution of the Euler-Lagrange equations-x,;(¢)) plus the deviation from it (y(t)).So_ that the trajectory can be calculated as (x(t) = X,)(t) + y(t)). Describe in words the result obtained |

b) Given that the amplitude K (x,, t;, x;,¢;) for the particle to travel from x; at time f; to x, at time x, in the path integral formalism (the propagator) is given by the

expression:

K (2x, T|21.0) my \ Im (2 4.2) coswT — 20429]

rq, 1/21,0) = | ———_’°. exp 4 =| (2 rg) coswl — 21442

are 27ihsinwT P OhsinwT is

Generalize this expression for an arbitrary number of dimensions