Quantum particle of mass m m is bound in the ground state of the one-dimensional parabolic potential well K 0 x 2 2 K0x22 until time t = 0 t=0. Between time moments of t = 0 t=0 and t = T t=T the stiffness of the spring is ramped-up as K ( t ) = K 0 + t T [ K 1 − K 0 ) 0 ≤ t ≤ T ( 3 ) K(t)=K0+tT[K1−K0)0≤t≤T(3) and it stays equal to K 1 K1 afterwards, at t > T t>T. The overall change of the stiffness is small: | K 1 − K 0 | ≪ K 0 |K1−K0|≪K0.

(a) Does any work need to be done to exercise this ramp-up? Can this work be dependent on the duration T of the ramp-up and why?

(b) If the answer to the latter question is positive, evaluate the needed work in the lowestorder of the perturbation theory.

(c) Analyze the T-dependent part of the work, if any, and find out how much of the variation in the amount of work is achieved between the adiabatic ( T → [infinity] T→[infinity]) and nearly instantaneous ( T → 0 T→0)limits?