PH 216, Problem set 4, Due 4/27 1. A particle of mass M is in a one-dimensional harmonic oscillator potential V 1 = 1 2 kx 2 . (a) It is initially in its ground state. The spring constant is suddenly doubled ( k → 2 k ) so that the new potential is V 2 = kx 2 . The particle’s energy is then measured. What is the probability for Fnding that particle in the ground state of the new potential V 2 ? (b) The spring constant is suddenly doubled as in part (a), so that V 1 suddenly becomes V 2 , but the energy of the particle in the new potential is not measured. Instead, after a time T has elapsed since the doubling of the spring constant, the spring constant is suddenly restored back to the original value. ±or what values of T would the initial ground state in V 1 be restored with 100% certainty? 2. An electron is in the n = 1 eigenstate of a one-dimensional inFnite square-well potential which extends from x = − a/ 2to x = a/ 2. At t = 0 a uniform electric Feld E is applied in the x − direction. It is left on for a time τ and then removed. Use time-dependent
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2k, Heisenberg, Time-dependent perturbation theory, Heisenberg position operators