Question

19. Suppose that an electron in a one-dimensional harmonic-oscillator potential muo2 is subjected to an oscillating electric field o) cos wt in the x direction (a) If the electron is initially in the ground state, what is the proba- bility that the electron will be in the nth excited state at time t? (b) I , perturbation theory will fail at some time t. What is the critical time?

Answer 1

1 - Dimensional harmonic - oscillator potential = 1/2 m w^2_0 x^2 oscillating electric field e = e(0) cos wt - nabla phi = E phi = -epsilon (0) times cos wt thus H = p^2/2m + 1/2 m w^2 x^2 - epsilon(0) times cos wt harmonic is H prime(H) = H prime cos wt H prime = -x epsilon Xi(0) = -e Xi(0) < n Vertical-bar x Vertical-bar n prime rang H prime = -rho epsilon_0/squareroot 2 beta [squareroot n delta_n, n - 1 + squareroot n + 1 delta_n, n + 1] where beta = squareroot m w-0/planck The probability at time at P_1 rightarrow n (t) = Vertical-bar H^prime Vertical-bar ^2/planck^2 Vertical-bar integral^infinity_-infinity e^i w_0 t^prime f(t prime) dt prime Vertical-bar ^2 P_1 rightarrow n (t) = 2 (t^prime)^2/planck^2 Vertical-bar integral^x_0 e^j w_n t prime cos w t prime dt prime Vertical-bar ^2 \r\n integral^t_0 e^i w_n t prime cos w t prime dt prime = e^i(w_n - w)t/2 sin 1/2(w_n - w)t/w_n - w + e^i(w_n + w)t/2