Question

6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds?

Answer 1

Consider an electron bound in a potential v = k/z x^2 ----(1) the ground state was function of an electron is psi(u, t) = 1/pi 1/4 squareroot a^e - x^2/2a^2 e^-1 w, t where a = (u^2/m k)^1/4 (a) The expectation value of x is (x) = integral psi (n, t) psi(u, t)dt Where psi(x, t) is complex conjugate of psi, that is psi^* (u, t) = 1/pi 1/4 squareroot a^e -x^2/2a^2 e^+ I w, t Now < x > = integral^infinity_-infinity psi.(x, t) x psi(v, t) dx 2 pi < x > = 1/pi 1/2(a) integral^infinity_infinity x e^-2 pi^2/2a^2 e^w, t - 1w, t dx = 1/pi 1/2 a integral^infinity x e^-x^2/a^2 dx i.e < x > = 2/squareroot pi(a) integral_0 4 e^-x^2/a^2 dx = 2/squareroot pi a integral^infinity_0 a^2/2 e^-t dt = a/squareroot pi integral^infinity_0 e^-t dt = a/squareroot pi integral^infinity_0 t^1 - 1 e^-t dt = a/squareroot pi ----(1) (b) expectation value of u^2 in this state