Question

5. (20 pts) Show that (x3) 0 for a simple harmonic oscillator wave functions with unspecified n. (Hint: Use Symmetry properties of the Hermite Polynomials) a. (15 pts) a. Show that the solution to the free particle Schrodinger equation dx is: ψ Ae-ikx + Beikx where k = b. (5 pts) Which term vanishes (blows up) if the particle for xO region. (note: k >0)

Step by step typed format preferred. If hand written, PLEASE write clearly and legibly. Thanks!

Answer 1

The second term in the above solution blows since at x = -infinity to x < 0 B e^-i k (-infinity) = B e^+ infinity = infinity rightarrow which is un physical since psi psi^* = density \r\n For harmonic oscillator the wave function is given by psi_n (k) = 1/squareroot 2^n n! squareroot pi x_0 e^-x^2/2x_0 H_n (x/x_0) x_0 = squareroot hwithstroke/mw H_n rightarrow Hermite polynomial of order 'n' The pairty of Hermite polynomial depends on n (-1)^n i middot e for odd value of n Hermite polynomial is odd & for even value of n it is even < x^3 > = integral^+ infinity_-infinity psi_n x^3 psi_n dx = 1/(s^n n! squareroot pi x_0) integral^+ infinity_-infinity e^-x^2/2 x_0 H_n (x/x_0) x^3 e^-x^2/2x_0 H_n (x/x_0) dx = 1/(2n n! squareroot pi x_0) integral^+ infinity_-infinity integral e^-x^2/x_0 [H_n (x/x_0)]^2 x^3 dx The considering f (x) = e^-x^2/x_0 middot [H_n (x/x_0)]^2 x^3 The parity of f (x) i middot e f (-x) f (x) = e^-x^2/x_0 (-1)^2n