Consider the harmonic oscillator wave function 1/4 where α = (-)"*. Here k, is the stiffness...

Psi_0 (x) = N e -x^2/2 alpha^2 Now Normalization integral^infinity_- infinity psi*_0 (x) pai_0 (x) dx = 1 Because N^2 integral^infinity_-infinity -x^2/alpha^2 dx = 1 Here b = 1/alpha^2 [integral^infinity_-infinity e^bx^2 dx = squareroot pi/b] because N^2 squareroot pi/(1/alpha)^2 = 1 N = 1/squareroot alpha^2 pi = > N = 1/alpha squareroot pi probability DensityVertical-bar psi* (x) psi (x) Vertical-bar = N^2 e^-x^2/alpha^2Vertical-bar psi Vertical-bar ^2 = 1/alpha^2 pi e^-x^2/alpha^2 \r\n delta x = squareroot < x^A2 > - < x > ^2 Because < x > = integral^infinity_-infinity psi* (x) x psi (x) dx/integral^infinity_infinity Vertical-bar psi Vertical-bar ^2 dx = N^2 integral^infinity_-infinity xe^-x^2/alpha^2 dx/1 < x > = 1/alpha^2 pi integral^infinity_-infinity k e^-x^2/alpha^2 dx using integral^infinity_-infinity x^2 b^-bx^2 dx = squareroot pi/b/2b Here b = 1/alpha^2 < x^2 > = 1/alpha^2 pi . squareroot pi (1/alpha^2)/2 (1/alpha^2) < x^2 > = 1/2pi squareroot alpha^2 pi = > < x^2 > = alpha/2 squareroot pi using (2) & (3) delta x = squareroot < x^2 > - < x > ^2 delta x = (alpha/2 squareroot pi)^1/2 \r\n {p^^ > = integral^infinity_-infinity psi* (x) (-ihd/dx) psi (x) dx = -ih/alpha^2 pi integral^infinity_-infinity e^-x^2/2 alpha^2