Question

1. For a harmonic oscillator with unit mass and unit frequency, the Schrödinger equation for its eigenfunction is given by where n 0, 1, 2, . . .. Answer the following questions. Given a trial wave function, ?(x)-?000CnUn(x), where expression for the expectation value is is assumed to be real, the Prove that Eo2 h/2 2. Assume that the trial wave function for the ground state eigenfunction in Eq. (1) is ?(x) = cos Xx, for-2a<x<2 and ?(x) 0 otherwise. Use variational theorem to calculate the optimum value of ? and the best approximation for Eo

Quantum Chemistry. Thx in Advance!

Answer 1

Given that A harmonic oscillator with unit mass and unit frequency the Schrodinger eqn for eigen function is (h^2 /-2 d^2/dx^2 + x^2 /2) psi_n (x) = h(n + 1/2) psi_n(x) where n = 0, 1, 2, the trial wave function phi(x) = sigma^infinity_n = 0 c_n psi_n(x) when c_n is assumed to be real the the expression for the expectation value is E_phi = sigma^infinity_n = 0 C^2_n h(n + 1/2)/sigma^infinity_n = 0 c^2_n the wave function of the first executed state of harmonic oscillator is psi (x) = [4 alpha^3/ pi]^1/4 x_c - alpha_x^3 taking c_1 = [4 alpha^3/ pi]^1/4 psi(x) = c_1 x_e^- alpha x^2 the Schrodinger time independent wave eqn is d^2 psi(x)/dx^2 + 2 m/ h^2 c(- v) psi(x) = 0 d^2 (c_1 x e^-alpha x^2)/ d x^2 +_2m/h^2 [E - 1/2 m w^2 x^2] x(c_1 x_e^-alpha x^2] = 0