Question

P.18.26 Verify that the wave function for the first excited state of the harmonic oscillator is a solution of the Schroedinger equation for that system. Determine the energy eigenvalue.

Answer 1

Wave function of the harmonic oscillation is T_n(x) = squareroot alpha/2^n n! squareroot pi e^- 1/2 alpha^2 x^2 H_n(alpha x) 1st excited state n = 1 T_1 (x) = squareroot alpha/2^n n! squareroot pi e^- 1/2 alpha^2 x^2 H_1 (alpha x) Here alpha = squareroot mw/h H_1 (alpha x) = 2 alpha x T_1 (x) = A e^- 1/2 alpha^2 x^2. 2 alpha x Schrodinger equation H T_1 = E_1 T_1 Rightarrow - h^2/2m d^2 T_1/dx^2 + V T_1 = E T_1 ---- (1) Here d^2 nsim_1/d x^2 = ? d T_1/dx = A(- 1/2 alpha^2 x^2)e^- 1/2 alpha^2 x^2. 2 alpha x + A e^- 1/2 alpha^2 x^2. 2 alpha d^2 T_1/dx^2 = A(- 1/2 alpha^2. 2)e^- 1/2 alpha^2 x^2.2 alpha x + A(- 1/2 alpha^2 x^2)^2 e^- 1/2 alpha^2 x^2. 2 alpha x + A(- 1/2 alpha^2 x 2)e^- 1/2 alpha^2 x^2.2 alpha + A(- 1/2 alpha^2 2 x)e^- 1/2 alpha^2 x^2. 2 x = -alpha^2 T + (- alpha^2 x)^2 T + (- alpha^2 x) times 1/x T + (- alpha^2 x) times 1/x T = (- alpha^2 + alpha^4 x^2 - alpha^2 - alpha^2)