Question

Assume t=0 for the following wavefunction, , then , and show with the potential energy function V = that the wavefunction has definite energy

Answer 1

Given psi(n, t0 = A e^- b x^2, A = (2b/pi)^1/2 Using Time Independent sch. equation. - h^2/2 m partial differential^2/partial differential x^2 psi + v psi = E psi - h^2/2 m A partial differential^2/partial differential x^2 (e^- b x^20 + A v e^- b x^2 = E A e^- b x^2 - h^2/2 m [partial differential/partial differential x (- 2 b x e^- b x^2)] + v e^- b x^2 = E e^- b x^2 - h^2/2 m [(- 2b0 [I, e^- b x^2 + x. (- 2 b x) e^- b x^2}] = v e6- b x^2 = E e^- b x^2 h^2 b/m (e^- b x^2 - 2 b x^2 e^- b x^2) + v e^- b x^2 = E e^- b x^2 h^2 b/m[1 - 2 b x^2]