Question

2. (a) Are the eigenfunctions of H for the particle in the one-dimensional box also eigenfunctions of the position operator ? (b) Calculate the average value of x for the case where n 4. Explain your result by comparing it with what you would expect for a classical particle. Repeat your calculation for n = 6 and, from these two results, suggest an expression valid for all values of n. How does your result compare with the prediction based on classical physics?

Answer 1

The wave function of Given wave function is psi (x) = squareroot 2/L sin (n pi x/L) Apply the Hamiltonian operated on wave function psi (x), H^cap psi (x) = (negative h^2/2 m d^2/dx^2 + v_2) (squareroot 2/L sin (n pi x/L)) For free particle V_0 = 0 H^cap psi (x) = negative h^2/2 m d^2/dx^2 (squareroot 2/L sin (n pi x/L)) = h^2/2 m [(n pi/L)^2 squareroot 2/L sin (n pi x/L)] = h^2/2m (n pi/L)^2 psi (x) = (h^2/2m (n pi/L)^2) psi (x) Therefore, H^cap is Eigen function with Eigen value h^2 /2 m (n pi/L)^2