Question

Parity (please answer from part a to part d)

Consider Infinite Square Well Potential,

V(x) = 0 for |x| < 1/2a and V(x) = infinity for |x| > 1/2a

a) Find energy eigenstates and eigenvalues by solving eigenvalue equation using appropriate boundary conditions. And show orthogonality of eigenstates.

For rest of part b to part d please look at the image below:

Problem 1 . Parity Consider an infinite square well potential, V(x) = 0 for lxl 〈 a and V(x) = oo for l 2 > a 2 a) Find the energy eigenstates and eigenvalues by solving eigenvalue equation using appropriate boundary conditions.(>kw *hytyf b) Consider the initial wave function Ψ(x,0)-A ( eenstafes) -x2) 4 Normalize and find y(x, t) c) Find 〈x) at t = 0 and at tヂ0. Are they same or not? Explain your result. d) Show that the uncertainty Δx < ½ a in any energy eigenstate. Using the uncertainty principle for position and momentum find the lower bound of <HE in an energy eigenstate. (This problem explicitly demonstrates that the lowest energy is positive due to uncertainty principle.)

Answer 1

"Potential v(x) = {0 - 4 < x < a infinity otherwise} this is symmetric potential, so the soln of this problem will be either even or either cold schrodinque eqn d^2 psi/d x^2 + partial differential m/h^2 [E - v] psi (x) = 0 In region - a < x < a, d^2 psi/d x^2 + partial differential mE/h^2 psi = 0 (D^2 + k^2) psi (x) = 0 Because D^2 = d^2/dx^2 & k^2 = 2 ME/h^2 straightline (1) D^2 + k^2 = 0 D = plusminus ik psi_even = a (km) & psi_odd (x) = b sin Kx Boundary condition psi_even (a) = 0 & psi_odd (a) = 0 psi_even (- a) = 0 & psi_odd (- a) = 0 for even psi_even (a) = 0 A = 0 A plusminus 0, so, k^a = n pi/2 k = n pi/2a straightline (2) psi_even = A (n pi x/2a) Normal equation integral^a_- a (A (n pi x/2a) dx = 1