Question

Calculate the momentum space wave function for the lowest energy eigenstate of the infinite square well. Plot the momentum probability density

Please make sure you go through the steps and not just write down the formula. Need some conceptual understanding.

Thank you!

Answer 1

Ground state of infinite square well is psi(x) = squareroot 2/L sin(pi/L)x {Length of well) Momentum space wave function obtain by fourier transform phi(P) = 1/squareroot 2 pi h integral^a_0 e^-iPx/h psi(x) dx = 1/squareroot 2pi h squareroot 2/L integral^a_0 e^-ipx/h sin(pi/L)x dx since, integral e^ax sin bx dx = 1/a^2 + b^2 e^ax {a sin bx - b cos bx} = 1/squareroot pi hL 1/[(-iP/h)^2 + (pi/L)^2 e^-iPx/h [(-iP/h) sin(pi x/L) - (pi/L) cos pi x/L] Vertical-bar ^a_0 = 1/squareroot pi hL 1/[-P^2/h^2 + pi^2/L^2] {e^-iPa/h (pi/L) - e^0 (-pi/L)} = pi/L squareroot 1/pi hL h^2 L^2/[-P^2 L^2 + pi^2 h^2] {1 + e^0iPa/h} = (pi L h^3)^1/2/(pi^2 h^2 - P^2 L^2) [1 + e^iPa/h] phi(P) = squareroot pi L h^3/(pi^2 h^2 - P^2 L^2) [1 + e^-iPa/h] Momentum probability density P = Vertical-bar phi(P) Vertical-bar ^2 = phi(P) phi(P) = pi Lh^3/(pi^2 h^2 - P^2 L^2)^2 [(1 + e^+ iPa/h) (1 + e^iPa/h)] = 2pi L h^3/(pi^2 h^2 - P^2 L^2)^2 [1 + cos(Pa/h)]