Question

6. Quantum waves. Suppose a quantum particle with speed v and mass m has energy E- mv*. a. Find the phase velocity of the quantum wave representing this particle. b. Is this velocity different from the velocity at which the particle transports mass and energy?
Please write the unit don’t write like this 10^ 7

Answer 1

The time independent Schrodinger equation can be written as -hstrok^2/2 m d^2 Psi/dx^2 = E Psi implies d^2 Psi/dx^2 + k^2 Psi = 0 where k = squareroot 2 mE/hstrok The general solution of the above equation can be written as Psi(x) = A e^i kx + Be^-ik x \r\n where A and B are contain multiplying the above solution to exap(-iEt/hstrok) we get, Psi(x, t) = Ae^(k x - E t/hstrok) + Be^-i(kx + E t/hstrok) The first part n right hand side represent wave travelling to right and the second part in right hand side represents wave travelling to left in general we can written Psi_k(x, t) = A e^I (k x - E/hstrok t) where k = plusminus squareroot 2 mE/hstrok {k > 0 implies right propagating wave k < 0 implies left propagating wave \r\n (a) phase velocity of the quantum wave in v_p = w/k comparing Psi_k (x, t) with e^i(kx - wt) we get w = epsilon/hstrok thus