Question

mechani mie The potential energy barrier shown below is a simplified model of thec electrons in metals. The metal workfunction (Ew), the minimum energy required to remove an electron from the metal, is given by Ew-,-E where 1s the height of the potential energy barrier and E is the energy of the electrons near the surface of the metal. The potential energy barrier is = 5 eV V(x) V=0 (a) The wavefunction of an electron on the surface (x< 0) of the metal is, Ψ(x, t) = Ae+,(kr-5.3x10-st) Calculate the energy (E) of the electron and the metal workfunction (Ew). Determine the value of the wavenumber (k) (b) By direct substitution into the time-independent Schröedinger's equation, show that is the wavefunction for the particle in the barrier region (x20). Write down the equation for the probability density of finding the electron in the barrier region (x 2 0) By defining the tunneling depth δ as the distance from the boundary at which the probability density has decreased by a factor of e-1, calculate δ for this metal.