Question

Question 2: A particle of mass m moves in a potential energy U(x) that is zero forェ* 0 and is-oo at r-0. This is am attractive delta function, very odd. Do not worry about the physical meaning of the potential, just roll with it for now. The system is described by the wave function Afor <0 where a is a real, positive constant with dimensions of 1/Length, and A is the normalization constant, treat it as a unknown complex-number for the time being. a) Determine the energy E of the particle in terms of h, m, a, and numbers. Hint: Since the energy is a constant by definition it doesnt matter where in the avaliable space you evaluate for it! b) Sketch the probability density. Indicate on your plot where is located. Also, indicate on your plot where you would expect the particle to be located if you had to guess.

Answer 1

Given A particle of mass 'm' moves in a potential energy u(x) that is zero for x notequalto 0 and is - infinity at x = 0 psi (x) {Ae^alpha x for x < 0 Ae^alpha x for x greaterthanorequalto 0 (A) A particle of mass m moves in a potential energy U(x) = S(x) psi (x) = {Ae^alpha x for x < 0 Ae^- xx for x greaterthanorequalto 0 Here _h^2/gm .. nabla^2 psi + s (x) psi (x) = E psi (x) for x notequalto 0, - h^2/gm nabla^2 psi (x) = E psi (x) for x < 0 rightdoublearrow - h^2/gm (A alpha^2 e^alpha x) = E.Ae^- alpha x \r\n rightdoublearrow - h^2/gm A alpha^2 = E for x greaterthanorequalto 0 rightdoublearrow - h^2/gm (A^2 alpha^2 e^- alpha x) = F.Ae^- alpha x rightdoublearrow E = -h^2/gm A alpha^2 The energy of particle is E = -h^2/gm A alpha^2 (B) at x = 1/alpha wave function reduced by 1/e Now if we Guess, particle must be located at the x = 0