Question

Question 21 Consider a free electron in one dimension (i.e. an electron free to move along say the x-direction on (a) The time-independent Schrödinger equation is Αψη (x)-Εηψη (x), where is the Hamiltonian (total energy) operator, and ψη (x) are the electron wave functions associated with energies En Assuming the electron's energy entirely comprises kinetic energy (as it is 'free' there is no potential energy term), write down the Schrödinger equation given that the momentum operator in one- dimension is i- -ih, where i -I and h h/(2T) is the reduced Planck constant. Show that your result can be simplified to: equation 1 dx2 and write out the resulting expression for k,. (2 marks) (b) By inspection, write down the four functions for ^n(x) that are solutions to equation 1. Only two of these solutions are correct for this particular problem, which are they? [Hint: remember that the wave function is related to the probability of finding the electron at position x, and since there are no regions of space where the free electron cannot be, the correct electron wave functions for this problem cannot ever be zero]. (2 marks) (c) By comparing the classical expression for kinetic energy, with the relationship between E and k found above, show that the electron's momentum is given by hk. (1 mark) (d) Sketch a graph of electron energy vs momentum; this is known as a dispersion curve Demonstrate by reference to the relationship between E and k found above that the electron mass is related to the second derivative of the energy with respect to the k, ie the curvature of the dispersion curve. (2 marks) max 7 marks]

Answer 1

"(a_ans): Given: - H^^Psi_n (x) Now, H^^= K^2/2^m (for electron V = 0) Therefore H^^= 1/2.m (- i h/1 d/dx) = d^2 h^2/2 m d^2/dx^2 = -h^2/2 m. d^2/dx^2 H^^Psi_m