1. Perturbation in lOne Dimension A free, non-interacting electron gas in one dimension is perturbed by a zero-mean potential of the form we) = 1..., 5(3: no). 1']. (a) Indicate on a carefully labeled sketch how the free electron dispersion will be modied by this weak interaction. (b) Find an expression for the energy 6,1 about which the nth gap occurs. 'Write down an approxima- tion to the wavefunction at the gap. using two unperturbed wavefunctions as a basis. Just give the general form. (c) Using the two-component wavefunction from b}. derive an expression for the magnitude an of the nth gap. ((1) Find the specic forms of the wavefunctions at the gap and indicate which has the higher energy. Explain physically why the energy of one is higher than that of the other. (e) Calculate the density of states as a function of energy. BE}, in the absence of the perturbation. Sketch how 19(6) will be modied by the interaction