ENEE 313, Fall. ’08Homework I - Due March 2, 20091. (15 pts.) Examine the hypothetical 2-D crystal in the figure above. The large black circlesand the smaller gray circles represent two different types of atoms. The atom arrangementcontinues indefinitely in both dimensions x and y, and only a portion is shown here for clarity.(a) Which of the five types of 2-D lattices is this crystal based on?(b) Identify the primitive cell and primitive vectors and draw them on the figure. Write theprimitive cell in terms of the unit vectors, ˆx and ˆy. The axis and the scale (in arbitraryunits) are included as a convenience.(c) Remember, ”Crystal = Lattice + Basis”, that is to say, a crystal is formed by placing anatom or arrangement of atoms, called the basis, at each lattice point. What is the basisfor this crystal with the lattice you identified? Describe it in terms of atoms and the unitvectors. Not e that one than more correct answer is possible.2. (60 pts.) Solve the following problems from the end-of-chapter problems of the textbook,Streetman and Banerjee, 6th edition: Problems 1.3, 1.13, 2.2, 2.5, 2.10, 2.12.(Turn over for the rest.)13. (10 pts.) Find the probability of finding a particle between x = L/3 and x = L/2, if the particleis at the third energy level (n=3) of an infinite potential well, with boundaries at x = 0 andx = L.4. (15 pts.) This question is about a free (unbound) electron.(a) Using the method of separation of variables, solve the Schroedinger Wave Equation (SWE)for a free electron in one dimension. ”Free electron” means that there is no potential field;V (x) = 0. Therefore the equation reduces to:−¯h22me∂2Ψ(x, t)∂x2= i¯h∂Ψ(x, t)∂t. (1)Assume the electron is moving in the positive x direction. Solve the full SWE, includingthe time dependency (i.e. find both ψ(x) and φ(t) to obtain Ψ(x, t) = ψ(x)φ(t)). Use kdefined such thatE =¯h2k22me(2)where E is the separation constant, and also the total electron energy.(b) Show that the k above is equivalent in this case to defining k as the wavevector, k = 2π/λ,if E is the total electron energy. (Hint: The electron only has kinetic energy, classicallydefined as (1/2)mev2. Think of the de Broglie wavelength of the electron moving withspeed v.)(c) The quantum-mechanical momentum operator is p = −i¯h∂∂x. The average (or expected)momentum of the free electron is then< p >=R∞−∞Ψ∗(x, t)p(Ψ(x, t))dxR∞−∞Ψ∗(x, t)Ψ(x, t)dx(3)Show that this gives the same result as could be obtained from the definition of E and kabove, i.e. < p >= pclassical=
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