9780030839931, Chapter 12, Problem 6P

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Chapter 12, Problem 6P Problem The validity of the semiclassical result k(t) k(0) -eEt/h for an electron in a uniform electric field is strongly supported by the following theorem (which also provides a useful starting point for a rigorous theory of electric breakdown): Consider the time-dependent Schrödinger equation for an electron in a periodic potential U(r) and a uniform electric field: = + (12.78) 2m Suppose that at time 0, 0) is a linear combination of Bloch levels, all of which have the same wave vector k. Then at time t, t) will be a linear combination of Bloch levels, all of which have the wave vector eEt/h. Prove this theorem by noting that the formal solution to the Schrödinger equation is 0), (12.79) and by expressing the assumed property of the initial level and the property to be proved of the final level in terms of the effect on the wave function of translations through Bravais lattice vectors. Step-by-step solution Step of 4 The time-dependent Schrödinger equation is, 2m (1) This equation is a partial differential equation in four variables, the three position ordinates of the particle x, y, and the time 1. here the potential energy U is independent of time and depends only on position. Here, represents standing wave. So, U is independent of time, the position and time co-ordinates are separate in equation (1). Thus, may be expressed in the form of, (2) Here, is independent of time and u(t) is the independent of position, x, and Step of 4 Putting the value of from (2) in equation (1), we get, 2m at Dividing throughout by we get, 2m u(t) (3) In the above equation the left hand side is independent of time, while right hand side is independent of the co-ordinates (x,y,z). Step of 4 So, if above equation is to be satisfied, each side of equation (3) must be equal to a constant H (say), so that we have, 2m (4) Or, (5) From equations (3) and (4), ih (6) u(t) The solution of equation (6) may be written as, As we compare this equation with that for a plane wave, H must have the dimension of energy. Thus we can say that the constant H designates the energy of the particle which is described by the solution of Schrödinger equation. Step of 4 Putting the value of u (t) in equation (2), we get, (7) Here, is the wave function when = and 1=1 and is the wave function which is equal to when x=x,y=y,z=z = and So putting for and for in the equation we get, In general, Or, ( Since, and From the above observation we conclude that, the formal solution to the Schrödinger equation IS Therefore we can derive the result of semi-classical equation of motion directly from quantum mechanics.