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By Klaus Ziock
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If such a pulse is put through a circuit whose response is frequency dependent, the circuit will behave exactly as if it had been subjected to a superposition of sine waves covering the frequency range indicated by the Fourier integral. 3. 32 thus expresses a connection between the spatial eigenfunctions 1p(x) and the momentum eigenfunctions cp(k) or, in other words, between the probability that the particle is in a certain place x ± ~x and that it has a certain momentum Px ± ~Px' Obviously there must, then, exist a connection between Eq.
We assume that they are stationary and that they provide a periodic potential with a period d in which the electrons move. (b) The electrons are very nimble, and able to move out of each others way. We assume that the electric potential created by the electrons is the same everywhere in the crystal. These assumptions reduce our problem to the study of the motion of an individual electron in a superposition of a constant and a periodic potential. To further ease the solution, we make some mathematical simplifications.
46) This is usually called the asymptotic solution of Eq. 44. 47) x2 /2J(x) into Eq. 44, hoping that this will simplify our problem. 48 ) 6 We can easily convince ourselves that this has, indeed, the dimension of a length. 4. 40 SOME SIMPLE PROBLEMS Substituting Eqs. 48 into Eq. f(x) = f(x) + 2xf'(x) or f"(x) - 2xf'(x) - f"(x) - x2f(x) + f(x)(J.. 52) ao f"(x) =! 2 ao + 2)(k + 1)Ak+2Xk k(k - I)A kx k- 2 = ! 53) We substitute Eq. 53 into Eq. 50. ao ! (k + 2)(k + 1)Ak+2xk - 0 ao 2! kAkXk 0 + (J.. - ao I)!