By Bunge M.

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**Example text**

For a typical 1-D energy band, sketch graphs of the relationships between the wave vector, k, of an electron and its: (a) energy, E k 0 (b) group velocity, Vg k 0 (c) and effective mass. m* k 0 10 - 2 d. The approximate density-of-states D(1) (E) ) for the energy band of part a aboveis (1) D (E) k 0 10-4. The E(k x) vs. k x dependence for an electron in the conduction band of a one-dimensional semiconductor crystal with lattice constant a = 4 Å is given by: E (kx ) = E 2 −( E 2 − E1 )cos 2[k x a /2] ; E 2 > E1 (a) .

An electrical charged particle with a spin angular momentum will have a magnetization proportional to the spin angular momentum. Suppose the averaged expectation value of the magnetization of r r M = N Trace [ ρˆ (γ Sˆ ) ] . (a) the medium considered in Problem 11-1 above is The three Cartesian components of the magnetization in terms of the appropriate density- matrix elements as in Problem 11-1 above are: Mz = N γh ( ρ11 − ρ22 ) 2 , My = i N γh (ρ12 − ρ 21 ) 2 , 11 - 1 Mx = (b) N γh (ρ12 + ρ 21 ) 2 .

Suppose the averaged expectation value of the magnetization of r r M = N Trace [ ρˆ (γ Sˆ ) ] . (a) the medium considered in Problem 11-1 above is The three Cartesian components of the magnetization in terms of the appropriate density- matrix elements as in Problem 11-1 above are: Mz = N γh ( ρ11 − ρ22 ) 2 , My = i N γh (ρ12 − ρ 21 ) 2 , 11 - 1 Mx = (b) N γh (ρ12 + ρ 21 ) 2 . 16), the dynamic equations describing the precession of the magnetization r M around such a magnetic field are: d i (ρ11 − ρ22 ) = − 2 [H12 ρ 21 − ρ12 H21] = γ [i H x (−ρ12 + ρ21)+ H y ( ρ12 + ρ 21)] dt h , which can be shown to be [ r r d M z = γ [−H x M y + H y M x ] = γ M × H dt ] , z making use of the results in (a) above.