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By André Moliton; Wiley InterScience (Online service)

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E. 11]. 4. e. 1. 3. 3. Molecular wire of length L The Free Electron and State Density Functions 25 For a point with coordinate x, the probability is the same whatever the number of turns made, so we can write \ (x ) \ (x  L ) . Generally, after making n turns of length L we would end up at the same point, so we can write \ (x ) \ (x  nL ) where n is an integer. 16] is called the periodic boundary condition (PBC) or the Born-von Karman condition. When x = 0, it can be simplified so that: \ (0) \ (L ).

E. inclusively between E and E + dE in energy space). The upshot is that if F(E) is the occupation probability of a level denoted E, then the number N(E) dE of electrons distributed in the energy space between E and E + dE is equal to N(E) dE = Z(E) F(E) dE. 2. 1. 1a, the Hamiltonian is such that H (x) = H(– x), because V(x) = V(– x) and d² d² dx ² d ¬ª  x ² ¼º . If I denotes the inversion operator, which changes x to – x, then IH(x) = H(– x) = H(x). H(x) being invariant with respect to I, the proper functions of I are also the proper functions of H (see Chapter 1).

1st asymm. 1st symm. 1st symm. 7. Evolution of energy levels and electronic states on going from one isolated well to two close wells The example given shows how bringing together the discrete levels of the isolated atoms results in the creation of energy bands. The levels permitted in these bands are such that: – two wells induce the formation of a “band” of two levels; – n wells induce the formation of a “band” of n levels. 1. 1. 1. It can also be described by V = V0 = 0. For a free electron placed in a potential V0 = 0 with an electronic state described by its proper function with energy and amplitude denoted by E0 and \ 0 , respectively, the Schrödinger equation for amplitude is: '\ 0  2m =² E 0\0 0.

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