Chain with a Basis of Two Identical Atoms

Consider the chain shown in Fig.2.13 containing N unit cells with lattice parameter a and nearest-neighbour distances b and a - b where b a - b. Such a structure is a caricature of a hydrocarbon chain with alternating single and double bonds. If we put m = m = m in the chain of Fig.2.1, we obtain the model considered here. 

Linear chain with basis of two identical atoms of masses m and nearest-neighbour force constants f and g. a is the lattice parameter and b a - b 

We have put x(l) = 0 and x(2) = b in (2.53). Based on these normal coordinates, make a figure of the patterns of atomic displacements. Note that the displacements in Q( ) and Q( 2 ) are such that only the bonds with force constants g and f, respectively, are changed. 

3 Probability Densities of a Classical and Quantum Mechanical Qsci11ator 

We derive the results shown in Fig.2.8b. We require that the classical oscillator has the same total energy E as the quantum mechanical oscillator in its ground state n = 0. From (2.79,84), we therefore have 

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