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1. Tight-binding approximation Consider a wave function (x) in an atom. When the atoms form a solid (lattice constant a), we assume the wave function (x) = Σeiknap(x-na). (1) where the atomic wave function is copied to places na (with integer n) multiplied by the phase factor eina. Calculate the energy expectation value in state (1) using where the definitions Ek and the approximations n h² d² H = +U(x). (3) 2m dr² The denominator of (2) takes into account that (1) is not normalized. In the calculation use the normalization [ ø*(x)ø(x)dx = 1, S*(x) Hy(x) dx &*(x)(x) dx' [$(x± [ ¢*(x)Hø(x)dx = =-α o* (x±a) Ho(x) dx = -3 (2) [ ø*(x ±na)Hø(x)dx = 0, n≥2 [ ó*(x±na)ó(x)dx=0, n≥1. (4) (5) (6)

1. Tight-binding approximation Consider a wave function (x) in an atom. When the atoms form a solid (lattice constant a), we assume the wave function (x) = Σeiknap(x-na). (1) where the atomic wave function is copied to places na (with integer n) multiplied by the phase factor eina. Calculate the energy expectation value in state (1) using where the definitions Ek and the approximations n h² d² H = +U(x). (3) 2m dr² The denominator of (2) takes into account that (1) is not normalized. In the calculation use the normalization [ ø*(x)ø(x)dx = 1, S*(x) Hy(x) dx &*(x)(x) dx' [$(x± [ ¢*(x)Hø(x)dx = =-α o* (x±a) Ho(x) dx = -3 (2) [ ø*(x ±na)Hø(x)dx = 0, n≥2 [ ó*(x±na)ó(x)dx=0, n≥1. (4) (5) (6)

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1. Tight-binding approximation Consider a wave function (x) in an atom. When the atoms form a solid (lattice constant a), we assume the wave function (x) = Σeiknap(x-na). (1) where the atomic wave function is copied to places na (with integer n) multiplied by the phase factor eina. Calculate the energy expectation value in state (1) using where the definitions Ek and the approximations n h² d² H = +U(x). (3) 2m dr² The denominator of (2) takes into account that (1) is not normalized. In the calculation use the normalization [ ø*(x)ø(x)dx = 1, S*(x) Hy(x) dx &*(x)(x) dx' [(x ± [ ¢*(x)Hø(x)dx = =-α o* (x±a) Ho(x) dx = -3 (2) [ø*(x ±na)Hø(x)dx = 0, n ≥2 [o¹(x ± *(x±na)o(x)dx=0, n ≥ 1. (4) (5) (6) [Hint: calculate the numerator and the denominator of (2) separately. Write both in the double-sum form Σn' En An'n and calculate all An'n making use of the periodicity of the lattice potential U(r). Assume that the number of atoms N in the chain is large, which allows you to neglect all possible difficulties caused by the ends of the chain. You are expected to get the result Ek = -a -26 cos(ka).]
Transcribed Image Text:1. Tight-binding approximation Consider a wave function (x) in an atom. When the atoms form a solid (lattice constant a), we assume the wave function (x) = Σeiknap(x-na). (1) where the atomic wave function is copied to places na (with integer n) multiplied by the phase factor eina. Calculate the energy expectation value in state (1) using where the definitions Ek and the approximations n h² d² H = +U(x). (3) 2m dr² The denominator of (2) takes into account that (1) is not normalized. In the calculation use the normalization [ ø*(x)ø(x)dx = 1, S*(x) Hy(x) dx &*(x)(x) dx' [(x ± [ ¢*(x)Hø(x)dx = =-α o* (x±a) Ho(x) dx = -3 (2) [ø*(x ±na)Hø(x)dx = 0, n ≥2 [o¹(x ± *(x±na)o(x)dx=0, n ≥ 1. (4) (5) (6) [Hint: calculate the numerator and the denominator of (2) separately. Write both in the double-sum form Σn' En An'n and calculate all An'n making use of the periodicity of the lattice potential U(r). Assume that the number of atoms N in the chain is large, which allows you to neglect all possible difficulties caused by the ends of the chain. You are expected to get the result Ek = -a -26 cos(ka).]
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