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).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f68f18d-fa59-4aa0-aa9f-781b110a4410%2F7ab78821-c217-4a08-b7e8-a5e2f2bb54af%2F22kp1xb_processed.png&w=3840&q=75)
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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