Question

How to find approximate eigenstates of the Hamiltonian with equation.

Answer 1

To find approximate eigenstates of the Hamiltonian, we can use a linear combination of the atomic orbitals

$|k\rangle =\frac{1}{\sqrt{N}}\sum_{n=1}^N e^{inka} |n\rangle$

where N = total number of sites and k is a real parameter with -

$\frac{\pi}{a}\leqq k\leqq\frac{\pi}{a}$

can be expressed as

$\langle n|H|n\rangle= E_0 = E_i - U$

$\langle n\pm 1|H|n\rangle=-\Delta$

$\langle n|n\rangle= 1 \ ;   \langle n \pm 1|n\rangle= S \ .$

The overlap between states on neighboring atoms is S. We can derive the energy of the state |k\rangle using the above equation:

equation:

$H|k\rangle=\frac{1}{\sqrt{N}}\sum_n e^{inka} H |n\rangle$

$\langle k| H|k\rangle =\frac{1}{N}\sum_{n,\ m} e^{i(n-m)ka} \langle m|H|n\rangle$

for example,

$\frac{1}{N}\sum_n \langle n|H|n\rangle = E_0 \frac{1}{N}\sum_n 1 = E_0 \ ,$

Answer 2

Answer:

bhai baat nhi karni thi to bata dete block mar diya