Exploring the Meaning of k, k1, and k2 in Energy Gap Theory

[jasum]

Hi, I just get confused on the some definations in the energy gap theory.

in the equation
$$\cos {k_1a} \cos{k_2b} - \frac{k_1^2 + k_2^2}{2k_1k_2} \sin {k_1a} \sin{k_2b} = \cos {kd}$$
where
$$\frac {\hbar ^2 k_1^2}{2m} = E$$
$$\frac {\hbar ^2 k_2^2}{2m} = E - V$$

however the eigenergies
$$\frac {\hbar ^2 k^2}{2m} = E$$

Here I just get confused
what is the physical meaning of k1,k2 and k?
and what is the difference of the energy calculated from k and k_1?

 

[Jhero]

Hello,

I understand your confusion with the definitions and equations involved in the energy gap theory. Allow me to explain the physical meaning of k1, k2, and k in the context of this theory.

In this equation, k1 and k2 represent the wave vectors of two different particles in a crystal lattice. These wave vectors are related to the momentum of the particles and are used to describe their behavior in the crystal. k, on the other hand, represents the total wave vector of the system, which is the sum of k1 and k2.

The difference between the energies calculated from k and k1 lies in their physical interpretation. The energy calculated from k represents the total energy of the system, while the energy calculated from k1 only represents the energy of one of the particles in the system. This is because k1 and k2 represent the individual behaviors of each particle, while k represents the overall behavior of the system as a whole.

I hope this helps to clarify the definitions and their physical meanings in the energy gap theory. If you have any further questions or concerns, please don't hesitate to ask. it is my job to help others understand complex theories and concepts. Best of luck in your studies!