Wave eqtion of electron in a periodic potential

In summary, the potential energy U(x) is equal to the sum of UG times e^iGx and e^-iGx, which is necessary for it to be a real function. This is explained in Kittel text 8th edition on pages 170-171.
  • #1
JayKo
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quoted from Kittel text 8th edition page 170,171
the potential energy is U(x)=[tex]\Sigma[/tex][tex]U_{G}e^{iGx}[/tex]

but why it is also equal to-> U(x)=[tex]\Sigma[/tex][tex]U_{G}e^{iGx}+e^{-iGx}[/tex]


from text explanation, i couldn't get it why there is an extra term of [tex]e^{iGx}[/tex]

any 1 care to enlight me on this? thanks.
 
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  • #2
JayKo said:
quoted from Kittel text 8th edition page 170,171
the potential energy is U(x)=[tex]\Sigma[/tex][tex]U_{G}e^{iGx}[/tex]

but why it is also equal to-> U(x)=[tex]\Sigma[/tex][tex]U_{G}e^{iGx}+e^{-iGx}[/tex]


from text explanation, i couldn't get it why there is an extra term of [tex]e^{iGx}[/tex]

any 1 care to enlight me on this? thanks.

There is an extra term e-iGx to make potential energy U(x) a real function.
 

What is the wave equation of an electron in a periodic potential?

The wave equation of an electron in a periodic potential is known as the Schrödinger equation. It describes the behavior of a particle in a periodic potential and is a fundamental equation in quantum mechanics.

What is the significance of solving the wave equation of an electron in a periodic potential?

Solving the wave equation of an electron in a periodic potential allows us to understand the behavior of electrons in crystalline materials, such as metals and semiconductors. This provides important insights into the properties of these materials and their applications in technology.

How is the wave equation of an electron in a periodic potential solved?

The wave equation is typically solved using mathematical techniques such as perturbation theory, which involves approximating the solution by breaking it down into smaller, solvable parts. Other methods, such as the WKB approximation, can also be used to solve the wave equation.

What are some real-world applications of the wave equation of an electron in a periodic potential?

The wave equation is used in a wide range of applications, including the design and development of electronic devices such as transistors, lasers, and solar cells. It is also used in materials science to understand and predict the properties of materials at the atomic level.

Are there any limitations to the wave equation of an electron in a periodic potential?

Like any mathematical model, the wave equation has its limitations. It assumes a perfect, periodic potential, which may not accurately reflect the complexities of a real material. Additionally, it does not take into account relativistic effects, which are important for high-energy particles. However, despite these limitations, the wave equation remains a powerful tool for understanding the behavior of electrons in periodic potentials.

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