Kronig-Penney model only with cosh

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In summary, the conversation discusses the use of A'cosh(Qx)+B'sinh(Qx) as the approach for eigenfunctions in regions of high potential energy for bound states. It is suggested that using Acosh(Qx+m) instead may be a better approach, but the exp(ikx) term in the wavefunction breaks the symmetry for states with different wavevector k. It is then questioned whether only the probability density needs to be mirror symmetric, not the wavefunction itself. The speaker also mentions that sinh^2 is mirror symmetric around the origin and asks for clarification on the specific Hamiltonian and wavefunctions being discussed.
  • #1
8Apeiron8
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Hey!

Why i can't use as approach for the eigenfunctions in the regions of high potential energy for bound states only Acosh(Qx+m) instead of A'cosh(Qx)+B'sinh(Qx), where m is the midpoint of the wall?

If i put the origin of the coordinate system in this midpoint and use the mirror-symmetry of the chrystal, the sinh term must vanish.

Many thanks,

Felix
 
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  • #2
You can do so for the state of lowest energy or k=0. For states with different wavevector k, the symmetry is broken by the exp(ikx) term.
 
  • #3
Sry but i don't understand this. The exp(ikx) term is part of the approach for the region with low potential energy, why it breaks the symmetry in the other region?

Could the reason be, that only the probability density must be mirror symmetric but not the wavefunction?

sinh^2 is mirror symmetric around the origin!
 
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  • #4
Maybe you could specify exactly the hamiltonian and the wavefunctions you are talking about?
 

Related to Kronig-Penney model only with cosh

1. What is the Kronig-Penney model?

The Kronig-Penney model is a mathematical model used to describe the behavior of electrons in a periodic potential. It was developed by physicists Ralph Kronig and Walter Penney in the 1930s.

2. How is the Kronig-Penney model different when using cosh instead of cos?

The Kronig-Penney model with cosh is a modified version of the original model, where the potential function is replaced with a hyperbolic cosine function (cosh) instead of a cosine function (cos). This modification allows for a more accurate representation of certain physical systems, such as ionic crystals.

3. What are the advantages of using cosh in the Kronig-Penney model?

Using cosh in the Kronig-Penney model allows for a more realistic description of the electron behavior in certain systems, as it takes into account the repulsive forces between ions in a crystal lattice. It also allows for better predictions of energy band structures and electronic properties.

4. What are the limitations of the Kronig-Penney model with cosh?

One limitation of the Kronig-Penney model with cosh is that it assumes a perfectly periodic potential, which may not be accurate for all physical systems. It also does not take into account other factors that may affect electron behavior, such as electron-electron interactions or external fields.

5. How is the Kronig-Penney model with cosh used in practical applications?

The Kronig-Penney model with cosh is often used in condensed matter physics to study the electronic properties of materials, such as semiconductors and metals. It can also be used in the design and optimization of electronic devices, such as transistors and solar cells.

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