Energy Gap of Electron in Dirac Comb Potential

[hulet]

Pg 202 of Griffiths Introduction to Quantum Mechanics

Homework Statement

Hi, I am trying to solve a question involving band structure of solids... The electron is placed in an area filled with positive ions which is represented by a dirac delta potential...

Homework Equations

The problem asks me to find the equivalent of the Kronig-Penney equation which I found to be Cos[2 n Pi /N] = Cos[ka] - (Beta/ka) Sin[ka] and then asks me to:
a) find values for Beta for which half the states in the first band have negative energy
b) find values for Beta for which all states in the first band have negative energy
c) for Beta=5/2 what is the size of the energy gap between the 1st and 2nd energy band

The Attempt at a Solution

I have tried plotting the above equation for different values of Beta...

I understand that there are forbidden energies for any value of the RHS below -1 or above +1... I also understand that each value of n gives us a different energy level.

but I don't understand how the graph of the RHS of the above equation tells us whether the energy of the particle is positive or negative??

i have read kronig penney threads on this forum but none describe positive or negative energies... thanks for the help

 

[Jhero]

!

Dear student,

Thank you for your question. The Kronig-Penney equation is a useful tool for understanding the band structure of solids, but it can also be a bit confusing at first. Let me try to clarify some of your concerns.

Firstly, the graph of the right-hand side (RHS) of the Kronig-Penney equation does not directly tell us whether the energy of the particle is positive or negative. Rather, it tells us about the allowed energy levels for the particle. As you correctly noted, the Kronig-Penney equation has forbidden energies for any value of the RHS below -1 or above +1. These forbidden energies correspond to energy levels that are not allowed for the particle. This is because the particle cannot exist in regions where the potential is infinite (represented by the delta function in this case).

Now, to answer your specific questions:
a) To find values for Beta for which half the states in the first band have negative energy, you need to find values for Beta that give you a negative energy for half of the energy levels in the first band. This can be done by substituting different values of n into the Kronig-Penney equation and solving for Beta. You will need to keep in mind that for each value of n, there are two energy levels (one positive and one negative). So, for example, if you have a total of 8 energy levels in the first band, you will need to find values of Beta that give you a negative energy for 4 of those levels.

b) Similarly, to find values for Beta for which all states in the first band have negative energy, you need to find values of Beta that give you a negative energy for all the energy levels in the first band.

c) For Beta=5/2, the size of the energy gap between the 1st and 2nd energy band can be found by substituting Beta=5/2 into the Kronig-Penney equation and solving for ka. The energy gap can then be calculated by taking the difference between the energy levels at ka=0 (the bottom of the 1st band) and ka=π (the top of the 2nd band).

I hope this helps clarify some of your questions. If you need further assistance, please don't hesitate to ask for help. Good luck with your studies!