Calculating Transmissivity and Reflectivity Coefficients of a Potential

[bspoka]

Hey people,

So I have a problem where i have to find the transmitivity and reflectivity coefficients of a potential:

V=-(n^2-1)E where (E is positive, constant and is the energy of the particle) Note: V IS NEGATIVE
for n=1,2,...N.

For each "n" the region is of fixed length "a" except first and last one which go off to infinities.

I was thinking of approximating the staircase of the potential as a harmonic oscillator given N is large but i don't really know how that problem should be solved, it's definitely not easier :)
i.e. V=0 for x<0
=1/2m(w^2)(x^2) for 0<x<Na
=(N^2-1)E for x>0

Any help is appreciated!

Cheers

 

[Jhero]

Hello,

Thank you for sharing your problem with us. I understand the importance of finding the correct coefficients for a potential. Let's break down the problem and see if we can come up with a solution together.

Firstly, let's define the transmitivity and reflectivity coefficients. The transmitivity coefficient, denoted as T, is the ratio of the transmitted wave to the incident wave. In other words, it tells us how much of the wave is able to pass through the potential. The reflectivity coefficient, denoted as R, is the ratio of the reflected wave to the incident wave. This tells us how much of the wave is reflected back.

In order to find these coefficients, we need to understand the behavior of the wave as it encounters the potential. As you mentioned, the potential has a staircase shape with fixed length regions of a, except for the first and last regions which go off to infinity. This can be approximated as a harmonic oscillator, but we need to keep in mind that the potential is negative for all values of n.

To solve this problem, we can use the Schrödinger equation, which describes the behavior of a quantum particle in a potential. We can solve this equation for each region (x<0, 0<x<Na, x>Na) and then match the solutions at the boundaries to find the coefficients. This approach will give us a set of equations that we can solve for T and R.

Another approach could be to use the transfer matrix method, which is a powerful tool for solving problems involving potentials. This method involves breaking down the potential into smaller regions and using matrix operations to find the coefficients. However, this method may be more complex for your specific potential, so I would recommend starting with the Schrödinger equation approach first.

I hope this helps get you started on solving your problem. Remember to always consider the physical interpretation of the coefficients and make sure they make sense in the context of your potential. Good luck!