How to Determine the Transmission Coefficient for a Rectangular Barrier?

In summary, the task at hand is to determine the transmission coefficient for a rectangular barrier, considering three cases: E<Vo, E>Vo, and E=Vo. This can be solved by solving the Schrodinger equation in three regions and ensuring continuity of solutions and their derivatives at the boundaries. While this method may be tedious, it is a common approach and can be found in many resources. Alternatively, one can also consider treating the even and odd solutions separately.
  • #1
maggicmike660
4
0

Homework Statement


Determine the transmission coefficient for a rectangular barrier. Treat seperately the three cases E<Vo, E>Vo, and E=Vo.

Homework Equations


V(x)= +Vo if -a<xa
V(x)= 0 otherwise

Transmission coefficient=(amplitude of transmited wave)2/(amplitude of incoming wave)2

I am also using the Time independent Schrodinger equation

The Attempt at a Solution



What I did for each case was to solve the Schrodinger equation in the following regions:

a) x<-a
b) -a<x<a
c) x>a

After doing this I exploited the fact that my solutions had to be continuous at the boundaries. The derivatives also had to be continuous at the boundaries.

ie)[tex]\Psi[/tex](a)-=[tex]\Psi[/tex](a)+ and,

d/dx([tex]\Psi[/tex](a)-)=d/dx([tex]\Psi[/tex](a)+

After doing this I get a bunch of equations and my goal is to isolate the amplitude of transmiited wave function and the amplitude of the initial wave function. I think that after a bunch of tedious algebra I will be able to isolate for the desired things.

Is there an easier way to isolate my amplitudes? My professor mentioned something about treating the even and odd solutions of the wavefunctions seperately, but I do not know what he means.

If any of this is unclear please let me know!
 
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  • #2
maggicmike660 said:
I think that after a bunch of tedious algebra I will be able to isolate for the desired things.

This method will work. I solved this problem before, and if I recall correctly, it did require very tedious algebra.

This problem is a common one in introductory QM. Many texts and websites give the final result without derivation, so you can consult those for guidance toward the final answer should get stuck.
 
  • #3
Thanks for the tip. I just got home after working on it and yes it does just require tedious algebra. Not difficult otherwise.
 

Related to How to Determine the Transmission Coefficient for a Rectangular Barrier?

What is a finite square well/barrier?

A finite square well/barrier is a physical system in which a potential energy barrier or well of finite width is present. It is often used as a simplified model in quantum mechanics to study the behavior of particles trapped inside the well or barrier.

What is the difference between a finite square well and a finite square barrier?

A finite square well refers to a potential energy well, where the potential energy inside the well is lower than the surrounding energy. On the other hand, a finite square barrier refers to a potential energy barrier, where the potential energy inside the barrier is higher than the surrounding energy.

What are the key features of a finite square well/barrier?

The key features of a finite square well/barrier include the width and height of the potential energy, the energy levels of the particles inside the well/barrier, and the probability of transmission and reflection of the particles at the boundaries of the well/barrier. They are also affected by the mass and momentum of the particles and the strength of the potential energy.

What is the significance of studying finite square wells/barriers?

Studying finite square wells/barriers is important because it allows us to better understand the behavior of particles in confined systems and how they interact with potential energy barriers. This has implications in various fields such as quantum mechanics, solid state physics, and material science. It also helps in the development of new technologies, such as quantum computing and nanotechnology.

What are some real-world applications of finite square wells/barriers?

Finite square wells/barriers are used in various technologies, such as diodes and transistors, which are essential components in electronic devices. They are also used in optical fibers and waveguides for telecommunications, as well as in the design of quantum dots for solar cells. In addition, they play a crucial role in the study of nuclear reactions and fusion processes.

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