Electron in a box, size of the box

In summary, the conversation is about a test in a physics class and the professor giving a sample test and solutions as a study guide. The student is having trouble with one question involving the size of a box and the energy of an electron decaying. The professor is referencing the concept of a particle in a box and introduces a pi squared term in the solution. After further explanation and clarification, the student understands the process and thanks the group for their help.
  • #1
warfreak131
188
0

Homework Statement



I have a test in my physics class on tuesday, and as a study guide, our professor gave us a sample test and the solutions. I worked through most of it, except for this one question. I don't understand his method to solving it, I was hoping you guys had a solution. I have already email him about it, but he has yet to answer.

When an electron decays from the n=2 state to the n=1 state, it emits a photon. Determine the size of the box, L, for which the energy of this photon equals the energy of the analogous n=2 to n=1 transition in hydrogen.

The Attempt at a Solution



HIS attempt at the solution is in the attached jpeg.

I understand the process of finding the energy of the photon, that's very easy, but in finding the size of the box, he introduces this pi2 term, and instead of doing the 1/n2 - 1/m2, he simply does n2-m2
 

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  • #2
Are you familiar with the infinite square well? That's the box your professor is talking about.
 
  • #3
warfreak131 said:
I understand the process of finding the energy of the photon, that's very easy, but in finding the size of the box, he introduces this pi2 term, and instead of doing the 1/n2 - 1/m2, he simply does n2-m2

Your professor is considering the energy levels for a particle in a box. look here

Code:
[PLAIN]http://en.wikipedia.org/wiki/Particle_in_a_box
[/PLAIN] [Broken]

For one dimensional box of length L, the energy levels are

[tex]E_n=\frac{n^2\hbar^2 \pi^2}{2mL^2}[/tex]

so

[tex]\Delta E=E_2-E_1=\frac{\hbar^2 \pi^2}{2mL^2}(2^2-1^2)[/tex]
 
Last edited by a moderator:
  • #4
diazona said:
Are you familiar with the infinite square well? That's the box your professor is talking about.

Yes, but I'm still not sure what he's doing.
 
  • #5
IssacNewton said:
Your professor is considering the energy levels for a particle in a box. look here

Code:
[PLAIN]http://en.wikipedia.org/wiki/Particle_in_a_box
[/PLAIN] [Broken]

For one dimensional box of length L, the energy levels are

[tex]E_n=\frac{n^2\hbar^2 \pi^2}{2mL^2}[/tex]

so

[tex]\Delta E=E_2-E_1=\frac{\hbar^2 \pi^2}{2mL^2}(2^2-1^2)[/tex]

ooooo, i see.

thanks!
 
Last edited by a moderator:

1. What is an "electron in a box"?

An electron in a box is a theoretical model used in quantum mechanics to study the behavior of electrons in a confined space. The "box" represents the boundaries of the space in which the electron is confined, and the model assumes that the electron is free to move within this space but cannot escape it.

2. How does the size of the box affect the behavior of the electron?

The size of the box affects the energy levels and wave function of the electron. A larger box allows for more energy levels and results in a more continuous distribution of energy, while a smaller box limits the energy levels and leads to a more discrete distribution. The size of the box also determines the probability of finding the electron in different regions of the space.

3. Can the size of the box be changed?

In the theoretical model of the "electron in a box", the size of the box is typically fixed. However, in practical applications, the size of the box can be altered by changing the physical dimensions of the confined space, such as by adjusting the length of a nanowire or the dimensions of a quantum dot.

4. How is the size of the box related to the uncertainty principle?

The uncertainty principle states that the more precisely we know the position of a particle, the less certain we are about its momentum, and vice versa. In the case of an electron in a box, the size of the box determines the possible values of the electron's momentum, which in turn affects our ability to determine its position. Therefore, the size of the box is related to the uncertainty principle in this model.

5. What are the real-world applications of studying "electron in a box"?

The "electron in a box" model has many practical applications in fields such as semiconductor technology, nanotechnology, and quantum computing. By understanding how electrons behave in confined spaces, scientists can design and control the properties of materials and devices at the nanoscale. This model also helps in the development of new technologies that utilize quantum effects for computing and communication.

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