How do you solve degeneracy for 2-D particle in a box?

In summary, the degeneracy of the energy level E = 65 E0 for a two dimensional particle in a box can be found by determining the number of solutions to the equation n1^2 + n2^2 = 2(65). This can be done through the guess and test method, where the positive integer values of n1 and n2 are determined. The degeneracy is at least 2, but may have more solutions depending on the values of n1 and n2.
  • #1
PhuongV
2
0

Homework Statement



What is the degeneracy of the energy level E =65 E0 of the two dimensional particle in a box?
Answer

Homework Equations



E=(h_^2/8mL^2)*(nx^2+ny^2)--> I think we use this eq.


The Attempt at a Solution

 
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  • #2
As far as I can tell, it's basically a mathematical question - you need to find the number of combinations of integers n1 and n2 such that n12 + n22 = 65 * 2 (I think the 2 should be there since for the ground state energy n1 = n2 = 1).
 
  • #3
Assuming E0 is the zero point energy, you need to determine how many solutions there are to the equation

[tex] E_0 \left(\frac{8mL^2}{h^2}\right) = n_x^2 + n_y^2 = 2(65)[/tex]

where n_x and n_y are positive integers. For example, one solution would be n_x = 9 and n_y = 7, so obviously n_x = 7 and n_y = 9 is also a solution, so the degeneracy is at LEAST 2. You just need to find all possible solutions, and then count them.

Fortunately the guess and test method works easily for this problem. You could solve it graphically, but the integers are so small I think it's easier to just guess here.
 
  • #4
Thank you everyone! I figured it out some time ago! I was just not thinking at the time :)
 
  • #5


To solve for degeneracy in a 2-D particle in a box, we use the equation E=(h^2/8mL^2)*(nx^2+ny^2) where E is the energy level, h is Planck's constant, m is the mass of the particle, and L is the length of the box. The degeneracy of an energy level is the number of different ways a particular energy level can be achieved. In this case, the energy level E=65E0 can be achieved in multiple ways, depending on the values of nx and ny. To determine the degeneracy, we can rearrange the equation to solve for nx and ny:

nx^2+ny^2=(8mL^2E)/(h^2)

We can then use this equation to find all possible combinations of nx and ny that result in the energy level E=65E0. The number of possible combinations is the degeneracy of the energy level.

For example, if we assume that nx and ny can only take on integer values, we can solve for nx and ny using a trial and error method. We can start with nx=1 and solve for ny, then increase nx and repeat the process until we reach the desired energy level. In this case, we would find that nx=8 and ny=1, or nx=1 and ny=8, both result in the energy level E=65E0. Therefore, the degeneracy of the energy level E=65E0 is 2.

In summary, to solve for degeneracy in a 2-D particle in a box, we use the equation E=(h^2/8mL^2)*(nx^2+ny^2) to find all possible combinations of nx and ny that result in the desired energy level. The number of combinations is the degeneracy of the energy level.
 

FAQ: How do you solve degeneracy for 2-D particle in a box?

1. How does degeneracy occur in a 2-D particle in a box system?

Degeneracy in a 2-D particle in a box system occurs when two or more different energy states have the same energy value. This means that multiple particles can occupy the same energy level, resulting in degenerate energy levels.

2. What causes degeneracy in a 2-D particle in a box?

Degeneracy in a 2-D particle in a box system is caused by the confinement of particles within a finite space. This confinement leads to quantization of energy levels, resulting in certain energy levels having the same value.

3. How do you solve degeneracy in a 2-D particle in a box system?

To solve degeneracy in a 2-D particle in a box system, one must consider the boundary conditions and the symmetry of the system. By taking into account these factors, one can determine the number of possible energy states and the degeneracy of each energy level.

4. What are the implications of degeneracy in a 2-D particle in a box?

The presence of degeneracy in a 2-D particle in a box system can affect the behavior and properties of particles within the system. For example, particles occupying degenerate energy levels may have identical or similar properties, making it difficult to distinguish between them.

5. Can degeneracy be broken in a 2-D particle in a box system?

Yes, degeneracy in a 2-D particle in a box system can be broken by introducing external factors such as an electric or magnetic field. These external fields can lift the degeneracy of energy levels and lead to different energy values for each level.

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