Particle in a 3D box (Quantum)

Let's try this again.The degeneracies are 1, 3, 6, and 10 for the first four energy levels, respectively. This is because the quantum numbers nx, ny, and nz each take on integer values, and the total number of states is given by adding these values together. For the 1st level, there is only one possible combination (1,1,1), giving a degeneracy of 1. For the 2nd level, there are three possible combinations (1,1,2), (1,2,1), and (2,1,1), giving a degeneracy of 3. For the 3rd level, there are six possible combinations (1
  • #1
breeg
1
0

Homework Statement



What are the degeneracies of the first four energy levels for a particle in a 3D box with a=b=1.5c?

Homework Equations



Exxnynz=h2/8m(nx2/a2+ny2/b2+nz2/c2)

For 1st level, the above = 3h2/8m
For 2nd level, the above = 6h2/8m
For 3rd level, the above = 9h2/8m
For 4th level, the above = 11h2/8m

The Attempt at a Solution



I think I finally grasped the basis of what the problem wants.

For the 1st level:

Exxnynz=(h2/(8m*1) + h2/(8m*1) + h2/8m*(1/1.5))*(1+1+(3/2))

So E1 1 3/2

3/2 was obtained because 1/1.5 is 2/3=c because a=b=1=1.5c so c=1/1.5=2/3

Is this correct? I'm just unsure about the rest of the energy levels. It seems like one can just pick numbers and force it to work so I'm lost on the proper degeneracy.
 
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  • #2
breeg said:
For the 1st level:

Exxnynz=(h2/(8m*1) + h2/(8m*1) + h2/8m*(1/1.5))*(1+1+(3/2))

Why are you multiplying by (1+1+3/2)? I don't think you've got the idea. The quantum numbers each take on an integer value, and that is why the energy levels are different.
 

1. What is a particle in a 3D box in quantum mechanics?

A particle in a 3D box is a theoretical model used in quantum mechanics to study the behavior of a particle confined to a three-dimensional space. It is often used as a simplified representation of more complex systems, such as atoms or molecules.

2. How does a particle behave in a 3D box?

In a 3D box, the particle's behavior is governed by the principles of quantum mechanics. This means that the particle can exist in multiple states simultaneously, and its position and momentum cannot be precisely determined at the same time. The particle's energy levels are also quantized, meaning they can only have certain discrete values.

3. What is the significance of a particle in a 3D box in quantum mechanics?

The model of a particle in a 3D box allows us to understand and predict the behavior of particles at a quantum level. It also serves as a building block for more complex systems and can help us gain insights into phenomena such as energy levels and wave-particle duality.

4. How does the size of the box affect the behavior of the particle?

The size of the box has a significant impact on the behavior of the particle. As the size of the box decreases, the energy levels become more closely spaced, and the particle becomes more confined. This leads to a higher probability of finding the particle in certain regions of the box, known as the "quantum confinement effect."

5. Can a particle in a 3D box exist in a state of zero energy?

In quantum mechanics, the lowest possible energy state of a system is known as the ground state. In a particle in a 3D box, the ground state has a non-zero energy due to the uncertainty principle. However, the ground state energy can approach zero as the size of the box increases, but it can never be exactly zero.

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