Probabilities Inside Cubic 3D Infinite Well

[erok81]

Homework Statement

An electron is trapped in a cubic 3D infinite well. In the states (n_x,n_y,n_z) = (a)(2,1,1), (b)(1,2,1) (c)(1,1,2), what is the probability of finding the electron in the region (0 ≤ x ≤ L, 1/3L ≤ y ≤ 2/3L, 0 ≤ z ≤ L)?

Homework Equations

My normalized wave function in the box is:

$$\psi _{(x,y,z)} = \left( \frac{2}{L}\right)^{\frac{3}{2}}sin\left(\frac{n_{x} \pi x}{L_{x}}\right)sin\left(\frac{n_{y} \pi y}{L_{y}}\right)sin\left(\frac{n_{z} \pi z}{L_{z}}\right)$$

And my probability is found by |ψ²|

The Attempt at a Solution

Without integrating I am not sure how to proceed on this problem. If I had exact values for L, obviously it would be fairly straight forward.

My problem is I don't know what to put in for the normalized L (since it isn't axis specific) and my axis specific L's give ranges.

 

[ideasrule]

Since this well is cubic, isn't Lx=Ly=Lz?

The first and last conditions are always going to be satisfied, so all you have to worry about is 1/3L ≤ y ≤ 2/3L. You find that by integrating the one-dimensional solution from 1/3L to 2/3L. It shouldn't be a hard integral, especially if you use Wolfram Alpha. For part b, you shouldn't need to integrate; think about the symmetry of the wavefunction and you'll get the answer.

 

[erok81]

Ugh...how soon we forget.

I don't have the probability right. You nailed it. I was looking at an example in the book and they skipped over the probability calculation so I wrong assumed there wasn't any integration. No wonder it didn't make sense.

What do you mean by "The first and last conditions are always going to be satisfied..."

And say I had the same problem my ranges were all like the center one where it isn't just 0 to L. Would I set up a triple integral over the different variable ranges?

 

[ideasrule]

What do you mean by "The first and last conditions are always going to be satisfied..."

I mean that 0 ≤ x ≤ L and 0 ≤ z ≤ L are always true, so you don't need to worry about them.

And say I had the same problem my ranges were all like the center one where it isn't just 0 to L. Would I set up a triple integral over the different variable ranges?

Yes.

 

[rwisz]

I would suggest that there are a few things to consider in this problem. First, it is important to note that the probability of finding the electron in a certain region is given by the square of the wavefunction, as stated in the homework equations. This means that the normalized L for the entire cubic 3D infinite well is given by the length of each side of the well, which is L for all three axes.

Next, we can use the given wave function to calculate the probability of finding the electron in the specified region. We can do this by first evaluating the wave function at each of the given states (a)(2,1,1), (b)(1,2,1), and (c)(1,1,2) and then finding the square of each of these values. This will give us the probability of finding the electron at each of these states.

Finally, to find the probability of finding the electron in the specified region, we can add together the probabilities for each of the states and then normalize by dividing by the total probability of finding the electron in the entire cubic 3D infinite well. This will give us the overall probability of finding the electron in the specified region.

It is also worth noting that without knowing the exact values for L, we can still make some general statements about the probabilities. For example, since the region in question is a subset of the entire cubic 3D infinite well, the probability of finding the electron in this region will be less than the probability of finding it in the entire well. Additionally, since the given states have different values for nx, ny, and nz, the probabilities for each state will also be different.