Question

We have examined the Particle in a Box problem in one dimension, meaning that we consider only one variable, x. We can go to higher dimensions, for example, we can consider what this would look like if we wanted to think about both x and y.

In that case we would need to make some changes. The potential energy V(x) would become V(x,y), but would behave in a similar way as before. It would equal infinity if either x or y was greater than L or less than zero. The potential energy would equal zero if both were between 0 and L. The second derivative would become the sum of two second derivatives, one with respect to x, the other with respect to y. And finally, the wavefunction itself would become a function of both x and y.

The key to this approach is to understand that the new wavefunction will be the product of two new wavefunction, one a function only of x (labelled as X(x)) and one a function only of y (with the obvious label of Y(y)). In other words, Psi(x,y) = X(x)Y(y).

Use this new wavefunction and go through the Particle in a Box derivation. Find the solutions for X(x) and Y(y) (you do not need to normalize them), along with the energy states. Take a photo and upload the file.

Answer 1

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