- #1

Pyrus96

- 3

- 0

_{0}.

We know the wave function of a rectangular potential well and through this I found the following formula for the Hamiltonian matrix elements:

$$H_{nm} = [(\frac{m_{1}}{a_{1}})^{2} +(\frac{m_{2}}{a_{2}})^{2}]\delta_{nm} + V_{0}v_{nm}$$

where v

_{nm}is given by:

$$v_{nm} = \int_{\Gamma^{'}}\phi_{n}\phi_{m}$$

where ##\Gamma^{'}## denotes the area in the rectangle which isn't part of the stadium. And the wave functions are given by:

$$\phi_{m_{1},m_{2}}(x_{1},x_{2}) = \int sin(\frac{\pi m_{1}x_{1}}{a_{1}})sin(\frac{\pi m_{2}x_{2}}{a_{2}})$$

but here's my problem, the system we work in is two dimensional and thus we get energy levels corresponding to the doublets: ##n = (n_{1},n_{2})## and ##m = (m_{1},m_{2})##. How am I supposed to interpret the matrix with this because I can't imagine a matrix from that can correspond to ##H_{(2,1)x(1,1)}##

Please note that I didn't bother to write several constants in front of the formula of the hamiltonian and the wave function because they don't matter to the answer.