Question

Write down energy in 2 dimensional for a particle.

Answer 1

We can solve the Schrodinger problem for a particle in 2 - dimension hole:

$\widehat H \Psi_{n,m} = E_{n,m}\Psi_{n,m}$

where :

$\widehat H \rightarrow -\frac{\hbar^{2}}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) + U(x,y)$

if we solve it for a hole with rectangle as a shape, then we get the level of energy fo such system:

$E_{n,m} = \frac{\pi^2\hbar^2}{2m}\left(\frac{n^2}{a^2} + \frac{m^2}{b^2}\right)$

n,m - quantum numbers

a,b - sides of the rectangle