Answer to Question #140819 in Quantum Mechanics for Afraid

Question #140819
Write down energy in 2 dimensional for a particle.
1
Expert's answer
2020-11-02T09:25:42-0500

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

H^Ψn,m=En,mΨn,m\widehat H \Psi_{n,m} = E_{n,m}\Psi_{n,m}

where :

H^22m(2x2+2y2)+U(x,y)\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:

En,m=π222m(n2a2+m2b2)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


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