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The Hamiltonian is:

[itex]H= \frac{p_{x}^{2}+p_{y}^{2}}{2m}[/itex]

In quantum Mechanics:

[itex]\hat{H}=-\frac{\hbar^{2}}{2m}(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial x^{2}})[/itex]

In polar coordinates:

[itex]\hat{H}=-\frac{\hbar^{2}}{2m}( \frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial \phi^{2}})[/itex]

Now I want to write this operator in matrix form. What is an appropriate basis? I thought a good would be {sin(x),sin(2x),sin(3x),....,sin(nx)} Now how do I do that in a two dimensional space? What is the basis for that? Something like {sin(x),sin(y),...sin(nx),sin(ny)} ?

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# Hamilton Operator for particle on a circle -- Matrix representation...

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