- #1
JonnyMaddox
- 74
- 1
Hey JO.
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)} ?
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)} ?