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Homework Statement
Let the angular part of a wave function be proportional to x2+y2
Show that the wave function is an eigenfunction of Lz and calculate the associated
eigenvalue.
Homework Equations
Lz = xpy-ypx
px = -i[tex]\hbar[/tex][tex]\frac{\partial}{\partialx}[/tex]
py = -i[tex]\hbar[/tex][tex]\frac{\partial}{\partialy}[/tex]
The Attempt at a Solution
Lz (x2+y2) = ([tex]\lambda[/tex]x2+y2) (1)
(xpy-ypx)(x2+y2) = ([tex]\lambda[/tex]x2+y2) (2)
= xpy(x2+y2) - ypx(x2+y2) (3)
= xpy(x2) + xpy(y2) - ypx(x2) - ypx(y2) (4)
= 0 - 2i[tex]\hbar[/tex]xy + 2i[tex]\hbar[/tex]xy + 0 (5)
= 0 (6)
Which can only be correct if [tex]\lambda[/tex] = 0 (?). (7)
Is [tex]\lambda[/tex] = 0 a valid solution?
I'm pretty confident that (1), (2) and (3) are correct but after that I feel as if I'm missing some kind of 'trick'.