- #1

- 27

- 0

## Homework Statement

Let the angular part of a wave function be proportional to x

^{2}+y

^{2}

Show that the wave function is an eigenfunction of L

_{z}and calculate the associated

eigenvalue.

## Homework Equations

L

_{z}= xp

_{y}-yp

_{x}

p

_{x}= -i[tex]\hbar[/tex][tex]\frac{\partial}{\partialx}[/tex]

p

_{y}= -i[tex]\hbar[/tex][tex]\frac{\partial}{\partialy}[/tex]

## The Attempt at a Solution

L

_{z}(x

^{2}+y

^{2}) = ([tex]\lambda[/tex]x

^{2}+y

^{2}) (1)

(xp

_{y}-yp

_{x})(x

^{2}+y

^{2}) = ([tex]\lambda[/tex]x

^{2}+y

^{2}) (2)

= xp

_{y}(x

^{2}+y

^{2}) - yp

_{x}(x

^{2}+y

^{2}) (3)

= xp

_{y}(x

^{2}) + xp

_{y}(y

^{2}) - yp

_{x}(x

^{2}) - yp

_{x}(y

^{2}) (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'.