Angular Momentum Operator Eigenfunction

[InsertName]

Homework Statement

Let the angular part of a wave function be proportional to x²+y²

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\hbar\frac{\partial}{\partialx}

p_y = -i\hbar\frac{\partial}{\partialy}

The Attempt at a Solution

L_z (x²+y²) = (\lambdax²+y²) (1)

(xp_y-yp_x)(x²+y²) = (\lambdax²+y²) (2)

= xp_y(x²+y²) - yp_x(x²+y²) (3)

= xp_y(x²) + xp_y(y²) - yp_x(x²) - yp_x(y²) (4)

= 0 - 2i\hbarxy + 2i\hbarxy + 0 (5)

= 0 (6)

Which can only be correct if \lambda = 0 (?). (7)

Is \lambda = 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'.

 

[kuruman]

It looks fine. Zero is a perfectly good eigenvalue. Yours is a "brute force" solution. The "trick" that you may have missed is to express x and y in spherical coordinates, so that

x²+y²=r²sin²θ and then operate on it with

L_{z}=\frac{\hbar}{i}\frac{\partial}{\partial \phi}

 

[InsertName]

Thanks for the quick reply!