# Momentum eigenvalues and eigenfunctions

1. Jan 25, 2014

### White_M

1. The problem statement, all variables and given/known data

For the following wave functions:
ψ$_{x}$=xf(r)
ψ$_{y}$=yf(f)
ψ$_{z}$=zf(f)

show, by explicit calculation, that they are eigenfunctions of Lx,Ly,Lz respectively, as well as of L^2, and find their corresponding eigenvalues.

2. Relevant equations

I used:
L$_{x}$=-ih(y$\partial/\partial z$-z$\partial/\partial y$)
L$_{y}$=-ih(z$\partial/\partial x$-x$\partial/\partial z$)
L$_{z}$=-ih(x$\partial/\partial y$-y$\partial/\partial x$)

for solving:

L$_{z}$|ψ$_{z}$>=lz|ψ$_{z}$>
L$_{x}$|ψ$_{x}$>=lx|ψ$_{x}$>
L$_{y}$|ψ$_{y}$>=ly|ψ$_{y}$>

and
L^2|ψ>=l^2|ψ>

where:
L^2=L$_{x}$^2+L$_{y}$^2+L$_{z}$^2

3. The attempt at a solution

For instance for Lz:
ψ$_{z}$(r)=<r|z>=zf(r)

L$_{z}$|ψ$_{z}$>=-ih(x$\partial/\partial y$-y$\partial/\partial x$) zf(r)=lz|ψ$_{z}$>

Is that correct?

2. Jan 25, 2014

### lightgrav

I think they want you to "explicitly" take partial derivatives of the function z f(r) .

3. Jan 25, 2014

### White_M

-ih (x($\partial$z\$\partial$y f(r)+z $\partial$f(r)/$\partial$y)-y($\partial$z\$\partial$x+z$\partial$f(r)\$\partial$x))=-ih (xz$\partial$f(r)\$\partial$y-yz $\partial$f(r)\$\partial$x)=-ih (x$\partial$\$\partial$y-y$\partial$\$\partial$x)zf(r)

Which is the same as the initial expression.

What am I missing?

Thanks

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