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Momentum eigenvalues and eigenfunctions

  • Thread starter White_M
  • Start date
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1. Homework Statement

For the following wave functions:
ψ[itex]_{x}[/itex]=xf(r)
ψ[itex]_{y}[/itex]=yf(f)
ψ[itex]_{z}[/itex]=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. Homework Equations

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

for solving:

L[itex]_{z}[/itex]|ψ[itex]_{z}[/itex]>=lz|ψ[itex]_{z}[/itex]>
L[itex]_{x}[/itex]|ψ[itex]_{x}[/itex]>=lx|ψ[itex]_{x}[/itex]>
L[itex]_{y}[/itex]|ψ[itex]_{y}[/itex]>=ly|ψ[itex]_{y}[/itex]>


and
L^2|ψ>=l^2|ψ>

where:
L^2=L[itex]_{x}[/itex]^2+L[itex]_{y}[/itex]^2+L[itex]_{z}[/itex]^2

3. The Attempt at a Solution

For instance for Lz:
ψ[itex]_{z}[/itex](r)=<r|z>=zf(r)

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

Is that correct?
 

lightgrav

Homework Helper
1,248
30
I think they want you to "explicitly" take partial derivatives of the function z f(r) .
 
9
0
-ih (x([itex]\partial[/itex]z\[itex]\partial[/itex]y f(r)+z [itex]\partial[/itex]f(r)/[itex]\partial[/itex]y)-y([itex]\partial[/itex]z\[itex]\partial[/itex]x+z[itex]\partial[/itex]f(r)\[itex]\partial[/itex]x))=-ih (xz[itex]\partial[/itex]f(r)\[itex]\partial[/itex]y-yz [itex]\partial[/itex]f(r)\[itex]\partial[/itex]x)=-ih (x[itex]\partial[/itex]\[itex]\partial[/itex]y-y[itex]\partial[/itex]\[itex]\partial[/itex]x)zf(r)

Which is the same as the initial expression.

What am I missing?

Thanks
 

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