- #1
White_M
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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.
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
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?