- #1

dark_matter_is_neat

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- 0

- Homework Statement
- For ##\psi(\vec{r}) = (x+y+3z)f(r)## where ##r = \sqrt{x^{2}+y^{2}+z^{2}}##

Determine ##\hat{n}## and m such that ##(\hat{n} \cdot L) \psi(r) = m\hbar\psi(r)## Where ##L=-i\hbar(\vec{r} \times \nabla)## is the angular momentum operator.

- Relevant Equations
- ##\psi(\vec{r}) = (x+y+3z)f(r)##

First I calculated ##(\vec{n} \cdot L) \psi(r) = -i\hbar(n_{x}(3y-z)+n_{y}(z-3x)+n_{z}(x-y))f(r)## and then tried to solve for ##n_{i}## such that I get (x+y+3z)f(r), and then divide ##n_{i}## by the magnitude of ##\vec{n}## to get the unit vector and m, but when I try doing this, I get the system of equations: ##3n_{x}-n_{z} = 1##, ##n_{y}-n_{x} = 3## and ##n_{z}-3n_{y} = 1##, but there is no solution to this system of equations.

I'm not sure how else I would determine ##\hat{n}## and m.

I'm not sure how else I would determine ##\hat{n}## and m.

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