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

Particle is in state

[tex]\psi=A(x+y+2z)e^{-\alpha r}[/tex]

[tex]r=\sqrt{x^{2}+y^{2}+z^{2}[/tex]

A and alpha are real constants.

a) Normalize angular part of wave function.

b) Find [tex]<\vec{L}^{2}> , <L_{z}>[/tex]

c) Find probability of finding [tex]L_{z}=+\hbar[/tex].

2. Relevant equations

[tex]{L}^{2}= \hbar^{2}l(l+1)|lm>[/tex]

[tex]L_{z}=m \hbar|lm>[/tex]

3. The attempt at a solution

I have found a) part

[tex]T(\theta,\phi)=\frac{1}{2\sqrt{3}}(1+i)Y^{-1}_{1}-\frac{1}{2\sqrt{3}}(1-i)Y^{1}_{1}+\frac{2}{\sqrt{6}}Y^{0}_{1}[/tex]

b)

Since [tex]Y^{m}_{l}=|lm>[/tex]

Using [tex]L_{z}=m \hbar|lm>[/tex]

<Lz>= 1/(4*3)*2<1-1|Lz|1-1> + 1/(4*3)*2<11|Lz|11> + 4/6<10|Lz|10> = 0

To find [tex]<\vec{L}^{2}>[/tex] I would apply operator of L^2 to angular part of wave function, just like I have done for Lz.

[tex]{L}^{2}= \hbar^{2}l(l+1)|lm>[/tex]

Is this is the way to find expectation values ?

c)

[tex]P(\hbar)=|-\frac{1}{2\sqrt{3}}(1-i)|^{2}=\frac{2}{12}[/tex]

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# Homework Help: Qunatum Angular Momentum

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