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natugnaro
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[SOLVED] Qunatum Angular Momentum
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].
[tex]{L}^{2}= \hbar^{2}l(l+1)|lm>[/tex]
[tex]L_{z}=m \hbar|lm>[/tex]
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]
Homework Statement
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].
Homework Equations
[tex]{L}^{2}= \hbar^{2}l(l+1)|lm>[/tex]
[tex]L_{z}=m \hbar|lm>[/tex]
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]