Phy5645/AngularMomentumProblem

Posted by Group 6:

A system is initally in the state:

$\psi (\theta ,\phi )=1/{\sqrt {5}}Y_{1},-_{1}(\theta ,\phi )+{\sqrt {3/5}}Y_{1},_{0}(\theta ,\phi )+1/{\sqrt {5}}Y_{1},_{1}(\theta ,\phi )$

Let us now find the value of the opperator $L_{z}$ acting on the system as well as the probability of finding each value.

We may first rewright the notation for the system as follows;

$|\psi >=1/{\sqrt {5}}|1,-1>+{\sqrt {3/5}}|1,0>+1/{\sqrt {5}}|1,1>$

$L_{z}$ acting on the system produces three values for $l_{z}$;

$l_{z}=-\hbar ,0,\hbar$

The probablity for finding the value $l_{z}=-\hbar$ is;

$P_{1}=|<1,-1|\psi >|^{2}=|1/{\sqrt {5}}<1,-1|1,-1>+{\sqrt {3/5}}<1-1|1,0>+1/{\sqrt {5}}<1,-1|1,1>|^{2}$

$=1/5$

This can easially be verified since;

$<1,-1|1,0>=<1,-1|1,1>=0$ and $<1,-1|1,-1>=1$

The probablites of measuring $l_{z}=\hbar$ are give as follows;

$P_{0}=|<1,0|\psi >|^{2}=|{\sqrt {3/5}}<1,0|1,0>|^{2}=3/5$

$P_{1}=|<1,1|\psi >|^{2}=|{\sqrt {1/5}}<1,1|1,1>|^{2}=1/5$