Phy5645/AngularMomentumProblem

We first rewrite the wave vector in Dirac notation:

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

We see that the possible results for a measurement of ${\hat {L}}_{z}$ are $-\hbar ,$ $0,\!$ and $\hbar .$

The probablity for obtaining $-\hbar$ is

$P(-\hbar )=|\langle 1,-1|\psi \rangle |^{2}=\left|{\frac {1}{\sqrt {5}}}\langle 1,-1|1,-1\rangle +{\sqrt {\frac {3}{5}}}\langle 1-1|1,0\rangle +{\frac {1}{\sqrt {5}}}\langle 1,-1|1,1\rangle \right|^{2}={\tfrac {1}{5}}.$

Similarly, the probablites of obtaining $0\!$ and $\hbar$ are

$P(0)=|\langle 1,0|\psi \rangle |^{2}={\tfrac {3}{5}}$

and

$P(\hbar )=|\langle 1,1|\psi \rangle |^{2}={\tfrac {1}{5}}.$

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