Phy5645/AngularMomentumProblem

Posted by Group 6:

A system is initally in the state:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 $l_{z}=\hbar ,0$ are give as follows;

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

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

Now we will calculate the uncertainties $\Delta L_{x}$ and $\Delta L_{y}$ and the product $\Delta L_{x}\Delta L_{y}$

After measuring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_z=-\hbar} the system will be in the eigenstate $|lm>=|1,-1>$, that is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(\theta,\phi)=Y_1,_-1(\theta,|\phi)} . We will first calculate the expectation values of $L_{x},L_{y},L_{x}^{2},L_{y}^{2}$ using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1,-1>} . Symmetry requires Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <1,-1|L_x|1,-1>=<1,-1|L_y|1,-1>=0} . Using the relation $l-1$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=-1} ;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <L^2_x>=<L^2_y>=1/2[<L^2>-<L^2_z>]=\hbar^2/2[l(l+1)-m^2]=\hbar^2/2}

$\Delta L_{x}={\sqrt {<L_{x}^{2}>}}=\hbar /{\sqrt {2}}=\Delta L_{y}$

Therefore;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta L_x \Delta L_y=\sqrt{<L^2_x><L^2_y>}=\hbar^2/2}