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| <math>P(\hbar)=|\langle 1,1|\psi\rangle|^2=\tfrac{1}{5}.</math> | | <math>P(\hbar)=|\langle 1,1|\psi\rangle|^2=\tfrac{1}{5}.</math> |
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| Now we will calculate the uncertainties <math>\Delta L_x</math> and <math>\Delta L_y</math> and the product <math>\Delta L_x \Delta L_y</math>
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| After measuring <math>l_z=-\hbar</math> the system will be in the eigenstate <math>|lm>=|1,-1></math>, that is <math>\psi(\theta,\phi)=Y_1,_-1(\theta,|\phi)</math>. We will first calculate the expectation values of <math>L_x, L_y, L^2_x, L^2_y</math> using <math>|1,-1></math>. Symmetry requires <math><1,-1|L_x|1,-1>=<1,-1|L_y|1,-1>=0</math>. Using the relation <math>l-1</math> and <math>m=-1</math>;
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| <math><L^2_x>=<L^2_y>=1/2[<L^2>-<L^2_z>]=\hbar^2/2[l(l+1)-m^2]=\hbar^2/2</math>
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| <math>\Delta L_x=\sqrt{<L^2_x>}=\hbar/\sqrt{2}=\Delta L_y</math>
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| Therefore;
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| <math>\Delta L_x \Delta L_y=\sqrt{<L^2_x><L^2_y>}=\hbar^2/2</math>
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| Back to [[Orbital Angular Momentum Eigenfunctions]] | | Back to [[Orbital Angular Momentum Eigenfunctions]] |
Revision as of 22:53, 29 August 2013
We first rewrite the wave vector in Dirac notation:
We see that the possible results for a measurement of 
are 
and 
The probablity for obtaining 
is
Similarly, the probablites of obtaining 
and 
are
and
Back to Orbital Angular Momentum Eigenfunctions