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| Posted by Group 6:
| | We first rewrite the wave vector in Dirac notation: |
| :A system is initally in the state:
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| :<math>\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)</math>
| | <math>|\psi\rangle=\frac{1}{\sqrt{5}}|1,-1\rangle+\sqrt{\frac{3}{5}}|1,0\rangle+1/\sqrt{5}|1,1\rangle</math> |
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| :Let us now find the value of the opperator <math>L_z</math> acting on the system as well as the probability of finding each value.
| | We see that the possible results for a measurement of <math>\hat{L}_z</math> are <math>-\hbar,</math> <math>0,\!</math> and <math>\hbar.</math> |
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| :We may first rewright the notation for the system as follows;
| | The probablity for obtaining <math>-\hbar</math> is |
| :<math>|\psi>=1/\sqrt{5}|1,-1>+\sqrt{3/5}|1,0>+1/\sqrt{5}|1,1></math>
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| :<math>L_z</math> acting on the system produces three values for <math>l_z</math>;
| | <math>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}.</math> |
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| :<math>l_z=-\hbar, 0, \hbar </math>
| | Similarly, the probablites of obtaining <math>0\!</math> and <math>\hbar</math> are |
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| :The probablity for finding the value <math>l_z=-\hbar</math> is;
| | <math>P(0)=|\langle 1,0|\psi\rangle|^2=\tfrac{3}{5}</math> |
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| :<math>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</math>
| | and |
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| :<math>=1/5</math>
| | <math>P(\hbar)=|\langle 1,1|\psi\rangle|^2=\tfrac{1}{5}.</math> |
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| :This can easially be verified since;
| | Back to [[Orbital Angular Momentum Eigenfunctions#Problem|Orbital Angular Momentum Eigenfunctions]] |
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| :<math><1,-1|1,0>=<1,-1|1,1>=0</math> and <math><1,-1|1,-1>=1</math>
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| :The probablites of measuring <math>l_z=\hbar,0</math> are give as follows;
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| :<math>P_0=|<1,0|\psi>|^2=|\sqrt{3/5}<1,0|1,0>|^2=3/5</math>
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| :<math>P_1=|<1,1|\psi>|^2=|\sqrt{1/5}<1,1|1,1>|^2=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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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