Phy5645/AngularMomentumProblem: Difference between revisions

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We may first rewrite the notation for the system as follows;
We first rewrite the wave vector in Dirac notation:


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


<math>L_z</math> acting on the system produces three values for <math>l_z</math>;
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>


<math>l_z=-\hbar, 0, \hbar </math>
The probablity for obtaining <math>-\hbar</math> is


The probablity for finding the value <math>l_z=-\hbar</math> is;
<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>


<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>
Similarly, the probablites of obtaining <math>0\!</math> and <math>\hbar</math> are


<math>=1/5</math>
<math>P(0)=|\langle 1,0|\psi\rangle|^2=\tfrac{3}{5}</math>


This can easily be verified since;
and


<math><1,-1|1,0>=<1,-1|1,1>=0</math> and <math><1,-1|1,-1>=1</math>
<math>P(\hbar)=|\langle 1,1|\psi\rangle|^2=\tfrac{1}{5}.</math>
 
The probablites of measuring <math>l_z=\hbar,0</math> are give as follows;
 
<math>P_0=|<1,0|\psi>|^2=|\sqrt{3/5}<1,0|1,0>|^2=3/5</math>
 
<math>P_1=|<1,1|\psi>|^2=|\sqrt{1/5}<1,1|1,1>|^2=1/5</math>


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>
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>

Revision as of 22:52, 29 August 2013

We first rewrite the wave vector in Dirac notation:

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\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 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 \hat{L}_z} are 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 -\hbar,} 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 0,\!} 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 \hbar.}

The probablity for obtaining 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 -\hbar} 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 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 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 0\!} 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 \hbar} are

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

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

Now we will calculate the uncertainties 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} 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 \Delta L_y} and the product 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}

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 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 |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 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_x, L_y, L^2_x, L^2_y} 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 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-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}

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=\sqrt{<L^2_x>}=\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}

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