Phy5645/AngularMomentumProblem: Difference between revisions
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We may first rewrite the notation for the system as follows; | |||
<math>|\psi>=1/\sqrt{5}|1,-1>+\sqrt{3/5}|1,0>+1/\sqrt{5}|1,1></math> | |||
<math>L_z</math> acting on the system produces three values for <math>l_z</math>; | |||
<math>l_z=-\hbar, 0, \hbar </math> | |||
The probablity for finding the value <math>l_z=-\hbar</math> is; | |||
<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> | |||
<math>=1/5</math> | |||
This can easily be verified since; | |||
<math><1,-1|1,0>=<1,-1|1,1>=0</math> and <math><1,-1|1,-1>=1</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> | |||
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>; | |||
<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> | |||
<math>\Delta L_x=\sqrt{<L^2_x>}=\hbar/\sqrt{2}=\Delta L_y</math> | |||
Therefore; | |||
<math>\Delta L_x \Delta L_y=\sqrt{<L^2_x><L^2_y>}=\hbar^2/2</math> | |||
Revision as of 22:44, 29 August 2013
We may first rewrite the notation for the system as follows;
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>=1/\sqrt{5}|1,-1>+\sqrt{3/5}|1,0>+1/\sqrt{5}|1,1>}
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_z} acting on the system produces three values for 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} ;
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, 0, \hbar }
The probablity for finding the value 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} 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_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}
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 =1/5}
This can easily be verified since;
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|1,0>=<1,-1|1,1>=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 <1,-1|1,-1>=1}
The probablites of 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,0} are give as follows;
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=|<1,0|\psi>|^2=|\sqrt{3/5}<1,0|1,0>|^2=3/5}
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_1=|<1,1|\psi>|^2=|\sqrt{1/5}<1,1|1,1>|^2=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}