Time: Difference between revisions

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\end{pmatrix} = 0</math>
\end{pmatrix} = 0</math>


<math>\Rightarrow</math> <math>\lambda_1</math> = 0, <math>\lambda_2 = {\hbar}^2(A + B)</math>, <math>\lambda_3 = {\hbar}^2(A - B)</math>
<math>\Rightarrow</math> <math> \lambda_{1} </math> = 0, <math>\lambda_2 = {\hbar}^2(A + B)</math>, <math>\lambda_3 = {\hbar}^2(A - B)</math>
 
To find the eigenstate <math>|n_(\lambda)</math> that corresponds to the eigenvalue <math>\lambda</math> we have to solve the following equation:
 
 
<math> {\hbar}^2\begin{pmatrix}
  A & 0 & B \\
  0 & 0 & 0 \\
  B & 0 & A\\
 
\end{pmatrix} \begin{pmatrix}
a\\
b\\
c
\end{pmatrix} \Rightarrow  \lambda _{c} \begin{pmatrix}
a\\
b\\
c
\end{pmatrix}</math>
 
For <math>\lambda_1 = 0</math>
 
<math> {\hbar}^2\begin{pmatrix}
  A & 0 & B \\
  0 & 0 & 0 \\
  B & 0 & A\\
 
\end{pmatrix} \begin{pmatrix}
a\\
b\\
c
\end{pmatrix}= 0 \Rightarrow a= 0 , c= 0</math> 
 
<math>|n_0 \rangle = \begin{pmatrix}
0\\
b \\
0
\end{pmatrix} (normalizing ) \Rightarrow |n_0 \rangle = \begin{pmatrix}
0\\
1 \\
0
\end{pmatrix}\Rightarrow  |n_0 \rangle = |1 , 0 \rangle</math>
 
In the same way for <math>\lambda_2 = {\hbar}^2(A + B)</math>
 
<math> \begin{pmatrix}
  A & 0 & B \\
  0 & 0 & 0 \\
  B & 0 & A\\
 
\end{pmatrix} \begin{pmatrix}
a\\
b\\
c
\end{pmatrix}= (A+B)\begin{pmatrix}
a\\
b\\
c
\end{pmatrix} \Rightarrow a= c , b= 0</math> 
 
 
 
<math>|n_{A+B} \rangle = \begin{pmatrix}
c\\
0 \\
c
\end{pmatrix} (normalizing ) \Rightarrow |n_0 \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}
1\\
0 \\
1
\end{pmatrix}\Rightarrow  |n_{A+B} \rangle = \frac{1}{\sqrt{2}}|1 , +1 \rangle + \frac{1}{\sqrt{2}}|1 , -1 \rangle</math>
 
For

Revision as of 23:43, 30 April 2010

For a spin 1 system l = 1 and m = -1 , 0 , 1. For the operator 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 S_{z}} we have

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 S_z |l,m \rangle} 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 \Rightarrow} 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 \langle l,n| S_z|l,m \rangle = m\hbar \langle l,n|l,m \rangle} 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 \Rightarrow ( S_z)_{nm} = m\hbar \delta _{nm}}

So


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 S_{z} = \hbar \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix}} 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 \Rightarrow} 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 S_{z}^{2} = {\hbar}^2\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}}

For the 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 S_x} operator we have



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 S_{x} = \frac{\hbar}{2}\begin{pmatrix} 0 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & \sqrt{2} \\ 0 & \sqrt{2} & 0\\ \end{pmatrix}} 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 \Rightarrow} 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 S_{x}^{2} = {\hbar}^2\begin{pmatrix} \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ \frac{1}{2}& 0 & \frac{1}{2}\\ \end{pmatrix}} 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 \Rightarrow} 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 S_{y} = \frac{\hbar}{2i}\begin{pmatrix} 0 & \sqrt{2} & 0 \\ -\sqrt{2} & 0 & \sqrt{2} \\ 0 & -\sqrt{2} & 0\\ \end{pmatrix}} 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 S_{y}^{2} = {\hbar}^2\begin{pmatrix} \frac{1}{2} & 0 & -\frac{1}{2} \\ 0 & 1 & 0 \\ -\frac{1}{2}& 0 & \frac{1}{2}\\ \end{pmatrix}}

Thus the Hamiltonian can be represented by the matrix

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 H = {\hbar}^2\begin{pmatrix} A & 0 & B \\ 0 & 0 & 0 \\ B & 0 & A\\ \end{pmatrix}}

To find the energy eigenvalues we have to solve the secular equation

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 det(H - \lambda I) = 0 } 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 \Rightarrow} 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 det \begin{pmatrix} A{\hbar}^2 - \lambda & 0 & B{\hbar}^2 \\ 0 & - \lambda & 0 \\ B{\hbar}^2 & 0 & A{\hbar}^2 - \lambda\\ \end{pmatrix} = 0}

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 \Rightarrow} 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 \lambda_{1} } = 0, 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 \lambda_2 = {\hbar}^2(A + B)} , 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 \lambda_3 = {\hbar}^2(A - B)}

To find 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 |n_(\lambda)} that corresponds to the eigenvalue 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 \lambda} we have to solve the following equation:


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}^2\begin{pmatrix} A & 0 & B \\ 0 & 0 & 0 \\ B & 0 & A\\ \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix} \Rightarrow \lambda _{c} \begin{pmatrix} a\\ b\\ c \end{pmatrix}}

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 \lambda_1 = 0}

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}^2\begin{pmatrix} A & 0 & B \\ 0 & 0 & 0 \\ B & 0 & A\\ \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix}= 0 \Rightarrow a= 0 , c= 0}

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 |n_0 \rangle = \begin{pmatrix} 0\\ b \\ 0 \end{pmatrix} (normalizing ) \Rightarrow |n_0 \rangle = \begin{pmatrix} 0\\ 1 \\ 0 \end{pmatrix}\Rightarrow |n_0 \rangle = |1 , 0 \rangle}

In the same way 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 \lambda_2 = {\hbar}^2(A + B)}

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 \begin{pmatrix} A & 0 & B \\ 0 & 0 & 0 \\ B & 0 & A\\ \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix}= (A+B)\begin{pmatrix} a\\ b\\ c \end{pmatrix} \Rightarrow a= c , b= 0}


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 |n_{A+B} \rangle = \begin{pmatrix} c\\ 0 \\ c \end{pmatrix} (normalizing ) \Rightarrow |n_0 \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\ 0 \\ 1 \end{pmatrix}\Rightarrow |n_{A+B} \rangle = \frac{1}{\sqrt{2}}|1 , +1 \rangle + \frac{1}{\sqrt{2}}|1 , -1 \rangle}

For

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