DetailedBalance: Difference between revisions
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Consider a transition from <math>i \rightarrow f</math> between two states of | Posted by student team #5 (Anthony Kuchera, Jeff Klatsky, Chelsey Morien) | ||
QUESTION: | |||
Consider a transition from <math>i \rightarrow f</math> between two states of a nucleus with spins <math>J_i </math> and <math>J_f</math>, respectively. The transition probability is proportional to the squared matrix element <math>|\langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle|^2</math> where <math>T_{\lambda \mu}</math> is a hermitian tensor operator of rank <math>\lambda</math> responsible for the process. Define the reduced transition probability | |||
<math>B(T_{\lambda}; i \rightarrow f)=\sum_{\mu M_f}|\langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle|^2</math> | <math>B(T_{\lambda}; i \rightarrow f)=\sum_{\mu M_f}|\langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle|^2</math> | ||
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a) Express <math>B(T_{\lambda}; i \rightarrow f)</math> in terms of the reduced matrix element | a) Express <math>B(T_{\lambda}; i \rightarrow f)</math> in terms of the reduced matrix element <math>(f|| T_{\lambda}|| i)</math> and show that it does not depend on the initial projection <math>M_i</math>. | ||
b) Establish the detailed balance between the reduced transition probabilities of the direct <math>i \rightarrow f</math>, and inverse <math>f \rightarrow i</math> processes. ''Hint: This is just the ratio between <math>B(T_{\lambda}; i \rightarrow f)</math> and <math>B(T_{\lambda}; f \rightarrow i)</math> '' | |||
SOLUTION: | |||
a) According to the Wigner-Eckert theorem, the entire dependence of the matrix element of a tensor operator on the magnetic quantum numbers is concentrated in the vector coupling coefficients, | |||
<math>\langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle = (-)^{J_f-M_f} | |||
\left( \begin{array}{lll} | |||
J_f & \lambda & J_i \\ | |||
-M_f & \mu & M_i | |||
\end{array} \right) | |||
(f|| T_{\lambda}|| i) </math> | |||
We obtain the rate by squaring this and summing over <math>\mu</math> and <math>M_f</math> | |||
<math>B(T_{\lambda}; i \rightarrow f)= (f|| T_{\lambda}|| i)^2 \sum_{\mu M_f} | |||
\left( \begin{array}{lll} | |||
J_f & \lambda & J_i \\ | |||
-M_f & \mu & M_i | |||
\end{array} \right) | |||
\left( \begin{array}{lll} | |||
J_f & \lambda & J_i \\ | |||
-M_f & \mu & M_i | |||
\end{array} \right)</math> | |||
Using the orthogonality condition: <math>\sum_{m_1 m_2} | |||
\left( \begin{array}{lll} | |||
j_1 & j_2 & j_3 \\ | |||
m_1 & m_2 & m_3 | |||
\end{array} \right) | |||
\left( \begin{array}{lll} | |||
j_1 & j_2 & j'_3 \\ | |||
m_1 & m_2 & m'_3 | |||
\end{array} \right)=\dfrac{\delta_{j_3 j'_3}\delta_{m_3 m'_3}}{2j_3+1}</math> | |||
Which leads us to our final result: <math>B(T_{\lambda}; i \rightarrow f)= \dfrac{(f|| T_{\lambda}|| i)^2}{2J_i+1}</math> | |||
It is obvious that this result does not depend on <math>M_i</math> | |||
b) All that is missing to find the detailed balance relation is <math>B(T_{\lambda}; f \rightarrow i)</math>. This is done in the same way as part a). | |||
<math>B(T_{\lambda}; f \rightarrow i)= (f|| T_{\lambda}|| i)^2 \sum_{\mu M_i} | |||
\left( \begin{array}{lll} | |||
J_f & \lambda & J_i \\ | |||
-M_f & \mu & M_i | |||
\end{array} \right) | |||
\left( \begin{array}{lll} | |||
J_f & \lambda & J_i \\ | |||
-M_f & \mu & M_i | |||
\end{array} \right)</math> | |||
Note, the only difference is the sum over <math>M_i</math> | |||
Thus, we have <math>B(T_{\lambda}; i \rightarrow f)= \dfrac{(f|| T_{\lambda}|| i)^2}{2J_f+1}</math> | |||
And the detailed balance relation is: <math>\frac{B(T_{\lambda}; i \rightarrow f)}{B(T_{\lambda}; f \rightarrow i)}=\frac{2J_f+1}{2J_i+1}</math> | |||
Latest revision as of 16:49, 12 April 2010
Posted by student team #5 (Anthony Kuchera, Jeff Klatsky, Chelsey Morien)
QUESTION:
Consider a transition from 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 i \rightarrow f} between two states of a nucleus with spins 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 J_i } 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 J_f} , respectively. The transition probability is proportional to the squared matrix element 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 J_f M_f| T_{\lambda \mu}| J_i M_i \rangle|^2} where 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 T_{\lambda \mu}} is a hermitian tensor operator of rank 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} responsible for the process. Define the reduced transition probability
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 B(T_{\lambda}; i \rightarrow f)=\sum_{\mu M_f}|\langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle|^2}
as a sum of squared matrix elements over final projections 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_f} and operator projections 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 \mu} .
a) Express 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 B(T_{\lambda}; i \rightarrow f)}
in terms of the reduced matrix element 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 (f|| T_{\lambda}|| i)}
and show that it does not depend on the initial projection 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_i}
.
b) Establish the detailed balance between the reduced transition probabilities of the direct 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 i \rightarrow f} , and inverse 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 f \rightarrow i} processes. Hint: This is just the ratio between 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 B(T_{\lambda}; i \rightarrow f)} 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 B(T_{\lambda}; f \rightarrow i)}
SOLUTION:
a) According to the Wigner-Eckert theorem, the entire dependence of the matrix element of a tensor operator on the magnetic quantum numbers is concentrated in the vector coupling coefficients,
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 J_f M_f| T_{\lambda \mu}| J_i M_i \rangle = (-)^{J_f-M_f} \left( \begin{array}{lll} J_f & \lambda & J_i \\ -M_f & \mu & M_i \end{array} \right) (f|| T_{\lambda}|| i) }
We obtain the rate by squaring this and summing over 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 \mu} 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_f}
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 B(T_{\lambda}; i \rightarrow f)= (f|| T_{\lambda}|| i)^2 \sum_{\mu M_f} \left( \begin{array}{lll} J_f & \lambda & J_i \\ -M_f & \mu & M_i \end{array} \right) \left( \begin{array}{lll} J_f & \lambda & J_i \\ -M_f & \mu & M_i \end{array} \right)}
Using the orthogonality condition: 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 \sum_{m_1 m_2} \left( \begin{array}{lll} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{array} \right) \left( \begin{array}{lll} j_1 & j_2 & j'_3 \\ m_1 & m_2 & m'_3 \end{array} \right)=\dfrac{\delta_{j_3 j'_3}\delta_{m_3 m'_3}}{2j_3+1}}
Which leads us to our final result: 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 B(T_{\lambda}; i \rightarrow f)= \dfrac{(f|| T_{\lambda}|| i)^2}{2J_i+1}}
It is obvious that this result does not depend on 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_i}
b) All that is missing to find the detailed balance relation 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 B(T_{\lambda}; f \rightarrow i)}
. This is done in the same way as part a).
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 B(T_{\lambda}; f \rightarrow i)= (f|| T_{\lambda}|| i)^2 \sum_{\mu M_i} \left( \begin{array}{lll} J_f & \lambda & J_i \\ -M_f & \mu & M_i \end{array} \right) \left( \begin{array}{lll} J_f & \lambda & J_i \\ -M_f & \mu & M_i \end{array} \right)}
Note, the only difference is the sum over 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_i}
Thus, 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 B(T_{\lambda}; i \rightarrow f)= \dfrac{(f|| T_{\lambda}|| i)^2}{2J_f+1}}
And the detailed balance relation 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 \frac{B(T_{\lambda}; i \rightarrow f)}{B(T_{\lambda}; f \rightarrow i)}=\frac{2J_f+1}{2J_i+1}}