DetailedBalance: Difference between revisions
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Consider a transition from <math>i \rightarrow f</math> between two states of the 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 | Consider a transition from <math>i \rightarrow f</math> between two states of the 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. | |||
Revision as of 15:25, 12 April 2010
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 the 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.