Editing Matrix Elements and the Wigner Eckart Theorem Example: Difference between revisions
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<math> \bold (a) </math> According to the Wigner-Eckart Theorem we have: | <math> \bold (a) </math> According to the Wigner-Eckart Theorem we have: | ||
<math> \langle J M | J_1^- | J' (M + 1)\rangle = \langle J', 1, M+1, -1 | J', 1, J, M\rangle \langle J || \vec J_1||J'\rangle</math> The Clebsch-Gordan coefficient unless the triangular relation among the vectors is satisfied, i.e. <math> |J'-1| \leq J\leq J'+1 </math> | <math> \langle J M | J_1^- | J' (M + 1)\rangle = \langle J', 1, M+1, -1 | J', 1, J, M\rangle \langle J || \vec J_1||J'\rangle</math> The Clebsch-Gordan coefficient unless the triangular relation among the vectors is satisfied, i.e. <math> |J'-1| \leq J\leq J'+1 </math> | ||
This implies that either <math> \bold J'= \bold J </math> or <math> J'=J \pm 1 </math> | |||
<math> \bold (b) </math> We use again the Wigner-Eckart Theorem | |||
<math>i) \langle J M | J_1^- | J (M + 1)\rangle = \langle J', 1, M+1, -1 | J, 1, J, M\rangle \langle J || \vec J_1||J'\rangle</math> | |||
but we know that <math>\langle J', 1, M+1, -1 | J, 1, J, M\rangle = \sqrt{(J+M+1)(J-M)} </math> | |||
therefore <math> \frac{\langle J M | J_1^- | J (M + 1)\rangle}{\sqrt{(J+M+1)(J-M)}} = \langle J || \vec J_1||J'\rangle</math> which does not depend on M. | |||
Revision as of 20:45, 6 April 2010
Let 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 \vec J_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 \vec J_2 } be two angular momentum operators, 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 \vec J = \vec J_1 + \vec J_2 } is the sum of these two vectors, 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 M \rangle } denotes the eigen states 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 \vec J^2 } 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 \bold J _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 \bold (a) } Show that the matrix elements 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 J_1^- , \langle J M | J_1^- | J' (M + 1)\rangle } , vanish, unless 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 \bold J' = \bold J } or 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 \bold J' = \bold J \pm 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 \bold (b) }
Show also that the following expressions are independent 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 \bold M }
:
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) \frac{\langle J M | J_1^- | J (M + 1)\rangle}{\sqrt{(J+M+1)(J-M)}} }
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 ii) \frac{\langle J M | J_1^- | (J-1)(M + 1)\rangle}{\sqrt{(J-M+1)(J-M)}} }
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 iii) \frac{\langle J M | J_1^- | (J+1)(M + 1)\rangle}{\sqrt{(J+M+2)(J+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 \bold (a) } According to the Wigner-Eckart Theorem 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 \langle J M | J_1^- | J' (M + 1)\rangle = \langle J', 1, M+1, -1 | J', 1, J, M\rangle \langle J || \vec J_1||J'\rangle} The Clebsch-Gordan coefficient unless the triangular relation among the vectors is satisfied, i.e. 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'-1| \leq J\leq J'+1 }
This implies that either 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 \bold J'= \bold J } or 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'=J \pm 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 \bold (b) } We use again the Wigner-Eckart Theorem
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) \langle J M | J_1^- | J (M + 1)\rangle = \langle J', 1, M+1, -1 | J, 1, J, M\rangle \langle J || \vec J_1||J'\rangle}
but we know that 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', 1, M+1, -1 | J, 1, J, M\rangle = \sqrt{(J+M+1)(J-M)} }
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 \frac{\langle J M | J_1^- | J (M + 1)\rangle}{\sqrt{(J+M+1)(J-M)}} = \langle J || \vec J_1||J'\rangle} which does not depend on M.