Editing Matrix Elements and the Wigner Eckart Theorem Example

From PhyWiki
Revision as of 21:45, 6 April 2010 by JustinVonmoss (talk | contribs)
Jump to navigation Jump to search

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.

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) \langle J M | J_1^- | (J-1) (M + 1)\rangle = \langle J-1, 1, M+1, -1 | J-1, 1, J, M\rangle \langle J || \vec J_1||J-1\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, 1, M+1, -1 | J-1, 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-1) (M + 1)\rangle}{\sqrt{(J-M-1)(J-M)}} = \langle J || \vec J_1||J-1\rangle} which does not depend on 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) \langle J M | J_1^- | (J+1) (M + 1)\rangle = \langle J+1, 1, M+1, -1 | J+1, 1, J, M\rangle \langle J || \vec J_1||J+1\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, 1, M+1, -1 | J+1, 1, J, M\rangle = \sqrt{(J+M+2)(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+1) (M + 1)\rangle}{\sqrt{(J+M+2)(J+M)}} = \langle J || \vec J_1||J+1\rangle} which does not depend on M.

Retrieved from "http://wiki.physics.fsu.edu/index.php?title=Editing_Matrix_Elements_and_the_Wigner_Eckart_Theorem_Example&oldid=7475"