Phy5645/Plane Rotator: Difference between revisions
No edit summary |
No edit summary |
||
| Line 15: | Line 15: | ||
<math> <L^2>=\frac{4\hbar}{3} </math> | <math> <L^2>=\frac{4\hbar}{3} </math> | ||
Back to [[Orbital Angular Momentum Eigenfunctions]] | |||
Revision as of 23:17, 29 August 2013
(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 A\!} can be determined from the normalization 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 1=\int_{-\pi}^{\pi}d\phi\,|\psi(\phi)|^2=A^2 \int_{-\pi}^{\pi}d\phi\,\sin^2{2\phi} = A^23\pi/4 }
Then, we could get 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 A= \frac{2}{\sqrt{3 \pi}} }
b) The probability to measure the angular momentum to be 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 m }
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 P_m = |<\psi_m|\psi>|^2 = |\int_{-\pi}^{\pi}d\phi \frac {e^{-im\phi}}{\sqrt {2\pi}} \psi(\phi)|^2 = \frac {2}{3} \delta_{m,0} + \frac {1}{6}(\delta_{m,2}+\delta_{m,-2}) }
Therefore the probability to measure 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 L_z = 0 } 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 {2}{3} } , the probability to measure 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 L_z = 2 } 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 {1}{6} } , and the probability to measure 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 L_z = -2 } is </math> 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 {1}{6} } . The probability to measure any other value is zero.
c) 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 <L_z>=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 <L^2>=\frac{4\hbar}{3} }