Phy5645/Plane Rotator: Difference between revisions
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'''(a)''' <math>A\!</math> can be determined from the normalization condition | '''(a)''' <math>A\!</math> can be determined from the normalization condition, | ||
<math>1=\int_{-\pi}^{\pi}d\phi\,|\psi(\phi)|^2=A^2 \int_{-\pi}^{\pi}d\phi\,\sin^ | <math>1=\int_{-\pi}^{\pi}d\phi\,|\psi(\phi)|^2=A^2 \int_{-\pi}^{\pi}d\phi\,\sin^4{\phi} = A^2\cdot\frac{3\pi}{4}.</math> | ||
Therefore, <math>A=\frac{2}{\sqrt{3\pi}}.</math> | |||
b) The probability to measure the angular momentum to be <math> \hbar m </math> is | '''(b)''' The probability to measure the angular momentum to be <math> \hbar m </math> is | ||
<math> P_m = | | <math> P_m = |\langle m|\psi\rangle|^2 = \left |\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi}d\phi\,e^{im\phi}\psi(\phi)\right |^2 = \tfrac{2}{3} \delta_{m,0}+\tfrac {1}{6}(\delta_{m,2}+\delta_{m,-2}) </math> | ||
Therefore the probability | Therefore the probability of measuring <math>L_z = 0\!</math> is <math>\tfrac{2}{3},</math> that of measuring <math>L_z = 2\hbar\!</math> is <math>\tfrac{1}{6}</math>, and that of measuring <math> L_z = -2\hbar</math> is also <math>\tfrac {1}{6}.</math> The probability of measuring any other value is zero. | ||
c) <math> | c) <math> \langle\hat{L}_z\rangle=\frac{4}{3\pi}\int_{-\pi}^{\pi}=0 </math> | ||
<math> <L^2>=\frac{4\hbar}{3} </math> | <math> <L^2>=\frac{4\hbar}{3} </math> | ||
Back to [[Orbital Angular Momentum Eigenfunctions]] | Back to [[Orbital Angular Momentum Eigenfunctions]] | ||
Revision as of 23:28, 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^4{\phi} = A^2\cdot\frac{3\pi}{4}.}
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 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 = |\langle m|\psi\rangle|^2 = \left |\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi}d\phi\,e^{im\phi}\psi(\phi)\right |^2 = \tfrac{2}{3} \delta_{m,0}+\tfrac {1}{6}(\delta_{m,2}+\delta_{m,-2}) }
Therefore the probability of measuring 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 \tfrac{2}{3},} that of measuring 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\hbar\!} 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 \tfrac{1}{6}} , and that of measuring 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\hbar} is also 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 \tfrac {1}{6}.} The probability of measuring 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 \langle\hat{L}_z\rangle=\frac{4}{3\pi}\int_{-\pi}^{\pi}=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} }