Phy5645/Plane Rotator
(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}d\phi\,\sin^2{\phi}\left (-i\hbar\frac{d}{d\phi}\sin^2{\phi}\right )=-\frac{8}{3\pi}i\hbar\int_{-\pi}^{\pi}d\phi\,\sin^3{\phi}\cos{\phi}=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 \langle\hat{L}_z^2\rangle=\frac{4}{3\pi}\int_{-\pi}^{\pi}d\phi\,\sin^2{\phi}\left (-\hbar^2\frac{d^2}{d\phi^2}\sin^2{\phi}\right )=-\frac{8}{3\pi}\hbar^2\int_{-\pi}^{\pi}d\phi\,\sin^2{\phi}(1-2\sin^2{\phi})=\tfrac{4}{3}\hbar^2}