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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 \chapter{Angular momentum}\begin{enumerate}    \item    Take $\hat{L}_x$ as an example,     \begin{align*}        \braket{f|\hat{L}_x|g} &= \jifen         f^{*} \dfrac{\hbar}{\imath}(y\pd{g}{z}-z\pd{g}{y}) \dif \tau \\        &= \dfrac{\hbar}{\imath}\left ( -\jifen gy\pd{f^{*}}{z} \dif \tau + \jifen         gz\pd{f^{*}}{y} \dif \tau \right )    \end{align*}    \begin{align*}        \braket{\hat{L}_xf|g} &= -\dfrac{\hbar}{\imath}\left ( \jifen         yg\pd{f^{*}}{z} \dif \tau -\jifen gz\pd{f^{*}}{y} \dif \tau \right )    \end{align*}    \item        \begin{align*}            \commu{\hat{L}_x}{x}\psi &= \hat{L}_xx\psi-x\hat{L}_x\psi \\            &= \dfrac{\hbar}{\imath}\psi        \end{align*}        $\commu{\hat{L}_x}{x}=\frac{\hbar}{\imath}$.        \begin{align*}            \commu{ \lx }{ \px } &= \commu{y \pz -z \py}{\px} \\            &= \commu{y \pz }{ \px } - \commu{z\py}{\px} \\            &= \commu{y}{ \px } \pz + y\commu{ \pz }{ \px }             - \commu{z}{ \px }\py - z\commu{ \py }{ \px } \\            &= 0        \end{align*}        \begin{align*}            \commu{ \lx }{y} &= \commu{y \pz }{y} - \commu{z \py             }{y} \\            &= -z \commu { \py }{y} \\            &= z\imath\hbar        \end{align*}        \begin{align*}            \commu{ \lx }{ \py } &= \commu{y \pz }{ \py             }-\commu{z \py }{ \py } \\            &= \commu{y}{ \py }\pz \\            &= \hbar^2 \pd{}{z}        \end{align*}    \item            \begin{align*}                \braket{ \lx }=\braket{lm| \lx | lm} &=                 \dfrac{1}{\imath\hbar}\left(\braket{lm| \ly \lz | lm}-                 \braket{lm | \lz \ly |lm }\right) \\                &= \dfrac{1}{\imath\hbar}\left(m\hbar \braket{lm| \ly                 | lm} - m\hbar\braket{lm| \ly | lm}\right) \\                &=0            \end{align*}            Similarily, $\braket{ \ly }=0$.     \item         As $\commu{ \lx }{ \ly }=\imath\hbar \lz$, we have        \begin{align*}            \Delta \lx \Delta \ly &\geq \dfrac{1}{2}|\braket{\hbar \lz }| \\            &= \dfrac{1}{2}\hbar \braket{ \lz } \\        \end{align*}        And         $\braket{lm | \lz | lm} = m\hbar$    \item        \begin{align*}            \commu{ \jf }{ \jx } &= \commu{ \jy }{ \jx }\jy + \jy \commu{ \jy }{ \jx } + \commu{ \jz }{ \jx } \jz+ \jz \commu{ \jz }{ \jx } \\            &= -\imath\hbar \jz \jy -\imath\hbar \jy \jz + \imath\hbar \jy \jz + \imath\hbar \jz \jy \\            &= 0        \end{align*}        Similarily we can prove that \jf commutes with \jy.                      \item        \begin{align*}            \braket{f | \jp | g} &= \braket{f | \jx + \imath \jy | g} \\            &= \braket{f | \jx | g} + \imath \braket{ f | \jy | g} \\            &= \braket{\jx f|g} - \braket{\imath \jy f | g}\\            &= \braket{(\jx - \imath \jy )f | g} \\            &= \braket{\jm f | g}        \end{align*}        where the fact that \jx and \jy is self-adjoint is used.    \item        \begin{enumerate}            \item                 \label{item:1}                \begin{align*}                    \commu{ \jz }{ \jp } &= \commu{ \jz }{ \jx + \imath \jy } \\                    &= \commu{ \jz }{ \jx } + \commu{ \jz }{ \imath \jy } \\                    &= \hbar ( \jx + \imath \jy )\\                    &= \jp                \end{align*}            \item                \begin{align*}                    \commu{ \jf }{ \jp } &= \commu{ \jf }{ \jx } + \commu { \jf }{ \imath \jy } \\                    &= 0 + 0\\                    &= 0                \end{align*}                            \item                \begin{align*}                    \commu{ \jp }{ \jm } &= \commu{ \jx + \imath \jy }{ \jx - \imath \jy } \\                    &= \commu{ \jx }{ \jx -\imath \jy} + \commu{\imath \jy}{\jx - \imath \jy} \\                    &= 2\hbar \jz                \end{align*}            \item                \begin{align*}                    \jp \jm &= (\jx + \imath \jy )(\jx - \imath \jy ) \\                    &= \jfx + \jfy + \hbar \jz \\                    &= \jf - \jfz + \hbar \jz                \end{align*}        \end{enumerate}    \item        \begin{align*}            \ket{ \jp \jm | \lambda m} &= \ket{ \jp \sqrt{\lambda-m(m-1)}\hbar |\lambda , m-1 } \\            &= \ket{\sqrt{\lambda-m(m-1)}\hbar \sqrt{\lambda-(m-1)m}\hbar | \lambda m } \\            &= (\lambda - m^2+m )\hbar^2        \end{align*}    \item        \begin{align*}            \braket{jm | \jx | jm} &= \dfrac{1}{2} \braket{jm | \jp | jm} + \dfrac{1}{2} \braket{ jm | \jp |jm}\\            &=\dfrac{1}{2} \braket{jm | \sqrt{j-m(m+1)} |j, m+1} +\frac{1}{2}\braket{jm | \sqrt{j-m(m-1)} |j, m-1} \\            &= 0        \end{align*}        \begin{align*}            \braket{ jm | \jy | jm} & = \frac{1}{2\imath} \braket{ jm | \jp - \jm | jm} \\            &= 0        \end{align*}    \item        \begin{align*}            \braket{jm | \jfx | jm} &= \frac{1}{4} \braket{jm | \jfp + \jfm + \jp \jm + \jm \jp | jm} \\            &= \dfrac{1}{4} \braket{jm | \jm \jp | jm } + \dfrac{1}{4} \braket{jm | \jp \jm | jm} \\            &= \dfrac{1}{2}[j(j+1)-m^2]\hbar^2        \end{align*}    \item        \begin{align*}            \commu{ \jx }{ \jp } \ket{jm} &= \dfrac{1}{2} \commu{ \jm }{ \jp} \ket{ jm} \\            &= -m \hbar^2 \ket{ jm}        \end{align*}        So the eigenvalue of \commu{ \jx }{ \jp } is $-m\hbar^2$.        \begin{align*}            \commu {\jy }{ \jp} \ket{ jm} &= m\hbar^2 \ket{ jm}        \end{align*}        The eigenvalue of \commu{ \jy }{\jp} is $m\hbar^2$.\end{enumerate}
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