現代の量子力学の行間埋め　第４章

\begin{eqnarray} [x,L_x] &=& [x,yp_z-zp_y] \\ \\ &=& 0 \\ \\ [x,L_y] &=& [x,zp_x-xp_z] \\ \\ &=& [x,zp_x]-[x,xp_z] \\ \\ &=& z[x,p_x]-0 \\ \\ &=& zi\hbar \\ \\ [y,L_x] &=& [y,yp_z-zp_y] \\ \\ &=& i\hbar\frac{\partial}{\partial p_y}(yp_z-zp_y) \\ \\ &=& -zi\hbar \\ \\ [y,L_y] &=& [y,zp_x-xp_z] \\ \\ &=& 0 \\ \\ [z,L_x] &=& [z,yp_z-zp_y] \\ \\ &=& [z,yp_z]-[z,zp_y] \\ \\ &=& y[z,p_z]-0 \\ \\ &=& yi\hbar \\ \\ [z,L_y] &=& [z,zp_x-xp_z] \\ \\ &=& i\hbar\frac{\partial}{\partial p_z}(zp_x-xp_z)\\ \\ &=& -xi\hbar \\ \\ [z,L_z] &=& [z,xp_y-yp_x] \\ \\ &=& 0 \\ \\ [p_x,L_x] &=& [p_x,yp_z-zp_y] \\ \\ &=& -i\hbar\frac{\partial}{\partial x}(yp_z-zp_y) \\ \\ &=& 0 \\ \\ [p_x,L_y] &=& [p_x,zp_x-xp_z] \\ \\ &=& -i\hbar\frac{\partial}{\partial x}(zp_x-xp_z) \\ \\ &=& i\hbar p_z \\ \\ [p_y,L_x] &=& [p_y,yp_z-zp_y] \\ \\ &=& -i\hbar\frac{\partial}{\partial y}(yp_z-zp_y) \\ \\ &=& -i\hbar p_z \\ \\ [p_y,L_y] &=& [p_y,zp_x-xp_z] \\ \\ &=& -i\hbar\frac{\partial}{\partial y}(zp_x-xp_z) \\ \\ &=& 0 \\ \\ [p_z,L_x] &=& [p_z,yp_z-zp_y] \\ \\ &=& -i\hbar\frac{\partial}{\partial z}(yp_z-zp_y) \\ \\ &=& i\hbar p_y \\ \\ [p_z,L_y] &=& [p_z,zp_x-xp_z] \\ \\ &=& -i\hbar\frac{\partial}{\partial z}(zp_x-xp_z) \\ \\ &=& -i\hbar p_x \\ \\ [p_z,L_z] &=& [p_z,xp_y-yp_x] \\ \\ &=& 0 \end{eqnarray} より、以上とp.289下、式(1.232b)をまとめると\([x_i,L_j]=i\hbar\varepsilon_{ijk}x_k,[p_i,L_j]=i\hbar\varepsilon_{ijk}p_k\)となる。

\begin{eqnarray} -(\boldsymbol{L}\times\boldsymbol{p})\cdot(\boldsymbol{p}\times\boldsymbol{L}) &=& - \left( \begin{array}{cccc} L_yp_z-L_zp_y \\ L_zp_x-L_xp_z \\ L_xp_y-L_zp_y \end{array} \right)\cdot \left( \begin{array}{cccc} p_yL_z-p_zL_y \\ p_zL_x-p_xL_z \\ p_xL_y-p_zL_y \end{array} \right) \\ \\ &=& - \left( \begin{array}{cccc} p_zL_y+i\hbar p_x-p_yL_z+i\hbar p_x \\ p_xL_z+i\hbar p_y-p_zL_x+i\hbar p_y \\ p_yL_x+i\hbar p_z-p_xL_y+i\hbar p_z \end{array} \right)\cdot \left( \begin{array}{cccc} p_yL_z-p_zL_y \\ p_zL_x-p_xL_z \\ p_xL_y-p_xL_y \end{array} \right) \\ \\ &=& - \left[ \left( \begin{array}{cccc} p_zL_y -p_yL_z \\ p_xL_z -p_zL_x \\ p_yL_x -p_xL_y \end{array} \right)+2i\hbar\boldsymbol{p} \right]\cdot \left( \begin{array}{cccc} p_yL_z-p_zL_y \\ p_zL_x-p_xL_z \\ p_xL_y-p_yL_x \end{array} \right) \\ \\ &=& \left[ \left( \begin{array}{cccc} p_zL_y -p_yL_z \\ p_xL_z -p_zL_x \\ p_yL_x -p_xL_y \end{array} \right)+2i\hbar\boldsymbol{p} \right]\cdot \left( \begin{array}{cccc} p_zL_y-p_yL_z \\ p_xL_z-p_zL_x \\ p_yL_x-p_xL_y \end{array} \right) \\ \\ &=& (p_zL_y -p_yL_z)(p_zL_y -p_yL_z)+(p_xL_z -p_zL_x)(p_xL_z -p_zL_x)+(p_yL_x -p_xL_y)(p_yL_x -p_xL_y)+2i\hbar\boldsymbol{p}\cdot(-\boldsymbol{p}\times\boldsymbol{L}) \\ \\ &=& (p_zL_yp_zL_y -p_zL_yp_yL_z -p_yL_zp_zL_y+p_yL_zp_yL_z)+(p_xL_zp_xL_z -p_xL_zp_zL_x -p_zL_xp_xL_z+p_zL_xp_zL_x)+(p_yL_xp_yL_x -p_yL_xp_xL_y -p_xL_yp_yL_x+p_xL_yp_xL_y)+2i\hbar0 \\ \\ &=& (p_zL_yp_zL_y+p_yL_zp_yL_z+p_xL_zp_xL_z+p_zL_xp_zL_x+p_yL_xp_yL_x+p_xL_yp_xL_y )-( p_zL_yp_yL_z +p_yL_zp_zL_y+p_xL_zp_zL_x +p_zL_xp_xL_z+p_yL_xp_xL_y +p_xL_yp_yL_x) \\ \\ &=& (p_z(p_zL_y+i\hbar p_x)L_y+p_y(p_yL_z-i\hbar p_x)L_z+p_x(p_xL_z+i\hbar p_y)L_z+p_z(p_zL_x-i\hbar p_y)L_x+p_y(p_yL_x+i\hbar p_z)L_x+p_x(p_xL_y-i\hbar p_z)L_y ) \\ &-& ( (L_yp_z-i\hbar p_x)p_yL_z +(L_zp_y+i\hbar p_x)p_zL_y+(L_zp_x-i\hbar p_y)p_zL_x +(L_xp_z+i\hbar p_y)p_xL_z+(L_xp_y-i\hbar p_z)p_xL_y +(L_yp_x+i\hbar p_z)p_yL_x) \\ \\ &=& (p_zp_zL_yL_y+p_yp_yL_zL_z+p_xp_xL_zL_z+p_zp_zL_xL_x+p_yp_yL_xL_x+p_xp_xL_yL_y ) \\ &-& ( L_yp_zp_yL_z +L_zp_yp_zL_y+L_zp_xp_zL_x +L_xp_zp_xL_z+L_xp_yp_xL_y +L_yp_xp_yL_x) \\ &+& (+i\hbar p_zp_xL_y-i\hbar p_yp_xL_z+i\hbar p_xp_yL_z-i\hbar p_zp_yL_x+i\hbar p_yp_zL_x-i\hbar p_xp_zL_y-i\hbar p_xp_yL_z+i\hbar p_xp_zL_y-i\hbar p_yp_zL_x+i\hbar p_yp_xL_z-i\hbar p_zp_xL_y+i\hbar p_zp_yL_x) \\ \\ &=& (p_zp_zL_yL_y+p_yp_yL_zL_z+p_xp_xL_zL_z+p_zp_zL_xL_x+p_yp_yL_xL_x+p_xp_xL_yL_y+{\color{red}p_xp_xL_xL_x+p_yp_yL_yL_y+p_zp_zL_zL_z-(p_xp_xL_xL_x+p_yp_yL_yL_y+p_zp_zL_zL_z)} ) \\ &-& ( L_yp_yp_zL_z +L_zp_zp_yL_y+L_zp_zp_xL_x +L_xp_xp_zL_z+L_xp_xp_yL_y +L_yp_yp_xL_x) \\ &+& 0 \\ \\ &=& (p_z^2(L_x^2+L_y^2+L_z^2)+p_y^2(L_x^2+L_y^2+L_z^2)+p_x^2(L_x^2+L_y^2+L_z^2)-(p_xL_xp_xL_x+p_yL_yp_yL_y+p_zL_zp_zL_z) ) \\ &-& ( L_yp_yp_zL_z +L_zp_zp_yL_y+L_zp_zp_xL_x +L_xp_xp_zL_z+L_xp_xp_yL_y +L_yp_yp_xL_x) \\ \\ &=& ((p_x^2+p_y^2+p_z^2)(L_x^2+L_y^2+L_z^2) ) \\ &-& ( p_yL_yp_zL_z +p_zL_zp_yL_y+p_zL_zp_xL_x +p_xL_xp_zL_z+p_xL_xp_yL_y +p_yL_yp_xL_x+{\color{red}p_xL_xp_xL_x+p_yL_yp_yL_y+p_zL_zp_zL_z}) \\ \\ &=& \boldsymbol{p}^2\boldsymbol{L}^2 \\ &-& ((p_xL_x)(p_xL_x+p_yL_y+p_zL_z)+(p_yL_y)(p_xL_x+p_yL_y+p_zL_z)+(p_zL_z)(p_xL_x+p_yL_y+p_zL_z)) \\ \\ &=& \boldsymbol{p}^2\boldsymbol{L}^2 - (p_xL_x+p_yL_y+p_zL_z)(p_xL_x+p_yL_y+p_zL_z) \\ \\ &=& \boldsymbol{p}^2\boldsymbol{L}^2 - (\boldsymbol{p}\cdot\boldsymbol{L})(\boldsymbol{p}\cdot\boldsymbol{L}) \\ \\ &=& \boldsymbol{p}^2\boldsymbol{L}^2 \end{eqnarray} と得られる。

\begin{eqnarray} \boldsymbol{p}\cdot\frac{\boldsymbol{x}}{r}-\frac{\boldsymbol{x}}{r}\cdot\boldsymbol{p} \\ \\ &=& \displaystyle\sum_i\left(p_i\frac{x_i}{r}-\frac{x_i}{r}p_i\right) \\ \\ &=& \displaystyle\sum_i[p_i,\frac{x_i}{r}] \\ \\ &=& \displaystyle\sum_i(-i\hbar\frac{\partial}{\partial x_i}\frac{x_i}{r})&...&\text{式(2.97b)より} \\ \\ &=& -\displaystyle\sum_ii\hbar\frac{\partial}{\partial x_i}\frac{x_i}{r} \\ \\ &=& -\displaystyle\sum_ii\hbar\frac{r-x_i\frac{\partial}{\partial x_i}r}{r^2} \\ \\ &=& -\displaystyle\sum_ii\hbar\frac{r-x_i\frac{x_i}{r}}{r^2}&...&r=\sqrt{x_i^2+x_j^2+x_k^2}\text{を用いた} \\ \\ &=& -\displaystyle\sum_ii\hbar\frac{r^2-x_i^2}{r^3}&\\ \\ &=& -i\hbar\frac{r^2-x_i^2}{r^3}-i\hbar\frac{r^2-x_j^2}{r^3}-i\hbar\frac{r^2-x_k^2}{r^3}&\\ \\ &=& -i\hbar\frac{3r^2-(x_i^2+x_j^2+x_k^2)}{r^3}\\ \\ &=& -i\hbar\frac{3r^2-r^2}{r^3}\\ \\ &=& -i\hbar\frac{2}{r}\\ \\ \end{eqnarray} と導出できる。

\begin{eqnarray} \frac{\boldsymbol{x}}{r}\cdot(\boldsymbol{p}\times\boldsymbol{L})-(\boldsymbol{L}\times\boldsymbol{p})\cdot\frac{\boldsymbol{x}}{r} &=& \frac{x}{r}(p_yL_z-p_zL_y)-(L_yp_z-L_zp_y)\frac{x}{r}+\ldots \\ \\ &=& \frac{1}{r}(xp_yL_z-xp_zL_y)-(L_y\underbrace{p_z\frac{1}{r}}_{(6)}-L_z\underbrace{p_y\frac{1}{r}}_{(6)})x+\ldots&\\ \\ &=& \frac{1}{r}(xp_yL_z-xp_zL_y)-(L_y\underbrace{(\frac{1}{r}p_z+i\hbar\frac{z}{r^3})}_{(6)}-L_z\underbrace{(\frac{1}{r}p_y+i\hbar\frac{y}{r^3})}_{(6)})x+\ldots&\\ \\ &=& \frac{1}{r}(xp_yL_z-xp_zL_y)-(L_y\frac{1}{r}p_zx+L_yi\hbar\frac{z}{r^3}x-L_z\frac{1}{r}p_yx-L_zi\hbar\frac{y}{r^3}x+\ldots&\\ \\ &=& \frac{1}{r}(xp_yL_z-xp_zL_y)-(\underbrace{\frac{1}{r}L_y}_{(7)}p_zx+i\hbar \underbrace{\frac1rL_y}_{(7)}\frac{zx}{r^2}-\underbrace{\frac{1}{r}L_z}_{(7)}p_yx-i\hbar \underbrace{\frac1rL_z}_{(7)}\frac{xy}{r^2}+\ldots&\\ \\ &=& \frac{1}{r}\left(xp_yL_z-xp_zL_y-L_yp_zx-i\hbar L_y\frac{zx}{r^2}+L_zp_yx+i\hbar L_z\frac{xy}{r^2}\right)+\ldots&\\ \\ &=& \frac{1}{r}\left(xp_yL_z-xp_zL_y-L_y{\color{red}xp_z}-i\hbar L_y\frac{zx}{r^2}+L_z{\color{red}xp_y}+i\hbar L_z\frac{xy}{r^2}\right)+\ldots&\\ \\ &=& \frac{1}{r}\left(xp_yL_z-xp_zL_y-L_yxp_z-i\hbar L_y\frac{zx}{r^2}+L_zxp_y+i\hbar L_z\frac{xy}{r^2}\right) \\ &+& \frac{1}{r}\left(yp_zL_x-yp_xL_z-L_zyp_x-i\hbar L_z\frac{xy}{r^2}+L_xyp_z+i\hbar L_x\frac{yz}{r^2}\right) \\ &+& \frac{1}{r}\left(zp_xL_y-zp_yL_x-L_xzp_y-i\hbar L_x\frac{yz}{r^2}+L_yzp_x+i\hbar L_y\frac{zx}{r^2}\right) &...&x\to y,y\to z,z\to x\text{として式が成立するため}\\ \\ &=& \frac{1}{r}\left(L_x[yp_z-zp_y]+L_y[zp_x-xp_z]+L_z[xp_y-yp_x]+[yp_z-zp_y]L_x+[zp_x-xp_z]L_y+[xp_y-yp_x]L_z+i\hbar(L_zxy-L_yzx-L_zxy+L_xyz-L_xyz+L_yzx)\frac{1}{r^2}\right) \\ \\ &=& \frac{1}{r}\left(L_xL_x+L_yL_y+L_zL_z+L_xL_x+L_yL_y+L_zL_z+i\hbar(0)\frac{1}{r^2}\right) \\ \\ &=& \frac{2}{r}\left(L_x^2+L_y^2+L_z^2\right) \\ \\ &=& \frac{2}{r}\boldsymbol{L}^2 \\ \\ \end{eqnarray} と導出できる。

\begin{eqnarray} [x,L_x] &=& [x,yp_z-zp_y] \\ \\ &=& 0 \\ \\ [x,L_y] &=& [x,zp_x-xp_z] \\ \\ &=& [x,zp_x]-[x,xp_z] \\ \\ &=& z[x,p_x]-0 \\ \\ &=& zi\hbar \\ \\ [y,L_x] &=& [y,yp_z-zp_y] \\ \\ &=& i\hbar\frac{\partial}{\partial p_y}(yp_z-zp_y) \\ \\ &=& -zi\hbar \\ \\ [y,L_y] &=& [y,zp_x-xp_z] \\ \\ &=& 0 \\ \\ [z,L_x] &=& [z,yp_z-zp_y] \\ \\ &=& [z,yp_z]-[z,zp_y] \\ \\ &=& y[z,p_z]-0 \\ \\ &=& yi\hbar \\ \\ [z,L_y] &=& [z,zp_x-xp_z] \\ \\ &=& i\hbar\frac{\partial}{\partial p_z}(zp_x-xp_z)\\ \\ &=& -xi\hbar \\ \\ [z,L_z] &=& [z,xp_y-yp_x] \\ \\ &=& 0 \\ \\ [p_x,L_x] &=& [p_x,yp_z-zp_y] \\ \\ &=& -i\hbar\frac{\partial}{\partial x}(yp_z-zp_y) \\ \\ &=& 0 \\ \\ [p_x,L_y] &=& [p_x,zp_x-xp_z] \\ \\ &=& -i\hbar\frac{\partial}{\partial x}(zp_x-xp_z) \\ \\ &=& i\hbar p_z \\ \\ [p_y,L_x] &=& [p_y,yp_z-zp_y] \\ \\ &=& -i\hbar\frac{\partial}{\partial y}(yp_z-zp_y) \\ \\ &=& -i\hbar p_z \\ \\ [p_y,L_y] &=& [p_y,zp_x-xp_z] \\ \\ &=& -i\hbar\frac{\partial}{\partial y}(zp_x-xp_z) \\ \\ &=& 0 \\ \\ [p_z,L_x] &=& [p_z,yp_z-zp_y] \\ \\ &=& -i\hbar\frac{\partial}{\partial z}(yp_z-zp_y) \\ \\ &=& i\hbar p_y \\ \\ [p_z,L_y] &=& [p_z,zp_x-xp_z] \\ \\ &=& -i\hbar\frac{\partial}{\partial z}(zp_x-xp_z) \\ \\ &=& -i\hbar p_x \\ \\ [p_z,L_z] &=& [p_z,xp_y-yp_x] \\ \\ &=& 0 \end{eqnarray} より、以上とp.289下、式(1.232b)をまとめると\([x_i,L_j]=i\hbar\varepsilon_{ijk}x_k,[p_i,L_j]=i\hbar\varepsilon_{ijk}p_k\)となる。

\begin{eqnarray} [L_x,\frac{1}{r}] &=& [yp_z-zp_y,\frac{1}{r}] \\ \\ &=& [yp_z,\frac{1}{r}]-[zp_y,\frac{1}{r}] \\ \\ &=& y[p_z,\frac{1}{r}]-z[p_y,\frac{1}{r}] \\ \\ &=& y(-i\hbar)\frac{\partial}{\partial z}\frac{1}{r}-z(-i\hbar)\frac{\partial}{\partial y}\frac{1}{r} \\ \\ &=& y(-i\hbar)\frac{-1}{r^2}\frac{\partial r}{\partial z}-z(-i\hbar)\frac{-1}{r^2}\frac{\partial r}{\partial y} \\ \\ &=& y(-i\hbar)\frac{-1}{r^2}\frac{z}{r}-z(-i\hbar)\frac{-1}{r^2}\frac{y}{r}&...&r=\sqrt{x^2+y^2+z^2}\text{を用いた} \\ \\ &=& 0& \\ \\ \\ [L_y,\frac{1}{r}] &=& [zp_x-xp_z,\frac{1}{r}] \\ \\ &=& [zp_x,\frac{1}{r}]-[xp_z,\frac{1}{r}] \\ \\ &=& z[p_x,\frac{1}{r}]-x[p_z,\frac{1}{r}] \\ \\ &=& z(-i\hbar)\frac{\partial}{\partial x}\frac{1}{r}-x(-i\hbar)\frac{\partial}{\partial z}\frac{1}{r} \\ \\ &=& z(-i\hbar)\frac{-1}{r^2}\frac{\partial r}{\partial x}-x(-i\hbar)\frac{-1}{r^2}\frac{\partial r}{\partial z} \\ \\ &=& z(-i\hbar)\frac{-1}{r^2}\frac{x}{r}-x(-i\hbar)\frac{-1}{r^2}\frac{z}{r}&...&r=\sqrt{x^2+y^2+z^2}\text{を用いた} \\ \\ &=& 0& \\ \\ \\ [L_z,\frac{1}{r}] &=& [xp_y-yp_x,\frac{1}{r}] \\ \\ &=& [xp_y,\frac{1}{r}]-[yp_x,\frac{1}{r}] \\ \\ &=& x[p_y,\frac{1}{r}]-y[p_x,\frac{1}{r}] \\ \\ &=& x(-i\hbar)\frac{\partial}{\partial y}\frac{1}{r}-y(-i\hbar)\frac{\partial}{\partial x}\frac{1}{r} \\ \\ &=& x(-i\hbar)\frac{-1}{r^2}\frac{\partial r}{\partial y}-y(-i\hbar)\frac{-1}{r^2}\frac{\partial r}{\partial x} \\ \\ &=& x(-i\hbar)\frac{-1}{r^2}\frac{y}{r}-y(-i\hbar)\frac{-1}{r^2}\frac{x}{r}&...&r=\sqrt{x^2+y^2+z^2}\text{を用いた} \\ \\ &=& 0& \\ \\ \\ \end{eqnarray} と示される。

\begin{eqnarray} [(p_yL_z-p_zL_y),(p_zL_x-p_xL_z)] &=& [p_yL_z,p_zL_x]-[p_yL_z,p_xL_z]-[p_zL_y,p_zL_x]+[p_zL_y,p_xL_z] \\ \\ &=& \underbrace{p_z[p_yL_z,L_x]}_{(1)}-\underbrace{([p_yL_z,p_x]L_z+p_x[p_yL_z,L_z])}_{(2)}-\underbrace{([p_zL_y,p_z]L_x+p_z[p_zL_y,L_x])}_{(2)}+\underbrace{p_z[L_y,p_xL_z]}_{(1)}&...&(1)p_z\text{と}p_i,L_z\text{は交換するため}(2)\text{式(1.232e)より} \\ \\ &=& -p_z[L_x,p_yL_z]-(-p_y[p_x,L_z]L_z+\underbrace{p_x[p_y,L_z]L_z}_{(3)})-(\underbrace{p_z[L_y,p_z]L_x}_{(3)}{\color{red}-}p_z[L_x,p_zL_y])+\underbrace{p_z([L_y,p_x]L_z+p_x[L_y,L_z])}_{(2)}&...&(3)p_z,L_z\text{同士は交換するため} \\ \\ &=& -p_z([L_x,p_y]L_z+p_y[L_x,L_z])-(-p_y(-i\hbar p_y)L_z+\underbrace{p_x(i\hbar p_x)L_z}_{(3)})-(\underbrace{-p_z[p_z,L_y]L_x}_{(3)}-p_z([L_x,p_z]L_y+p_z[L_x,L_y]))+p_z(-[p_x,L_y]L_z+p_xi\hbar L_x)&...&\\ \\ &=& -p_z(-[p_y,L_x]L_z+p_y(-i\hbar L_y))-(i\hbar p_y^2L_z+i\hbar p_x^2L_z)-(-p_z(-i\hbar p_x)L_x-p_z((-i\hbar p_y)L_y+p_zi\hbar L_z))+p_z(-i\hbar p_zL_z+p_xi\hbar L_x)&...&\\ \\ &=& -p_z(-(-i\hbar)p_zL_z-i\hbar p_yL_y)-(i\hbar p_y^2L_z+i\hbar p_x^2L_z)-(i\hbar p_z p_xL_x+i\hbar p_z p_yL_y-p_z^2i\hbar L_z)+p_z(-i\hbar p_zL_z+i\hbar p_x L_x)&...&\\ \\ &=& -i\hbar(p_x^2+p_y^2+ p_z^2)L_z\\ \\ \end{eqnarray} と導出できる。

\begin{eqnarray} &&[(p_yL_z-p_zL_y-i\hbar p_x),\frac{y}{r}]-[(p_zL_x-p_xL_z-i\hbar p_y),\frac{x}{r}] \\ \\ &=& [p_yL_z-p_zL_y,\frac{y}{r}]-i\hbar[ p_x,\frac{y}{r}]-[p_zL_x-p_xL_z,\frac{x}{r}]+i\hbar[ p_y,\frac{x}{r}] \\ \\ &=& [p_yL_z-p_zL_y,\frac{1}{r}y]-i\hbar(-i\hbar)\frac{\partial}{\partial x}\frac{y}{r}-[p_zL_x-p_xL_z,\frac{1}{r}x]+i\hbar(-i\hbar)\frac{\partial}{\partial y}\frac{x}{r} \\ \\ &=& [p_yL_z-p_zL_y,\frac{1}{r}]y+\frac{1}{r}[p_yL_z-p_zL_y,y]-i\hbar(-i\hbar)\frac{y(-x)}{r^3}-[p_zL_x-p_xL_z,\frac{1}{r}]x-\frac{1}{r}[p_zL_x-p_xL_z,x]+i\hbar(-i\hbar)\frac{x(-y)}{r^3} \\ \\ &=& ([p_yL_z,\frac{1}{r}]-[p_zL_y,\frac{1}{r}])y+\frac{1}{r}(-[y,p_yL_z-p_zL_y])-([p_zL_x,\frac{1}{r}]-[p_xL_z,\frac{1}{r}])x-\frac{1}{r}(-[x,p_zL_x-p_xL_z])-\hbar^2\frac{xy}{r^3}+\hbar^2\frac{yx}{r^3} \\ \\ &=& ([p_y,\frac{1}{r}]L_z-[p_z,\frac{1}{r}]L_y)y+\frac{1}{r}(-i\hbar\frac{\partial}{\partial p_y}(p_yL_z-p_zL_y))-([p_z,\frac{1}{r}]L_x-[p_x,\frac{1}{r}]L_z)x-\frac{1}{r}(-(i\hbar)\frac{\partial}{\partial p_x}(p_zL_x-p_xL_z)) \\ \\ &=& ((-i\hbar)(\frac{\partial}{\partial y}\frac{1}{r})L_z-(-i\hbar)(\frac{\partial}{\partial z}\frac{1}{r})L_y)y+\frac{1}{r}(-i\hbar\frac{\partial}{\partial p_y}(p_y(xp_y-yp_x)-p_zL_y))-((-i\hbar)(\frac{\partial}{\partial z}\frac{1}{r})L_x-(-i\hbar)(\frac{\partial}{\partial x}\frac{1}{r})L_z)x-\frac{1}{r}(-(i\hbar)\frac{\partial}{\partial p_x}(p_zL_x-p_x(xp_y-yp_x))) \\ \\ &=& ((-i\hbar)(\frac{-y}{r^3})L_z-(-i\hbar)(\frac{-z}{r^3})L_y)y+\frac{1}{r}(-i\hbar(p_yx+(xp_y-yp_x)))-((-i\hbar)(\frac{-z}{r^3})L_x-(-i\hbar)(\frac{-x}{r^3})L_z)x-\frac{1}{r}(-(i\hbar)(-(xp_y-yp_x)-p_x(-y))) \\ \\ &=& i\hbar\left[\frac{y}{r^3}L_zy-\frac{z}{r^3}L_yy-\frac{1}{r}(p_yx+xp_y-yp_x)-\frac{z}{r^3}L_xx+\frac{x}{r^3}L_zx+\frac{1}{r}(-xp_y+yp_x+p_xy)\right] \\ \\ &=& i\hbar\left[\frac{y}{r^3}(yL_z-i\hbar x)-\frac{z}{r^3}{\color{red}yL_y}-\frac{1}{r}(2xp_y-yp_x)-\frac{z}{r^3}{\color{red}xL_x}+\frac{x}{r^3}(xL_z+i\hbar y)+\frac{1}{r}(-xp_y+2yp_x)\right] \\ \\ &=& i\hbar\left[\frac{y^2}{r^3}L_z-i\hbar\frac{yx}{r^3}-\frac{yz}{r^3}(zp_x-xp_z)-\frac{xz}{r^3}(yp_z-zp_y)+\frac{x^2}{r^3}L_z+i\hbar\frac{xy}{r^3}+\frac{1}{r}(3(yp_x-xp_y))\right] \\ \\ &=& i\hbar\left[\frac{x^2+y^2}{r^3}L_z-\frac{yz^2}{r^3}p_x+\frac{yz}{r^3}(xp_z)-\frac{xz}{r^3}yp_z+\frac{xz}{r^3}zp_y+i\hbar\frac{xy-yx}{r^3}-\frac{1}{r}(3(xp_y-yp_x))\right] \\ \\ &=& i\hbar\left[\frac{x^2+y^2}{r^3}L_z+\frac{z^2}{r^3}(xp_y-yp_x)+\frac{xyz}{r^3}p_z-\frac{xyz}{r^3}p_z-\frac{3}{r}L_z\right] \\ \\ &=& i\hbar\left[\frac{x^2+y^2}{r^3}L_z+\frac{z^2}{r^3}L_z-\frac{3}{r}L_z\right] \\ \\ &=& i\hbar\left[\frac{x^2+y^2+z^2}{r^3}L_z-\frac{3}{r}L_z\right] \\ \\ &=& i\hbar\left[\frac{r^2}{r^3}L_z-\frac{3}{r}L_z\right] \\ \\ &=& -2i\hbar\frac{1}{r}L_z \\ \\ \end{eqnarray} と導出できる。