angular_momentum.saty

@require: stdjabook
@require: code
@require: itemize
@require: tabular
@require: math
@import: local

document (|
  title = {全角運動量演算子${L^2}};
  author = {hsjoihs};
  show-title = true;
  show-toc = false;
|) '<
  +section{概要}<
    +p{
      全角運動量演算子${L^2}をシュッと表す。
    }
  >
  +section{計算}<
    +math(${L^2\psi = 
      \paren{x\frac{\hbar}{i}\partial_y - y\frac{\hbar}{i}\partial_x}
      \paren{x\frac{\hbar}{i}\partial_y - y\frac{\hbar}{i}\partial_x} \psi + ...
    });
    +math(${= 
      x\frac{\hbar}{i}\partial_yx\frac{\hbar}{i}\partial_y\psi - x\frac{\hbar}{i}\partial_yy\frac{\hbar}{i}\partial_x\psi  -
      y\frac{\hbar}{i}\partial_xx\frac{\hbar}{i}\partial_y\psi + y\frac{\hbar}{i}\partial_xy\frac{\hbar}{i}\partial_x\psi  + ...
    });
    +math(${
      = -\hbar^2\paren{
        x\partial_yx\partial_y\psi - x\partial_yy\partial_x\psi  -
        y\partial_xx\partial_y\psi + y\partial_xy\partial_x\psi  + ...
      }
    });
    +math(${
      = -\hbar^2\paren{
        x^2\partial_y^2\psi + y^2\partial_x^2\psi - x\partial_y\paren{y\partial_x\psi}  -
        y\partial_x\paren{x\partial_y\psi}   + ...
      }
    });
    +pn{
      これの前半2項×3は
    }
    +math(${ 
      -\hbar^2\paren{
          x^2\partial_y^2\psi + y^2\partial_x^2\psi
        + y^2\partial_z^2\psi + z^2\partial_y^2\psi
        + z^2\partial_x^2\psi + x^2\partial_z^2\psi
      }
    });
    +math(${
      = \hbar^2\paren{
        x^2\partial_x^2\psi + y^2\partial_y^2\psi + z^2\partial_z^2\psi
      } - \hbar^2\paren{
        x^2+ y^2+ z^2
      }\paren{
        \partial_x^2\psi + \partial_y^2\psi + \partial_z^2\psi
      }
    });
    +math(${
      = \hbar^2\paren{
        x^2\partial_x^2\psi + y^2\partial_y^2\psi + z^2\partial_z^2\psi
      } - \hbar^2 \|x\vec\|^2 \nabla^2\psi
    });
    +pn{
      であり、これの後半2項は
    }
    +math(${ 
      \hbar^2\paren{ 
          x\partial_y\paren{y\partial_x\psi} 
        + y\partial_x\paren{x\partial_y\psi}   + ... 
      }
   
      = \hbar^2\paren{ 
          x\partial_x\psi + xy\partial_y\partial_x\psi
        + y\partial_y\psi + yx\partial_x\partial_y\psi   + ... 
      }
    });
    +math(${
      = \hbar^2\paren{ 
          x\partial_x\psi 
        + y\partial_y\psi
        + 2xy\partial_y\partial_x\psi  + ... 
      }
    });

    +math(${
       = \hbar^2\paren{ 
          2x\partial_x\psi 
        + 2y\partial_y\psi
        + 2z\partial_z\psi
        + 2xy\partial_x\partial_y\psi
        + 2yz\partial_y\partial_z\psi
        + 2zx\partial_z\partial_x\psi
      }
    });

    +math(${
      = 2\hbar^2\paren{ 
           x\partial_x\psi 
         + y\partial_y\psi
         + z\partial_z\psi
      } - \hbar^2\paren{
        x^2\partial_x^2\psi + y^2\partial_y^2\psi + z^2\partial_z^2\psi
      }
    });
    +math(${
      + \hbar^2\paren{
           2xy\partial_x\partial_y\psi
         + 2yz\partial_y\partial_z\psi
         + 2zx\partial_z\partial_x\psi
         + x^2\partial_x^2\psi + y^2\partial_y^2\psi + z^2\partial_z^2\psi
      }
    });

    +math(${
      = 2\hbar^2 x\vec \cdot \nabla\vec\psi - \hbar^2\paren{
        x^2\partial_x^2\psi + y^2\partial_y^2\psi + z^2\partial_z^2\psi
      }
    });
    +math(${
      + \hbar^2 x\paren{
          x\partial_x
        + y\partial_y
        + z\partial_z
      } \partial_x\psi + \hbar^2y\paren{
          x\partial_x
        + y\partial_y
        + z\partial_z
      } \partial_y\psi + \hbar^2z\paren{
          x\partial_x
        + y\partial_y
        + z\partial_z
      } \partial_z\psi
    });

    +math(${
      = 2\hbar^2 x\vec \cdot \nabla\vec\psi - \hbar^2\paren{
        x^2\partial_x^2\psi + y^2\partial_y^2\psi + z^2\partial_z^2\psi
      }
      + \hbar^2 x\paren{
        x\vec \cdot \nabla\vec
      } \partial_x\psi + \hbar^2y\paren{
        x\vec \cdot \nabla\vec
      } \partial_y\psi + \hbar^2z\paren{
        x\vec \cdot \nabla\vec
      } \partial_z\psi
    });

    +math(${
      = 2\hbar^2 x\vec \cdot \nabla\vec\psi - \hbar^2\paren{
        x^2\partial_x^2\psi + y^2\partial_y^2\psi + z^2\partial_z^2\psi
      }
      + \hbar^2 x\vec \cdot \paren {
        \paren{x\vec \cdot \nabla\vec}
        \nabla\vec\psi
      }
    });

    +pn{
      したがって
    }

    +math(${L^2\psi
      =  - \hbar^2 \|x\vec\|^2 \nabla^2\psi
      + 2\hbar^2 x\vec \cdot \nabla\vec\psi 
      + \hbar^2 x\vec \cdot \paren {
        \paren{x\vec \cdot \nabla\vec}
        \nabla\vec\psi
      }
    });

  >
  +section{極座標}<
    +math(${L^2\psi
      =  - \hbar^2 \|x\vec\|^2 \nabla^2\psi
      + 2\hbar^2 x\vec \cdot \nabla\vec\psi 
      + \hbar^2 x\vec \cdot \paren {
        \paren{x\vec \cdot \nabla\vec}
        \nabla\vec\psi
      }
    });
    +pn{に}
    +math(${
      x\vec = re\vec_r
    });
    +math(${
      r^2\nabla^2\psi = \partial_r\paren{r^2 \frac{\partial\psi}{\partial r} } + \frac{1}{\sin\theta} \partial_\theta\paren{\sin\theta \frac{\partial\psi}{\partial \theta}} + \frac{1}{\sin^2\theta} \partial_\phi^2 \psi
    });
    +math(${
      \nabla\vec\psi = e\vec_r\frac{\partial\psi}{\partial r} + \frac{1}{r} e\vec_\theta \frac{\partial\psi}{\partial \theta} + \frac{1}{r\sin\theta} e\vec_\phi \frac{\partial\psi}{\partial \phi}
    });
    +math(${
      x \vec \cdot \nabla\vec = r\partial_r
    });
    +pn{
      を代入すると
    }
    +math(${
      L^2\psi =  - \hbar^2 \paren{ 
        \partial_r\paren{r^2\partial_r \psi} + \frac{1}{\sin\theta} \partial_\theta\paren{\sin\theta \partial_\theta \psi} + \frac{1}{\sin^2\theta} \partial_\phi^2 \psi
      }
    });
    +math(${
      + 2\hbar^2 r \partial_r \psi 
      + \hbar^2 x\vec \cdot
        \paren{r \partial_r} \paren{e\vec_r\frac{\partial\psi}{\partial r} + \frac{1}{r} e\vec_\theta \frac{\partial\psi}{\partial \theta} + \frac{1}{r\sin\theta} e\vec_\phi \frac{\partial\psi}{\partial \phi}}
    });

    +math(${
      = - \hbar^2 \paren{ 
        \partial_r\paren{r^2\partial_r \psi} + \frac{1}{\sin\theta} \partial_\theta\paren{\sin\theta \partial_\theta \psi} + \frac{1}{\sin^2\theta} \partial_\phi^2 \psi
      }
      + 2\hbar^2 r \partial_r \psi 
      + \hbar^2 r \paren{r \partial_r} \frac{\partial\psi}{\partial r} 
    });

    +math(${
      = - \hbar^2 \paren{ 
        \frac{1}{\sin\theta} \partial_\theta\paren{\sin\theta \partial_\theta \psi} + \frac{1}{\sin^2\theta} \partial_\phi^2 \psi
      } - \hbar^2 \partial_r \paren{r^2\partial_r \psi}
      + 2\hbar^2 r \partial_r \psi 
      + \hbar^2 r^2
        
          \frac{\partial^2\psi}{\partial r^2} 
    });

    +math(${
      = - \hbar^2 \paren{ 
        \frac{1}{\sin\theta} \partial_\theta\paren{\sin\theta \partial_\theta \psi} + \frac{1}{\sin^2\theta} \partial_\phi^2 \psi
      }
    });

    +pn{
      となって、ラプラシアンの${\theta}・${\phi}絡みだけを残したやつが全角運動量演算子になることにもそこそこ納得がいきそうな気がしそうである。
    }
  >
>