angular_momentum3.saty

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

document (|
  title = {${\paren{A\cross B}\cdot \paren{C\cross D}}};
  author = {hsjoihs};
  show-title = true;
  show-toc = false;
|) '<
  +section{概要}<
    +p{
      ${
        \paren{A\cross B}\cdot \paren{C\cross D} = 
        \paren{A\cdot C}\paren{B\cdot D}- \paren{B\cdot C}\paren{A\cdot D}
      } はcommutableなときには成り立ちそうだが、そうでないときはどうなんだろう。
    }
  >
  +section{計算}<
    +math(${
      \paren{A\cross B}\cdot \paren{C\cross D}
      = \epsilon_{ijk}A_j B_k \epsilon_{ilm} C_l  D_m
      = \paren{\delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} }A_j B_k C_l  D_m
    });
    +math(${
      = A_j B_k C_j D_k - A_j B_k C_k D_j
    });
    +pn{あっなるほど。}
  >
  +section{全角運動量演算子}<
    +math(${
      L^2 = \paren{x\hat\cross p\hat}\cdot \paren{x\hat\cross p\hat}
      = x\hat_j p\hat_k x\hat_j p\hat_k - x\hat_j p\hat_k x\hat_k p\hat_j
      = -\hbar^2\paren{x_j \nabla_k x_j \nabla_k - x_j \nabla_k x_k \nabla_j}
    });
    +pn{ここで 
      ${\nabla_k x_j = x_j\nabla_k + \delta_{kj}} と
      ${\nabla_k x_k = x_k\nabla_k + 3} より、
    }
    +math(${
      L^2 = -\hbar^2\paren{
        x_j \paren{x_j\nabla_k + \delta_{kj}} \nabla_k - x_j \paren{x_k\nabla_k + 3} \nabla_j
      }
    });
    +math(${
      = -\hbar^2\paren{
        x_j x_j\nabla_k  \nabla_k 
        + x_j \delta_{kj} \nabla_k 
        - x_j x_k\nabla_k  \nabla_j
        - x_j 3 \nabla_j
      }
    
      = -\hbar^2\paren{
        x_j x_j\nabla_k  \nabla_k 
        - x_j x_k\nabla_k  \nabla_j
        - x_j 2 \nabla_j
      }
    });
    +math(${
      =  - \hbar^2 \|x\vec\|^2 \nabla^2
      + \hbar^2 x\vec \cdot \paren {
        \paren{x\vec \cdot \nabla\vec}
        \nabla\vec
      }
      + 2\hbar^2 x\vec \cdot \nabla\vec 
    });

  >
>