Homework for Matrix Components

  1. (MatrixElements) This practice exercise helps students understand that they can “pick out” a particular component of a matrix by taking the braket of the matrix with appropriate basis vectors. The “bra” vector picks out the row and the “ket” vector picks out the column. Part $( c)$ requires Dirac bra-ket notation.

    1. Carry out the following matrix calculations.

      $$ \pmatrix{0&1&0\cr}\pmatrix{a_{11} & a_{12} & a_{13} \cr a_{21} & a_{22} & a_{23} \cr a_{31} & a_{32} & a_{33}\cr}\pmatrix{1\cr0\cr0\cr}$$ and

      $$\pmatrix{0&1&0\cr}\pmatrix{a_{11} & a_{12} & a_{13} \cr a_{21} & a_{22} & a_{23} \cr a_{31} & a_{32} & a_{33}\cr}\pmatrix{0\cr1\cr0\cr}\quad $$

    2. What matrix multiplication would you do if you wanted the answer to be $a_{13}$?

    3. The bra-ket language for the calulations in part $(a)$ are

      $$\matrix{\langle2|A|1\rangle = ? & \hbox{and} & \langle2|A|2\rangle = ?}$$

      Write the question to part $(b)$ in bra-ket language.


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