Quantum Fundamentals: NoTerm-2022
Matrix Practice : Due Day 8 W 2/16 Mathbits

  1. Matrix Refresher S0 4365S Calculate the following quantities for the matrices: \[A\doteq \begin{pmatrix} 1&0&0\\ 0&0&1\\ 0&-1&0\\ \end{pmatrix} \hspace{2em} B\doteq \begin{pmatrix} a&b&c\\ d&e&f\\ g&h&j\\ \end{pmatrix} \hspace{2em} C\doteq \begin{pmatrix} \cos\theta&-\sin\theta\\\sin\theta&\cos\theta\\ \end{pmatrix} \] and the vector: \[\left|D\right\rangle\doteq \begin{pmatrix} 1\\ i\\ -1\\ \end{pmatrix} \hspace{2em} \]
    1. \(AB\)
    2. \({\rm tr} (B)\)
    3. \(A\vert D\rangle\)
    4. \(\det(\lambda{\cal I}-A)\) where \(\lambda\) is a scalar.
    5. \(C^{-1}\) (Hint: Geometrically, what is the \(C\) transformation? What transformation undoes what \(C\) does?)
  2. Pauli Practice S0 4365S The Pauli spin matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are defined by: \[\sigma_x= \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} \sigma_y= \begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} \sigma_z= \begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix} \] These matrices are related to angular momentum in quantum mechanics. Prove, and become familiar with, the identities listed below.
    1. Show that each of the Pauli matrices is hermitian. (A matrix is hermitian if it is equal to its hermitian adjoint.)
    2. Show that the determinant of each of the Pauli matrices is \(-1\).
    3. Show that \(\sigma_i^2={\cal I}\) for each of the Pauli matrices, i.e. for \(i\in\left\{x,y,z\right\}\).
  3. Hermitian Adjoints S0 4365S Calculate the following quantities for the matrices: \[A\doteq \begin{pmatrix} 1&0&0\\ 0&0&1\\ 0&-1&0\\ \end{pmatrix} \hspace{2em} B\doteq \begin{pmatrix} a&b&c\\ d&e&f\\ g&h&j\\ \end{pmatrix} \hspace{2em} C\doteq \begin{pmatrix} \cos\theta&-\sin\theta\\\sin\theta&\cos\theta\\ \end{pmatrix} \] and the vectors: \[\left|D\right\rangle\doteq \begin{pmatrix} 1\\ i\\ -1\\ \end{pmatrix} \hspace{2em} \left|E\right\rangle\doteq \begin{pmatrix} 1\\ i\\ \end{pmatrix} \hspace{2em} \left|F\right\rangle\doteq \begin{pmatrix} 1\\ -1\\ \end{pmatrix} \]
    1. \(A^{\dagger}\)
    2. \(\vert E\rangle^{\dagger}\equiv\langle E\vert\)
    3. \(\langle D\vert A\vert D\rangle\)
    4. \(\left(A\vert D\rangle\right)^{\dagger}\)
    5. Using explicit matrix multiplication (without using a theorem) verify that \(\left(A\vert D\rangle\right)^{\dagger} =\langle D\vert A^{\dagger}\)