YBasis

• Consider the kets \(\vert + \rangle_y\) and \(\vert - \rangle_y\) defined by \[\vert + \rangle_y\doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ i \end{pmatrix} \]

  \[\vert - \rangle_y\doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ -i \end{pmatrix} \] Show that the statements below are true. (In fancy language, these statements show that the kets \(\vert + \rangle_y\) and \(\vert - \rangle_y\) form an orthonormal basis for the vector space of two-component complex vectors, i.e. they act just like \(\hat{x}\) and \(\hat{y}\) for 2-d vectors in space.)

  1. Show that \(\vert + \rangle_y\) and \(\vert - \rangle_y\) are normalized.
  2. Show that \(\vert + \rangle_y\) and \(\vert - \rangle_y\) are orthogonal.
  3. Show that \(\vert + \rangle_y\) and \(\vert - \rangle_y\) are complete, i.e. that any vector in the vector space can be written as a linear combination of these two vectors.