Student handout: Representations for Finding Components

Quantum Fundamentals 2021
In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.
What students learn Geometric understanding of a projection. Projections of the same vector look different in different bases. Taking a dot product of a vector with basis vectors gives the projection of that vector in that basis. Taking an inner product of a state ket with basis bras gives the projection of that state in that basis. The results of these inner products are matrix elements for describing the vector.
  1. In the first figure, draw the projections of \(\vec{v}\) onto the rectangular basis vectors \(\hat{x}\) and \(\hat{y}\).
  2. In the second figure, draw the projections of \(\vec{v}\) onto the polar basis vectors \(\hat{s}\) and \(\hat{\phi}\).
  3. Write the components of \(\vec{v}\) as dot products: \begin{align*} v_x &= \hskip 100pt& v_s&=\\ \\ v_y &= & v_{\phi}&= \end{align*}
  4. Write the components of \(\vec{v}\) as bra/kets: (Warning: Often, particularly in quantum mechanics settings, we will assume that all kets are states and therefore normalized so that their probability is one. We are NOT making that assumption here. Write the vector \(\vec{v}=\left|{v}\right\rangle \) without worrying about its normalization.) \begin{align*} v_x &= \hskip 100pt& v_s&=\\ \\ v_y &= & v_{\phi}&= \end{align*}
Author Information
Corinne Manogue & Maggie Greenwood
Keywords
basis vectors dirac notion projections dot product inner product Math Methods
Learning Outcomes