Small Group Activity: 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.
  • Media
    • 2299/vecv.png
    • 2299/Vector_components.png
  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*}

Instructor's Guide

Students' Task

Students work in small groups to write the components of an arbitrary vector in both dot product notation and bra/ket notation.

Student Conversations

  • Direction of the polar basis vectors: The standard physics convention is that the tail of the vector is the point where the vector exists, but other convections exist (for example, Mathematica defaults to placing the arrow so that the middle of the arrow is at the location of the vector). Use the students to be explicit about where the vector is (tail, middle, point, etc) AND have them sketch the direction of the polar basis vectors.
  • Components as sides of a triangle: Some students will want to draw the components as sides of a triangle. This is also correct, but it's important that the get the direction correct based on the location of the vector (tail, middle, point).

Wrap-up

  • Different representations: Emphasize that the dot product notation and the bra/ket notation are just representations for the same thing.
  • Order of the vectors in the dot product: For real vectors, it doesn't matter which order you write the vectors (the dot product between real vectors is commutative), but this is not going to be true to complex valued quantum state. To preserve the sign of the component, the basis vector should come first (the Hermitian adjoint of the first vector will be taken, changing the sign of the imaginary part). It doesn't matter in this case, but students should start getting used to this more careful ordering. This case be brought out in the wrap up.
Author Information
Corinne Manogue & Maggie Greenwood
Keywords
basis vectors dirac notion projections dot product inner product Math Methods
Learning Outcomes