Kinesthetic: Inner Product of Spin-1/2 System with Arms

Students use their arms to act out two spin-1/2 quantum states and their inner product.
What students learn
  • Cartesian space and Hilbert space is different.
  • That inner products involve complex conjugation, multiplying vectors component-wise, and the sum of those products.

Prerequisite Understanding: Students should

  1. Know how to represent a spin-1/2 state with arms.
  2. Know how to multiple two complex numbers (with arms).
  3. Know how to add to vectors/complex numbers tip-to-tail.

Students' Task and Conversations:

  1. SetUp
    1. Ask 1 pair of students to represent an arbitrary quantum state.
    2. Ask a second pair of students (or members of the teaching team) to represent a different state - preferably one rotated by 90 degrees from the first.
  2. Ask the students: Are your states orthogonal to each other.
    1. Help students realize they need their inner product to be zero.
    2. Help students with determining whether their inner product is zero.
      1. First, they need to complex conjugate one of the vectors by reflecting their arms over the real axes.
      2. Second, they need to line up the components of the two vectors and multiply their arms - multiply the norms of their arms and add their phase angle.
        1. 1 person for each component-pair should represent the resulting product (another option is the write the product on the board).
        2. They can't stretch or shrink their arm, so they should pretend their arms is whatever length it needs to be to represent the norm.
      3. Third, they should add the resulting products.
        1. If they are using arms to represent the products, they need to orient themselves so they can do tip-to-tail addition.
        2. The result is the final result of the inner product.
  3. Wrap up the activity by doing an inner product calculation of the board and remind students of how the steps correspond with what they did with their arms, emphasizing complex conjugation, lining up the components, multiplying, and adding the products.


Keywords
Quantum Mechanics Arms Orthogonal States Inner Product Complex Conjugation Multiplying Complex Numbers Adding Complex Numbers.
Learning Outcomes