Kinesthetic: Inner Products with Arms

Students perform an inner product between two spin states with the arms representation.
What students learn Reinforces:
  • procedures for doing inner products between spins states, including taking the Hermitian adjoint
  • the difference between “perpendicular in 3D real space” and “orthogonal in the Hilbert state space”
  • vector “tip to tail” vector addition and multiplication of complex numbers in exponential form

Prerequisite Knowledge: Students should already be familiar with:

  1. Inner product for quantum states means:
    • take the Hermitian adjoint (i.e., complex conjugate the components) of one of the vectors,
    • find the product between the same component of each vector,
    • add the component-products together
  2. Add vectors tip-to-tail.
  3. How to multiply two complex numbers with exponential form/Arms (multiply arm lengths and add phase angles)

Prompts:

  1. Ask a pair of students to represent an arbitrary spin state \(|\psi_{student}\rangle\).
  2. Have the instructor and the TA represent the state \(e^{i\pi/2} |\psi_{student}\rangle\) (don't write it on the board)
  3. Ask the class: "Are the two states orthogonal?"
    1. If you've already discussed how the overall phase of the vector doesn't affect the quantum state, students might suggest (correctly) that they can't be orthogonal states because they are the same state!

      However, I recommend doing this activity before you talk about relative and overall phase. This activity can set hooks for that discussion.

    2. An appealing distractor is that the component pairs across the states (the spin up components or the spin down components) look perpendicular - the arms are perpendicular in real 3D space. However, the states are not orthogonal in the Hilbert (spin state) space.

    3. To really know if two states are orthogonal to each other, the inner product between the two vector must be zero.

  4. Lead a class discussion about the procedure for doing the inner product:

    1. Decide which order you want the vectors in the inner product

      (shouldn't matter if they're orthongal - changing the order complex conjugates the result).

    2. Complex conjugate the first vector

      (reflect arm of both people across the horizontal/real axis)

    3. Pair the components across the two vector

      (spin-up with spin up and spin down with spin down)

    4. Multiply like components

      (multiple arm lengths and add phase angle)

      A third person could represent the product for each like-component pair

    5. Add all the like-component products.

      (tip to tail - put one persons shoulder at the hand of the other person. Helps if someone kneels or crouches)

    For this pair of vectors, they should not get zero.

Possible FollowUps:

  1. Discuss how this procedure shows that \(|+\rangle_x\) and \(|-\rangle_x\) are orthongal. Could try representing \(|+\rangle_x\) (without naming it) and ask students to construct a state that is orthogonal to it.

  2. Try to discern a pattern for identifying two spins states with arms as orthongial

    (if the norm of each component is the same, i.e., all arms are the same length, then the relative relative phase between the two vectors should be different by \(e^{i\pi}\), i.e., spin-down arms are different by 180°).

  3. Discuss how this inner product procedure generalizes to higher-order spin states (spin-1, spin-3/2, etc.)

  4. Discuss how this inner product procedure matches the algebraic notations.


Keywords
Probability Amplitude Inner Product Arms Representation
Learning Outcomes