Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
1. << Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits) | Arms Sequence for Complex Numbers and Quantum States | Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems >>
This activity is part of the Arms Sequence for Complex Numbers and Quantum States.
Students should be familiar with complex numbers, particularly Argand diagrams and rectangular, polar, and exponential forms for complex numbers.
Students should also have been introduced to the idea that quantum states are represented by complex-valued vectors.
Students should be familiar with the spin basis kets: \(\left|{+}\right\rangle _z\), \(\left|{-}\right\rangle _z\), \(\left|{+}\right\rangle _x\), \(\left|{-}\right\rangle _x\), \(\left|{+}\right\rangle _y\), and \(\left|{-}\right\rangle _y\), written in the \(S_z\) basis in both matrix notation and Dirac Notation.
Set-Up
Optional Warm-Up Have each student represent a complex number given in rectangular, polar, or exponential form.
“Represent the state: spin up in the \(z-\)direction.” Write the \(\left|{+}\right\rangle _z\) ket on the board.
“Represent the state: spin down in the \(z-\)direction.” Write the \(\left|{-}\right\rangle _z\) ket on the board.
Be on the Lookout: Some incorrect answer to expect include:
Right person's arm straight up (pure imaginary). (This is a technically correct answer, but with a different overall phase. It's not worth getting into it at the moment.) Some students key in on the orthogonality of \(\left|{+}\right\rangle _z\) and \(\left|{-}\right\rangle _z\) and think the arms need to be perpendicular.
“Represent the state: spin up in the \(x-\)direction.” Write the \(\left|{+}\right\rangle _x = \frac{1}{\sqrt{2}} \left|{+}\right\rangle _z + \frac{1}{\sqrt{2}} \left|{-}\right\rangle _z\) ket on the board.
“Represent the state: spin down in the \(x-\)direction.” Write the \(\left|{-}\right\rangle _x= \frac{1}{\sqrt{2}} \left|{+}\right\rangle _z - \frac{1}{\sqrt{2}} \left|{-}\right\rangle _z\) ket on the board.
“Represent the state: spin up in the \(y-\)direction.” Write the \(\left|{+}\right\rangle _y = \frac{1}{\sqrt{2}} \left|{+}\right\rangle _z + \frac{i}{\sqrt{2}} \left|{-}\right\rangle _z\) ket on the board.
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