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## Two Spin System: Instructor's Guide

### Main Ideas

• Spin eigenvalue equations
• Spin matrices
• Uncoupled basis of two-spin system

• Recall Spin-1/2 eigenvalue equations for electron
• Extend to Spin-1/2 eigenvalue equations for proton
• Find possible states of two-spin system
• Determine matrix representations of electron and proton spin operators in uncoupled basis

Estimated Time: 30 min

### Prerequisite Knowledge

Spin-1/2 system eigenstates and matrices.

### Activity: Introduction

Small white board questions:

1. The electron has spin ½, with spin up and spin down eigenstates $|\pm\rangle_e$. For the electron, use the symbol $S$ for the spin.
• Write down the eigenvalue equations for the electron states.
2. The proton has spin ½, with spin up and spin down eigenstates $|\pm\rangle_p$. For the proton, use the symbol $I$ for the spin.
• Write down the eigenvalue equations for the proton states.
3. The system of electron and proton could be in the state: $|e^-\quad up\rangle |p^+\quad up\rangle\; =\;|+\rangle_e|+\rangle_p\;=\;|++\rangle$
• Using the compact $|++\rangle$ notation, what are the possible spin states of the electron-proton system?

Large white board activities:

1. Find the matrix representation of the electron spin component operator $S_z$.
2. Find the matrix representation of the proton spin component operator $I_z$.

### Activity: Wrap-up

Small whiteboard questions: Walk around the room as students are answering this question and pick up an example of each different representation or statement. Prop them on the chalkboard tray and give whatever review “lecture” you would normally give. You can add in any extra representations that the students haven't mentioned as you go along.

Large whiteboard questions: Facilitate their calculations as required.

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