Day 380: Ladder Operators — The Algebraic Method

Learning Objectives

• •Define the annihilation ($\hat{a}$) and creation ($\hat{a}^\dagger$) operators
• •Prove the fundamental commutation relation $[\hat{a}, \hat{a}^\dagger] = 1$
• •Express the Hamiltonian as $\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + \frac{1}{2})$
• •Derive $\hat{x}$ and $\hat{p}$ in terms of ladder operators
• •Understand why this method is called "algebraic" (no differential equations!)
• •Build matrix representations of ladder operators

Today's Schedule (7 hours)