Day 342: Continuous Spectra

Learning Objectives

• •**Define position eigenstates** $|x\rangle$ and express the eigenvalue equation $\hat{x}|x\rangle = x|x\rangle$ for continuous spectra
• •**Apply the Dirac delta function** $\delta(x-x')$ for orthonormality of continuous eigenstates
• •**Construct completeness relations** $\hat{I} = \int_{-\infty}^{\infty}|x\rangle\langle x|dx$ and explain their physical significance
• •**Interpret wave functions** as inner products $\psi(x) = \langle x|\psi\rangle$ in the position basis
• •**Derive the position-momentum overlap** $\langle x|p\rangle = (2\pi\hbar)^{-1/2}e^{ipx/\hbar}$ from first principles
• •**Connect Fourier transforms** to quantum mechanical basis transformations

Today's Schedule (7 hours)