Chapter 9 Key Takeaways

The Five Big Ideas

1. The eigenvalue equation $\hat{A}|a\rangle = a|a\rangle$ is the mathematical formulation of measurement. Every observable is a Hermitian operator. Its eigenvalues are the possible measurement outcomes (guaranteed real). Its eigenstates are the states of definite value. The probability of outcome $a_n$ is $|\langle a_n|\psi\rangle|^2$.

2. Discrete spectra use Kronecker deltas and sums; continuous spectra use Dirac deltas and integrals. This is the single most important distinction in the chapter. For discrete spectra: $\langle a_m|a_n\rangle = \delta_{mn}$, $\sum_n |a_n\rangle\langle a_n| = \hat{I}$. For continuous spectra: $\langle x|x'\rangle = \delta(x - x')$, $\int |x\rangle\langle x| dx = \hat{I}$.

3. The spectral theorem decomposes every Hermitian operator into eigenvalues times projectors. $\hat{A} = \sum_n a_n|a_n\rangle\langle a_n|$. This enables functions of operators: $f(\hat{A}) = \sum_n f(a_n)|a_n\rangle\langle a_n|$. The time-evolution operator $e^{-i\hat{H}t/\hbar} = \sum_n e^{-iE_n t/\hbar}|E_n\rangle\langle E_n|$ is the most important application.

4. The Fourier transform is a change of basis, not a trick. $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \psi(x) e^{-ipx/\hbar} dx$ follows from inserting the completeness relation for position into $\phi(p) = \langle p|\psi\rangle$. The sign conventions and normalization factors are determined by physics, not arbitrary choice.

5. The rigged Hilbert space $\Phi \subset \mathcal{H} \subset \Phi'$ justifies everything. Position and momentum eigenstates live in $\Phi'$ (not in $\mathcal{H}$), physical states live in $\Phi$, and Dirac's formalism is mathematically rigorous within this framework.

The Eigenvalue Problem at a Glance

Essential Formulas

Spectral decomposition and functions of operators

$$\hat{A} = \sum_n a_n |a_n\rangle\langle a_n| \qquad f(\hat{A}) = \sum_n f(a_n)|a_n\rangle\langle a_n|$$

Time-evolution operator

$$\hat{U}(t) = e^{-i\hat{H}t/\hbar} = \sum_n e^{-iE_n t/\hbar}|E_n\rangle\langle E_n|$$

Fourier transform pair

$$\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \psi(x) e^{-ipx/\hbar} dx \qquad \psi(x) = \frac{1}{\sqrt{2\pi\hbar}}\int \phi(p) e^{ipx/\hbar} dp$$

Position-momentum overlap

$$\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar} \qquad \langle p|x\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-ipx/\hbar}$$

Dirac delta function

$$\int f(x)\delta(x - a) dx = f(a) \qquad \delta(x) = \frac{1}{2\pi}\int e^{ikx} dk \qquad \delta(ax) = \frac{1}{|a|}\delta(x)$$

Parseval's theorem

$$\int|\psi(x)|^2 dx = \int|\phi(p)|^2 dp = \langle\psi|\psi\rangle$$

Operator representations

Spin-1/2 Eigenvalue Summary

All three spin components have eigenvalues $\pm\hbar/2$. The eigenstates in the $\hat{S}_z$ basis:

Spectral decomposition: $\hat{S}_i = \frac{\hbar}{2}(|+\rangle_i\langle+|_i - |-\rangle_i\langle-|_i)$ for $i = x, y, z$.

Common Pitfalls to Avoid

1. Confusing Kronecker and Dirac deltas. $\delta_{mn}$ is dimensionless and equals 0 or 1. $\delta(x - x')$ is a distribution with dimensions $1/[x]$. They play analogous roles in discrete and continuous bases, but are different objects.

2. Evaluating $\delta(0)$. The expression $\delta(0)$ is meaningless as a number. The delta function only has meaning inside an integral, acting on a test function.

3. Treating position eigenstates as physical states. $|x\rangle$ is not normalizable ($\langle x|x\rangle = \delta(0) = \infty$). It is a generalized eigenstate in $\Phi'$, not a physical state. Use it as a mathematical tool, not as a realistic state preparation.

4. Forgetting to conjugate when forming bras. $\langle\alpha\psi| = \alpha^*\langle\psi|$, not $\alpha\langle\psi|$. This carries over from Chapter 8 but remains the most common computational error.

5. Wrong Fourier sign convention. Position $\to$ momentum has $e^{-ipx/\hbar}$; momentum $\to$ position has $e^{+ipx/\hbar}$. Using Dirac notation (insert completeness, use $\langle x|p\rangle$) eliminates sign convention errors automatically.

6. Confusing the spectral decomposition with completeness. $\hat{A} = \sum a_n|a_n\rangle\langle a_n|$ is the spectral decomposition (eigenvalues as weights). $\hat{I} = \sum|a_n\rangle\langle a_n|$ is completeness (no weights). The spectral decomposition encodes the operator; completeness encodes the basis.

7. Assuming all spectra are discrete. Position, momentum, and the free-particle Hamiltonian have continuous spectra. Many realistic Hamiltonians (e.g., finite well, hydrogen in a magnetic field) have mixed spectra.

Self-Test Checklist

Before proceeding to Chapter 10, you should be able to:

• [ ] Solve the eigenvalue problem for any $2 \times 2$ Hermitian matrix (eigenvalues, eigenstates, verification)
• [ ] Write the spectral decomposition of $\hat{S}_x$, $\hat{S}_y$, $\hat{S}_z$ and compute $f(\hat{S}_i)$ for any function $f$
• [ ] Explain the difference between discrete and continuous spectra (orthonormality, completeness, normalizability)
• [ ] Apply the sifting property of $\delta(x-a)$ and the scaling property $\delta(ax) = |a|^{-1}\delta(x)$
• [ ] Derive the Fourier transform from Dirac notation by inserting a completeness relation
• [ ] Compute the Fourier transform of a Gaussian and verify the uncertainty relation
• [ ] Write the Schrodinger equation in momentum space
• [ ] Explain in one paragraph what the rigged Hilbert space is and why it matters
• [ ] Compute the time-evolution operator $\hat{U}(t)$ from the spectral decomposition of $\hat{H}$

If any item on this list feels uncertain, revisit the corresponding section before moving to Chapter 10 (Symmetry and Conservation Laws).