Overview

The overview section is the route map for coordinate-space quantum mechanics. It explains what belongs in this volume, how to turn a physical setup into a well-posed Schrödinger problem, and which canonical model isolates which physical idea.

Use these pages before diving into individual systems. They prevent three common mistakes: solving before the domain is specified, using the wrong normalization convention, and treating a canonical model as a disconnected textbook exercise rather than a reusable laboratory.

Overview Pages

Page	Main Question	Use It When
What This Volume Covers	What belongs in this volume, and what belongs elsewhere?	deciding the canonical home for a topic or planning a reading path
How to Solve a Wave-Mechanics Problem	What is the reusable workflow for coordinate-space problems?	starting a calculation from a Hamiltonian, potential, boundary condition, or initial packet
Map of Canonical Systems	Which standard model teaches which physical lesson?	choosing the simplest model that isolates an idea
Dependency Graph	Which pages should come before which models?	assigning prerequisites, planning a reading order, or avoiding duplicated derivations
Notation and Conventions Used Here	Which local symbols and conventions does this volume use?	checking wavefunction, Fourier, normalization, current, radial, or boundary-condition notation
How This Volume Connects to QFT	Which wave-mechanics structures are reused in field theory?	preparing for oscillator modes, propagators, S-matrix language, Fock space, and gauge-coupled systems
Problem-Solving Patterns	Which recurring method should I recognize?	deciding whether to match regions, exploit parity, use current, separate variables, or check a limit

First Reading Path

For a first pass, read:

1. What This Volume Covers
2. How to Solve a Wave-Mechanics Problem
3. Map of Canonical Systems
4. Dependency Graph
5. Notation and Conventions Used Here
6. How This Volume Connects to QFT
7. Coordinate Representation
8. Wavefunctions and Probability Density
9. Boundary Conditions

After that, the most efficient model sequence is usually free particle, infinite square well, finite square well, rectangular barrier, and harmonic oscillator. This path builds continuous spectra, boundary quantization, evanescent tails, current ratios, tunneling, and oscillator ladders in a compact order.

What the Overview Should Teach

The central translation is:

$$\text{physical setup} \longrightarrow \text{Hilbert space, Hamiltonian, domain, and measure} \longrightarrow \text{spectrum, states, probabilities, and currents}.$$

The Hamiltonian expression is only one part of the problem. A differential expression such as $-\hbar^2d^2/(2m\,dx^2)$ can describe a free particle, an infinite well, a ring, or a half-line problem depending on the domain and boundary conditions.

The overview pages also enforce the one-canonical-home rule. A page on a model should solve and interpret that model. A page on a mathematical tool should explain the tool. A reference table should summarize, not duplicate, derivations.

Quick Decisions

If You Need…	Start With
the scope boundary between this volume and other volumes	What This Volume Covers
a checklist for solving a coordinate-space problem	How to Solve a Wave-Mechanics Problem
the right canonical model for an idea	Map of Canonical Systems
prerequisite links for a page or roadmap	Dependency Graph
local wavefunction, Fourier, current, or radial conventions	Notation and Conventions Used Here
the bridge from canonical systems to field theory	How This Volume Connects to QFT
recurring methods and solution habits	Problem-Solving Patterns
compact model formulas and scales	Common Hamiltonians and Spectra and Eigenfunctions Table
practice problems and notebook validations	Exercise Sets and Benchmark Problems

Common Mistakes

• Starting with a formula instead of the physical domain.
• Treating $\lvert\psi(x)\rvert^2$ as a probability rather than a density.
• Forgetting that scattering states and bound states use different normalization conventions.
• Reading a model page without tracking the limiting cases that define its range of validity.
• Using a table as a derivation substitute.
• Importing advanced machinery from later volumes when the canonical wave-mechanics model has a simpler first explanation.

Where This Is Used

• Wave Mechanics Foundations starts the technical development after this orientation.
• Model Encyclopedia provides quick lookup tables for the standard models.
• Worked Examples Index turns overview habits into concrete examples.
• Reference supplies formula and symbol lookup after the conceptual home has been identified.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.

Exercises

1. A page draft solves the infinite square well but also includes a long general introduction to Sturm–Liouville theory. Where should the general theory live?

Solution

The infinite-well page should use the relevant Sturm–Liouville facts but should not become the canonical home for the general theory. The general mathematical theory belongs in the Mathematical Toolkit. The model page should link there and focus on the physical setup, boundary conditions, spectrum, normalization, and interpretation of the well.

2. A calculation starts from $\hat H=-\hbar^2d^2/(2m\,dx^2)$ and immediately writes plane waves. What missing question should be asked first?

Solution

Ask what the domain and boundary conditions are. The same differential expression gives plane waves on the full line, sine states in an infinite well, and discrete periodic momenta on a ring. The physical system is not determined by the local expression alone.