What This Volume Covers

This volume is the canonical home for coordinate-space wave mechanics and the standard exactly solvable quantum systems. It begins with wavefunctions, probability density, boundary conditions, normalization, and the Schrödinger equation in position representation. It then uses those tools to solve free particles, wave packets, wells, barriers, tunneling problems, harmonic oscillators, two-level models, separable three-dimensional systems, central potentials, rotors, and charged particles in magnetic fields.

The guiding question is simple:

Given a Hamiltonian and physical boundary conditions, what does the quantum system actually do?

The answer is usually a mixture of differential equations, operator language, normalization conventions, limiting cases, and physical interpretation. This volume keeps those pieces together for each canonical model.

Relationship to the formalism

Core Formalism gives the abstract language: states, observables, amplitudes, projectors, density operators, and unitary time evolution. This volume uses that language in the position representation. A ket $|\psi\rangle$ becomes a wavefunction

$$\psi(x)=\langle x|\psi\rangle,$$

and an abstract Hamiltonian $\hat H$ becomes a differential operator such as

$$\hat H =-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$

That replacement is not merely notation. Once a Hamiltonian is represented by differential operators, the domain and boundary conditions become part of the quantum problem. Two systems can have the same differential expression but different spectra because they impose different allowed behavior at the boundaries.

Relationship to the Mathematical Toolkit

This volume uses the mathematics developed in the Mathematical Toolkit. It does not reprove the general theory of Hilbert spaces, Fourier transforms, Sturm–Liouville problems, or spherical harmonics. It applies those tools to physical Hamiltonians.

Useful prerequisites include:

• Inner Products for amplitudes, norms, and expectation values.
• Hilbert Spaces and L2 Spaces for square-integrable wavefunctions.
• Position and Momentum Representations for the bridge between kets and wavefunctions.
• Fourier Transform and Delta Function for free particles, plane waves, and momentum states.
• Boundary Conditions and Sturm–Liouville Theory for bound-state eigenvalue problems.
• Spherical Harmonics and Angular Momentum Algebra for central potentials and rotors.

The Toolkit explains the reusable mathematical machinery. This volume explains what that machinery says about concrete quantum systems.

What belongs here

The core content belongs here when it is a coordinate-space solution of a standard Hamiltonian or a foundational tool for using such solutions.

Representative topics include:

• wavefunctions as position-space representations;
• probability density, probability current, and continuity;
• time-dependent and time-independent Schrödinger equations;
• normalization in discrete, continuous, box-normalized, and delta-normalized settings;
• boundary conditions and self-adjointness at a practical physics level;
• free particles, plane waves, and Gaussian wave packets;
• one-dimensional wells, barriers, tunneling, and scattering from simple potentials;
• harmonic oscillator spectra, wavefunctions, ladder methods, and coherent-state previews;
• two-level Hamiltonians as canonical finite-dimensional models;
• separable three-dimensional systems, boxes, central potentials, and hydrogenic atoms;
• angular systems such as particles on rings, particles on spheres, and rigid rotors;
• charged particles in electromagnetic fields, including Landau levels and gauge choices.

The best pages in this volume should give more than final answers. A mature canonical-system page should include the physical setup, Hilbert space, Hamiltonian, boundary conditions, natural scales, exact solution, spectrum, eigenfunctions, normalization, observables, limiting cases, common mistakes, references, and exercises.

What belongs elsewhere

Some topics are closely related but have better canonical homes elsewhere.

• Abstract postulates and measurement language belong in Core Formalism.
• General mathematical theorems belong in the Mathematical Toolkit, with rigorous versions reserved for a later rigorous quantum mechanics volume.
• Detailed perturbation theory, WKB, and scattering theory belong in approximation and scattering volumes.
• Full angular momentum representation theory belongs in the symmetry and spin volume; this volume uses it in coordinate-space systems.
• Detailed atomic spectra, fine structure, molecules, and light-matter coupling belong in atomic and molecular volumes.
• Quantum gates and algorithms belong in quantum information; two-level systems appear here as canonical Hamiltonians.
• Formula-only lookup tables belong in the Reference.

This separation preserves the one-canonical-home rule. For example, the harmonic oscillator page should solve the oscillator and explain its physical lessons, while the Hermite-polynomial machinery links to the Toolkit and field-mode quantization links forward to QFT-oriented material.

Main learning arc

A good reading path through this volume is:

1. Learn what a coordinate representation is.
2. Interpret $|\psi(x)|^2$ as a probability density, not as a probability by itself.
3. Use the Schrödinger equation to turn a Hamiltonian into an initial-value or boundary-value problem.
4. Understand how boundary conditions and normalization define the problem.
5. Study the free particle to see continuous spectra and wave packets.
6. Study wells to see discrete spectra, nodes, parity, and confinement.
7. Study barriers to see reflection, transmission, evanescent waves, and tunneling.
8. Study the harmonic oscillator to see zero-point energy, ladder structure, and universal local physics.
9. Move into three dimensions, central potentials, rotors, and magnetic-field systems.

Later pages should repeatedly connect back to this arc. The infinite square well is not only a box; it is the simplest laboratory for quantization from boundary conditions. The finite barrier is not only a rectangular potential; it is the simplest exact laboratory for tunneling. The oscillator is not only one solvable model; it is the local normal form of many stable quantum systems.

Common mistakes

• Treating a wavefunction as the state itself rather than a representation of the state.
• Reading $|\psi(x)|^2$ as a probability instead of a probability density.
• Forgetting that boundary conditions are part of the definition of the Hamiltonian problem.
• Assuming all energy eigenfunctions are normalizable in the same way; bound states and scattering states use different conventions.
• Using amplitude ratios instead of probability current ratios for reflection and transmission.
• Treating plane waves as ordinary physical states rather than generalized eigenstates or limiting idealizations.
• Forgetting the coordinate measure in three-dimensional and spherical problems.

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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.

Exercises

1. Classify the following as belonging primarily in this volume, Core Formalism, the Mathematical Toolkit, or the Reference: the Born rule, the Fourier transform convention, the infinite square well spectrum, Pauli matrix identities, and the radial Schrödinger equation.

Solution

The Born rule belongs primarily in Core Formalism. The Fourier transform convention belongs in Learn, while transform theory belongs in the Mathematical Toolkit. The infinite square well spectrum belongs in this volume. Pauli matrix identities belong in the Reference, with explanatory algebra in the Mathematical Toolkit. The radial Schrödinger equation belongs in this volume because it is the coordinate-space reduction of central-potential problems.

2. Explain why the differential expression $-\hbar^2 d^2/(2m dx^2)$ is not enough to define a complete one-dimensional quantum problem.

Solution

The expression gives the local kinetic-energy operator, but a complete problem also needs a Hilbert space, a domain, and boundary conditions. A particle on the full line, an interval with hard-wall boundaries, an interval with periodic boundaries, and a half-line problem can all use the same differential expression while having different allowed wavefunctions and different spectra.