Physics Map

Quantum mechanics is not learned in isolation from earlier physics. Classical mechanics supplies Hamiltonians, energy, action, phase space, angular momentum, and limiting cases. Waves supply superposition, modes, phase, and interference. Electromagnetism supplies potentials, fields, gauge freedom, radiation, and light-matter coupling. Statistical mechanics supplies ensembles, entropy, temperature, and thermodynamic limits. Relativity becomes essential for spin, relativistic wave equations, and the bridge to field theory.

This page is a readiness map. It does not replace a mechanics, electromagnetism, or statistical mechanics course.

Classical Mechanics

Classical mechanics is the main source of model-building language in nonrelativistic quantum mechanics. You should understand:

• generalized coordinates and momenta,
• kinetic plus potential energy models,
• Lagrangians and Hamiltonians,
• conservation laws,
• angular momentum,
• harmonic oscillators,
• central forces,
• action principles,
• small oscillations and normal modes.

The most important habit is to recognize the Hamiltonian as a model of energy and dynamics, not just a symbol named $H$. Useful review pages include Lagrangian Mechanics Review, Hamiltonian Mechanics Review, Phase Space, and Action Principles.

Diagnostic questions:

• Can you write the classical Hamiltonian for a particle in a potential?
• Can you identify conserved energy or angular momentum from a symmetry?
• Can you linearize a stable equilibrium into a harmonic oscillator?
• Can you explain why action has the same dimensions as $\hbar$?

Waves and Oscillations

Wave ideas enter quantum mechanics before the formalism is fully abstract. You should understand:

• sinusoidal waves,
• wavelength, frequency, phase velocity, and group velocity,
• superposition,
• interference,
• standing waves,
• normal modes,
• dispersion,
• wave packets.

Quantum waves are not simply classical material waves, but the mathematics of waves prepares you for amplitudes, Fourier decompositions, uncertainty, stationary states, scattering, and phonons.

Useful review pages include Fourier Series, Fourier Transform, Wave Packets, and Gaussian Wave Packets.

Diagnostic questions:

• Can you distinguish phase velocity from group velocity?
• Can you explain how adding two amplitudes can create interference?
• Can you describe a localized wave packet as a superposition of modes?
• Can you connect narrowness in position to spread in wave number?

Electromagnetism

Electromagnetism becomes essential for atoms, molecules, light, spin interactions, magnetic fields, spectroscopy, condensed matter, and relativistic extensions. You should understand:

• electric and magnetic fields,
• scalar and vector potentials,
• Lorentz force,
• electromagnetic waves,
• polarization,
• energy and momentum in fields,
• gauge freedom at an introductory level,
• dipoles and radiation at a qualitative level.

In quantum mechanics, potentials are not merely computational conveniences. The vector potential enters charged-particle Hamiltonians, and gauge choices can change representations without changing observable predictions.

Diagnostic questions:

• Can you state the difference between fields and potentials?
• Can you recognize when a magnetic field should affect orbital motion, spin, or both?
• Can you explain why light-matter interaction requires both quantum states and electromagnetic fields?
• Can you distinguish a gauge choice from a physical change?

Statistical Mechanics

Statistical mechanics is needed for density matrices, open systems, many-body physics, quantum matter, thermodynamics, and information-theoretic language. You should understand:

• probability distributions over states,
• ensembles,
• entropy,
• temperature,
• partition functions,
• chemical potential,
• Fermi-Dirac and Bose-Einstein statistics at least qualitatively,
• thermodynamic limits.

The key conceptual bridge is that a mixed quantum state is not simply a sloppy pure state. It is the correct object for ensembles, subsystems, thermal states, and noisy devices. Useful starting points include Density Operators, Entropy Overview, and Classical vs Quantum Probability.

Diagnostic questions:

• Can you distinguish a pure state from a statistical mixture?
• Can you explain why temperature is not a property of a single isolated energy eigenstate in elementary thermodynamics?
• Can you state what a partition function is used for?
• Can you recognize when many-particle statistics matter?

Special Relativity

Most early pages use nonrelativistic quantum mechanics, but relativity enters several major routes:

• spin and representation theory,
• relativistic wave equations,
• antiparticles and the limits of fixed-particle quantum mechanics,
• quantum fields,
• high-energy scattering,
• relativistic notation and units,
• causality in quantum information and foundations.

For the bridge to field theory, you should know spacetime intervals, Lorentz transformations, four-vectors, energy-momentum relations, and why particle number is not generally fixed in relativistic quantum theory.

Diagnostic questions:

• Can you write the relativistic energy-momentum relation?
• Can you distinguish Galilean from Lorentz symmetry?
• Can you explain why $c$ is kept explicit unless a unit convention is declared?
• Can you state why a relativistic many-particle theory naturally leads toward fields?

Path-Specific Physics Needs

Path	Physics background to prioritize
First quantum mechanics	waves, oscillators, energy, basic measurement ideas
Undergraduate physics	classical mechanics, waves, electromagnetism, angular momentum
Graduate quantum mechanics	Hamiltonian mechanics, scattering, angular momentum, approximation methods
Quantum information	finite systems, measurement, entropy, basic relativity for causality claims
Quantum chemistry	Coulomb forces, atoms, molecular geometry, oscillators, rotations
AMO physics	electromagnetism, angular momentum, spectroscopy, radiation
Condensed matter	statistical mechanics, lattices, electromagnetism, transport
Bridge to QFT	special relativity, classical fields, harmonic modes, scattering

How Much Review Is Enough

Review enough to use the concepts accurately in quantum problems. For example, you do not need a complete electromagnetism course before reading a basic spin page, but you should know that a magnetic moment couples to a magnetic field. You do not need advanced thermodynamics before density matrices, but you should know what an ensemble is.

The right test is operational: can you identify the physical assumptions in the quantum model and predict which classical limit or physical regime it is meant to approximate?

References

• H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2001.
• J. R. Taylor, Classical Mechanics, University Science Books, 2005.
• D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge University Press, 2017.
• D. J. Morin, Introduction to Classical Mechanics, Cambridge University Press, 2008.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier, 2011.
• E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed., W. H. Freeman, 1992.