Quantum Dynamics and Formulations

Quantum dynamics studies how quantum states, observables, amplitudes, and probability distributions change with time. Its central object is the Hamiltonian-generated time evolution of a closed system.

In the Schrödinger picture,

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle =H(t)\lvert\psi(t)\rangle,$$

and the same evolution can be written using a time-evolution operator:

$$\lvert\psi(t)\rangle =U(t,t_0)\lvert\psi(t_0)\rangle.$$

This volume explains the many equivalent languages built around that idea. For the local orientation chapter, start with Overview and Map of Quantum Dynamics, then use Foundations of Time Evolution, Pictures of Quantum Mechanics, and Operator Dynamics for the common equations behind the later formulations.

Why Multiple Formulations Exist

No single representation makes every problem simple. The Schrödinger picture is natural for wavefunctions and numerical state propagation. The Heisenberg picture is natural for operator motion, symmetries, and constants of motion. The interaction picture isolates perturbations. Propagators compute transition amplitudes. Green functions expose spectra and boundary conditions. Path integrals connect amplitudes to the action. Phase-space methods clarify the classical limit.

The formulations are not competing theories. They are different coordinate systems on the same quantum mechanics.

Main Formulations at a Glance

Need	Useful formulation	First page
Track a state in time	Schrödinger picture	Schrödinger Picture
Track observables in time	Heisenberg picture	Heisenberg Picture
Separate solvable and perturbing dynamics	Interaction picture	Interaction Picture
Evolve observables, density operators, and operator algebras	Operator dynamics	Operator Dynamics
Compute transition amplitudes	Propagators and kernels	Propagators and Kernels
Analyze spectra and boundary conditions	Green functions and resolvents	Green Functions and Resolvents
Connect to action principles	Path integrals	Path Integral Formulation
Study the classical limit	Phase-space and semiclassical methods	Phase-Space Formulation
Validate numerical evolution	Reproducible notebooks	Notebooks and Worked Examples
Prepare for field theory	Time ordering, propagators, path integrals	Why Dynamics Matters for QFT

Some linked pages are launch targets for this volume and may appear after the overview pages.

Boundaries

This volume owns closed-system time evolution and equivalent formulations. It does not replace:

• Core Formalism for states, observables, Born rule, and postulates;
• Wave Mechanics and Model Systems for solving standard one-particle Hamiltonians;
• future approximation and scattering pages for Fermi golden rule, WKB, and detailed scattering applications;
• future open-system pages for Lindblad dynamics, decoherence, and quantum trajectories.

Suggested Reading Paths

For a first dynamics pass, read Foundations of Time Evolution, then Pictures of Quantum Mechanics through Schrödinger picture, Heisenberg picture, and Ehrenfest theorem.

For a graduate formalism pass, add operator dynamics, interaction picture, time ordering, Dyson expansion, propagators, Green functions, and path integrals.

For a computational pass, use Notebooks and Worked Examples to pair analytic benchmarks with Fourier grids, propagators, product formulas, imaginary-time projection, Wigner transforms, and Floquet spectra.

For a field-theory bridge, focus on Heisenberg picture, interaction picture, time ordering, propagators, Green functions, path integrals, and correlation functions.

Cross-Links

• Unitary Time Evolution
• Foundations of Time Evolution
• Pictures of Quantum Mechanics
• Operator Dynamics
• Notebooks and Worked Examples
• Dynamics Reference
• Propagators and Kernels
• Green Functions and Resolvents
• Hamiltonians
• Time-Dependent Schrödinger Equation
• Harmonic Oscillator
• QFT Bridge Index
• Further Reading

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.

Exercises

1. Name two formulations that give the same expectation values but move the time dependence to different objects.

Solution

The Schrödinger and Heisenberg pictures are the standard pair. In the Schrödinger picture, states carry the main time dependence. In the Heisenberg picture, states are fixed and observables carry the time dependence. Expectation values agree when the pictures are transformed consistently.