Overview

This overview chapter is the routing layer for quantum dynamics. The volume landing page explains the whole volume; this chapter helps the reader choose a formulation and translate between them.

For the visual/conceptual companion, see Map of Quantum Dynamics.

The central message is simple:

Schrödinger, Heisenberg, interaction, propagator, Green-function,
operator-dynamics, path-integral, phase-space, algebraic, and geometric languages are
different formulations of the same quantum dynamics, not different theories.

The right formulation is the one that makes the question transparent while keeping the assumptions visible.

Core Object

For a closed system, the common object behind the volume is the time-evolution operator $U(t,t_0)$. It maps states at $t_0$ to states at $t$:

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

It obeys

$$i\hbar\frac{\partial}{\partial t}U(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

When $H$ is time independent,

$$U(t,t_0) = \exp \left[ - \frac{i}{\hbar}H(t-t_0) \right].$$

When $H(t)$ is time dependent, this exponential is generally not valid unless the Hamiltonians commute at different times. The general solution uses time ordering, introduced in Time Ordering.

Same Prediction, Different Location of Time Dependence

The Schrödinger and Heisenberg pictures give the same expectation values while placing time dependence on different objects:

$$\langle A\rangle_t = \langle\psi_S(t)\rvert A_S\lvert\psi_S(t)\rangle = \langle\psi_H\rvert A_H(t)\lvert\psi_H\rangle.$$

This equality is not a convention trick. It is the consistency condition that makes the pictures physically equivalent.

For a time-independent Hamiltonian,

$$A_H(t) = U^\dagger(t,t_0)A_SU(t,t_0),$$

and

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

The detailed comparison belongs to Translation Table of Formulations.

Reading Path

Use this order if the volume is new to you:

Stage	Goal	Start here
Map	See how the languages connect	Map of Quantum Dynamics
Foundations	Understand $U(t,t_0)$ and time ordering	Foundations of Time Evolution
Pictures	Move time dependence between states and observables	Pictures of Quantum Mechanics
Operator dynamics	Use commutators, density operators, and product formulas	Operator Dynamics
Translation	Decide which language fits a task	Which Formulation Should I Use?
Amplitudes	Compute transition amplitudes and kernels	Propagator Kernel
Spectra	Use resolvents and Green functions	What Is a Green Function?
Actions	Relate amplitudes to histories and stationary phase	Path Integral Formulation
Classical bridge	Compare Hilbert-space and phase-space dynamics	Phase-Space Formulation
QFT bridge	See which language survives into field theory	Why Dynamics Matters for QFT

The path is not linear. A reader doing spectroscopy may jump from time evolution to Green functions. A reader doing perturbation theory may go first to the interaction picture. A reader preparing for QFT may prioritize time ordering, propagators, and path integrals.

Formulation Map

Formulation	Primary object	Best for
Schrödinger picture	$\lvert\psi(t)\rangle$ or $\rho(t)$	state propagation, wavefunctions, numerical evolution
Heisenberg picture	$A_H(t)$	operator equations, symmetries, constants of motion
Interaction picture	transformed states and perturbations	driven systems, scattering setup, light-matter coupling
Operator-dynamics language	commutators, density operators, superoperators, product formulas	conservation laws, mixed-state dynamics, split evolutions
Propagator language	$\langle f\rvert U\lvert i\rangle$ and kernels	transition amplitudes, boundary-value evolution
Green-function language	$(z-H)^{-1}$ and boundary prescriptions	spectra, density of states, response previews
Path-integral language	histories weighted by $e^{iS/\hbar}$	action principles, stationary phase, QFT bridge
Phase-space language	Wigner functions and Weyl symbols	classical limit, quasi-probability, semiclassical intuition
Floquet language	one-period unitary evolution	periodically driven closed systems
Algebraic and geometric language	automorphisms, projective geometry	structural viewpoints and rigorous bridges

The map is a guide to useful starting points. It does not mean one formulation owns a topic exclusively.

Boundaries

This volume owns closed-system formulations and their bridges. It does not own:

• basic postulates, Born rule, or state-update rules, which belong to Core Formalism;
• solving canonical Hamiltonians, which belongs to Wave Mechanics and Model Systems;
• perturbation-theory applications and scattering cross sections, which belong to Approximation and Semiclassical Methods;
• Lindblad equations, decoherence, measurement records, and quantum trajectories, which belong to Measurement and Open Quantum Systems;
• field-theoretic path integrals and relativistic quantum fields, which continue on the QFT side.

When a dynamics page needs one of those topics, it should cross-link rather than duplicate.

Common Mistakes

• Treating formulations as competing interpretations rather than equivalent calculational languages.
• Applying $e^{-iH(t-t_0)/\hbar}$ to a time-dependent Hamiltonian without checking commutators.
• Calling a propagator a probability instead of an amplitude.
• Treating a Green function, propagator kernel, and response function as interchangeable without specifying boundary conditions.
• Using the interaction picture as if it were already an approximation. It is an exact rewriting before a series is truncated.
• Reading a Wigner function as an ordinary probability density.
• Importing QFT terminology before the Hilbert space, particle-number structure, and relativistic setting have changed.

Exercises

Where is the time dependence?

In the Schrödinger picture the state evolves as $\lvert\psi_S(t)\rangle=U(t,t_0)\lvert\psi(t_0)\rangle$. Write the corresponding Heisenberg operator $A_H(t)$ that gives the same expectation value.

Solution

For the same expectation value, define

$$A_H(t) = U^\dagger(t,t_0)A_SU(t,t_0).$$

Then

$$\langle\psi_S(t)\rvert A_S\lvert\psi_S(t)\rangle = \langle\psi(t_0)\rvert A_H(t)\lvert\psi(t_0)\rangle.$$

The state carries the time dependence in the Schrödinger picture, while the operator carries it in the Heisenberg picture.

Which language?

A calculation asks for the amplitude for a particle prepared at $x_i$ and time $t_i$ to be found at $x_f$ and time $t_f$. Which formulation is the natural starting point?

Solution

Use the propagator kernel:

$$K(x_f,t_f;x_i,t_i) = \langle x_f\rvert U(t_f,t_i)\lvert x_i\rangle.$$

It is a transition amplitude. Probabilities require using this amplitude with the initial wavefunction and then taking appropriate absolute squares or probability densities.

Cross-Links

• Quantum Dynamics
• Map of Quantum Dynamics
• Foundations of Time Evolution
• Which Formulation Should I Use?
• Translation Table of Formulations
• Time-Dependent Schrödinger Equation
• Time-Evolution Operator
• Time Ordering
• Pictures of Quantum Mechanics
• Schrödinger Picture
• Heisenberg Picture
• Interaction Picture
• Operator Dynamics
• Liouville–von Neumann Equation
• Trotter Product Formula
• Propagator Kernel
• What Is a Green Function?
• Why Path Integrals?
• Wigner Function
• Why Dynamics Matters for QFT

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).
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley (1977).
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley (1981).
• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover (1931).