Map of Quantum Dynamics

Quantum dynamics can look fragmented because the same time evolution appears as moving states, moving observables, transition amplitudes, kernels, Green functions, path integrals, phase-space flows, or algebraic automorphisms. The map below organizes those languages by what they do to the same underlying object.

The backbone is:

Hamiltonian H
  -> time-evolution operator U(t,t0)
  -> pictures of motion
  -> transition amplitudes and propagators
  -> Green functions and spectral information
  -> path integrals and action phases
  -> phase-space and classical-limit language
  -> QFT bridge: time ordering, propagators, sources, fields

The arrows are not a required reading order. They show how one language can be built from, or translated into, another.

Layer 1: Hamiltonian to Time Evolution

The Hamiltonian is the generator of closed-system time evolution. Given $H(t)$, define $U(t,t_0)$ by

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

For time-independent $H$,

$$U(t,t_0) = e^{-iH(t-t_0)/\hbar}.$$

For time-dependent $H(t)$, the map is generally time ordered:

$$U(t,t_0) = \mathcal T \exp \left[ - \frac{i}{\hbar} \int_{t_0}^{t}dt'\,H(t') \right].$$

That is the root of the dynamics tree. The rest of the volume asks how to represent, use, approximate, and reinterpret this object.

Layer 2: Pictures of Motion

The pictures of quantum mechanics answer the question: where should the time dependence live?

Picture	Time-dependent object	Useful when
Schrödinger	states $\lvert\psi_S(t)\rangle$	evolving wavefunctions or state vectors
Heisenberg	observables $A_H(t)$	deriving operator equations and conservation laws
Interaction	interaction states and transformed perturbations	separating solvable motion from an interaction

The same expectation value can be written as

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

The interaction picture inserts an intermediate split $H=H_0+V(t)$. It is exact before any approximation is made:

$$i\hbar \frac{d}{dt} \lvert\psi_I(t)\rangle = V_I(t)\lvert\psi_I(t)\rangle.$$

Start with Schrödinger Picture, Heisenberg Picture, and Interaction Picture.

Layer 3: Amplitudes and Kernels

Once $U(t,t_0)$ exists, transition amplitudes are matrix elements of it:

$$\mathcal A_{fi} = \langle f\rvert U(t_f,t_i)\lvert i\rangle.$$

In position space this becomes the propagator kernel:

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

The kernel evolves wavefunctions by integration:

$$\psi(x_f,t_f) = \int dx_i\, K(x_f,t_f;x_i,t_i)\psi(x_i,t_i).$$

This layer is useful when boundary conditions, wave-packet propagation, and transition amplitudes are central. The canonical entry is Propagator Kernel.

Layer 4: Spectra and Green Functions

Green functions and resolvents package spectral information. The resolvent is

$$G(z) = (z-H)^{-1},$$

defined away from the spectrum of $H$. In an energy eigenbasis,

$$G(z) = \sum_n \frac{ \lvert n\rangle\langle n\rvert } {z-E_n},$$

with integrals or generalized spectral measures replacing the sum for continuous spectra.

This expression explains why poles, branch cuts, and boundary prescriptions matter. They encode energies, resonances, density of states, and response behavior. The entry points are What Is a Green Function?, Resolvent Operator, and Spectral Representation.

Layer 5: Path Integrals and Action

The path-integral formulation rewrites propagator kernels as a formal sum over histories:

$$K(x_f,t_f;x_i,t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal D x\, e^{iS[x]/\hbar}.$$

This language makes the action central. It is especially useful for:

• stationary phase and the classical limit;
• tunneling and instanton previews;
• imaginary-time and Euclidean methods;
• generating functionals and correlation functions;
• the bridge to field-theory path integrals.

The Path Integral Formulation page maps the whole chapter. The correct starting sequence is Why Path Integrals?, From Propagators to Path Integrals, and Time Slicing.

Layer 6: Phase Space and Classical Limits

Phase-space formulations translate operators and density matrices into functions on $(q,p)$, with important caveats. For a pure one-dimensional wavefunction, a common Wigner function convention is

$$W(q,p) = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} dy\, e^{-ipy/\hbar} \psi^*\!\left(q-\frac{y}{2}\right) \psi\!\left(q+\frac{y}{2}\right).$$

The Wigner function has correct marginal distributions but need not be nonnegative. Its equation of motion uses the Moyal bracket, whose classical limit becomes the Poisson bracket under suitable assumptions.

Use this layer when the question is about the classical limit, semiclassical intuition, phase-space pictures, Gaussian states, or quasiprobability. The Phase-Space Formulation page maps the chapter; its first technical sequence is Why Phase Space in Quantum Mechanics?, Wigner Function, and Moyal Bracket.

Layer 7: QFT Bridge

Many field-theory tools are dynamical tools first:

Quantum mechanics	QFT continuation
$U(t,t_0)$	time-ordered evolution and S-matrix language
$\mathcal T$	time-ordered products of fields
propagator kernel	propagators and correlation functions
source coupled to $x(t)$	source coupled to fields
path integral over paths	functional integral over field configurations
oscillator modes	field modes with infinitely many degrees of freedom

The bridge is conceptual, not merely notational. QFT changes the degrees of freedom and the role of locality, particle number, and relativistic causality. The local starting point is Why Dynamics Matters for QFT.

Roadmaps

First dynamics pass

Read:

1. Time-Evolution Operator
2. Schrödinger Picture
3. Heisenberg Picture
4. Heisenberg Equations of Motion
5. Which Formulation Should I Use?

Graduate formalism pass

Add:

1. Time Ordering
2. Interaction Picture
3. Propagator Kernel
4. Resolvent Operator
5. Why Path Integrals?

QFT-bound pass

Prioritize:

1. time ordering;
2. interaction picture;
3. propagators and Green functions;
4. correlation functions;
5. path integrals;
6. sources and generating functionals;
7. Why Dynamics Matters for QFT.

Common Mistakes

• Reading the arrows in the map as a historical order rather than a translation structure.
• Forgetting that $U(t,t_0)$ is the shared root of many formulations.
• Treating path integrals as more fundamental than operators in ordinary nonrelativistic quantum mechanics.
• Treating Green functions as only a many-body tool; they already organize one-particle spectra and boundary conditions.
• Treating Wigner functions as ordinary probability densities.
• Assuming the QFT bridge is only about notation, rather than a change in degrees of freedom.

Exercises

Locate a calculation on the map

You want $\langle X(t)\rangle$ for a harmonic oscillator and know that the operator equation of motion closes on $X$ and $P$. Which layer and formulation should you try first?

Solution

Use the pictures-of-motion layer, specifically the Heisenberg picture. For the harmonic oscillator, the Heisenberg equations close on $X_H(t)$ and $P_H(t)$, making expectation values efficient to compute without first solving the full wavefunction.

Kernel or Green function?

You need the amplitude to propagate from $x_i$ at time $t_i$ to $x_f$ at time $t_f$. In a different calculation, you need to identify bound-state energies as poles. Which object belongs to each task?

Solution

The propagation amplitude is the propagator kernel

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

The pole structure is naturally described by a resolvent or Green function,

$$G(z) = (z-H)^{-1}.$$

The two objects are related, but they answer different questions and require different boundary or analytic prescriptions.

Cross-Links

• Overview
• Which Formulation Should I Use?
• Translation Table of Formulations
• Time-Evolution Operator
• Time Ordering
• Schrödinger Picture
• Heisenberg Picture
• Interaction Picture
• Propagator Kernel
• Resolvent Operator
• 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).
• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover (1931).
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley (1981).
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific (2009).