Dynamics Glossary

This glossary defines the terms that recur across quantum time evolution and its equivalent formulations. It is an orientation page: definitions are compact, conventions are explicit, and links lead to the canonical pages that own derivations, examples, and qualifications.

For formulas collected without full derivations, see the Formula Sheet. For mistakes that span several formulations, see Common Pitfalls.

Dyson Series

The Dyson series is the iterated-integral expansion of a time-ordered evolution operator. For an interaction-picture Hamiltonian $H_I(t)$,

$$\begin{aligned} U_I(t,t_0) &= I + \sum_{n=1}^{\infty} \left( -\frac{i}{\hbar} \right)^n \\ &\quad\times \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2 \cdots \int_{t_0}^{t_{n-1}}dt_n \\ &\quad\times H_I(t_1)H_I(t_2)\cdots H_I(t_n). \end{aligned}$$

The nested limits enforce chronological order. Equivalently, one can integrate over the full hypercube and insert a time-ordering operator with a factor $1/n!$. See Dyson Expansion as Formal Evolution.

Common mistake: treating the Dyson series as automatically convergent for every unbounded Hamiltonian and every time interval. In many applications it is a formal or asymptotic perturbative organization whose mathematical status requires additional hypotheses.

Floquet Operator

The Floquet operator is the one-period evolution operator of a periodic Hamiltonian:

$$U_F(t_0) = U(t_0+T,t_0), \qquad H(t+T)=H(t).$$

It is unitary for a closed system. Its eigenvectors are Floquet modes at the selected drive phase $t_0$, and its eigenphases define quasienergies. Powers $U_F^n$ generate same-phase stroboscopic dynamics. Shifting $t_0$ conjugates $U_F$, preserving eigenvalues while transporting eigenvectors. See Floquet Operators.

Common mistake: assuming $U_F$ determines motion inside the period. It determines boundary-to-boundary evolution; micromotion requires additional intra-period data.

Green Function

A Green function is an inverse kernel together with a boundary, support, or ordering prescription. For a time-independent Hamiltonian, an energy Green function can be written

$$G(x,x';z) = \left\langle x\left\rvert (z-H)^{-1} \right\lvert x'\right\rangle,$$

so that

$$(z-H_x)G(x,x';z) = \delta(x-x').$$

Retarded, advanced, incoming, outgoing, time-ordered, and Euclidean Green functions solve related inverse problems with different global prescriptions. See Green Functions and Resolvents.

Common mistake: saying “the Green function” without naming the operator and prescription. The local source equation generally does not select a unique solution.

Heisenberg Picture

In the Heisenberg picture, a reference-time state is fixed while operators carry the unitary time dependence:

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

For an operator without explicit Schrödinger-picture time dependence and a time-independent Hamiltonian,

$$\frac{dA_H}{dt} = \frac{i}{\hbar} [H_H,A_H].$$

Expectation values agree with the Schrödinger picture when states and operators are transformed consistently. See Heisenberg Picture.

Common mistake: dropping the explicit derivative term when $A_S$ depends directly on time, or evolving both the state and operator as though each belonged to a different picture.

Interaction Picture

The interaction picture splits a Hamiltonian as

$$H(t)=H_0(t)+V(t)$$

and uses the evolution generated by $H_0$ to transfer part of the time dependence from states to operators. When $H_0$ is time independent,

$$V_I(t) = e^{iH_0(t-t_0)/\hbar} V(t) e^{-iH_0(t-t_0)/\hbar}.$$

The interaction-picture state evolves under $V_I(t)$ and is organized by the Dyson series. The picture change is exact; perturbative truncation is a separate decision. See Interaction Picture.

Common mistake: calling the interaction picture itself an approximation. It becomes approximate only when its exact transformed dynamics is truncated or simplified.

Kernel

An operator kernel is a continuous-basis matrix element that represents an operator as an integral transform. In the position basis,

$$A(x,x') = \langle x\rvert A\lvert x'\rangle,$$

and

$$(A\psi)(x) = \int dx'\, A(x,x')\psi(x').$$

The integration measure, generalized-basis normalization, operator domain, and boundary conditions are part of the representation. Kernels are often distributions rather than ordinary square-integrable functions. See Propagator Kernel for the time-evolution case.

Common mistake: treating a kernel as basis independent or forgetting the measure in curvilinear coordinates and constrained spaces.

Moyal Bracket

The Moyal bracket is the phase-space image of the operator commutator. For Weyl symbols $A_W$ and $B_W$,

$$\{A_W,B_W\}_{\rm M} = \frac{1}{i\hbar} \left( A_W\star B_W - B_W\star A_W \right),$$

where $\star$ is the star product. Its expansion begins with the classical Poisson bracket:

$$\{A_W,B_W\}_{\rm M} = \{A_W,B_W\}_{\rm P} + O(\hbar^2)$$

for sufficiently smooth symbols under the standard convention. See Moyal Bracket.

Common mistake: replacing the Moyal bracket by the Poisson bracket without identifying the smoothness, scale, state, and observable conditions that suppress quantum corrections.

Path Integral

A path integral is a regulated representation of a quantum amplitude as a sum over histories. In real time, a coordinate kernel is written formally as

$$K(q_f,t_f;q_i,t_i) = \int_{q(t_i)=q_i}^{q(t_f)=q_f} \mathcal Dq\, e^{iS[q]/\hbar}.$$

The continuum notation abbreviates a regulator such as time slicing, including normalization, endpoint conditions, and an ordering convention. In imaginary time, the oscillatory weight becomes $e^{-S_E/\hbar}$ under suitable continuation. See Path Integral Formulation.

Common mistake: interpreting $e^{iS/\hbar}$ as a probability distribution over paths. Real-time histories contribute complex amplitudes that interfere.

Propagator

In ordinary closed-system quantum mechanics, a propagator is an object that carries data between times. At operator level it is the time-evolution operator $U(t,t_0)$. In a basis it is a matrix element such as

$$K(x,t;x',t_0) = \langle x\rvert U(t,t_0)\lvert x'\rangle.$$

The word is broader in many-body physics and QFT, where it may denote a time-ordered correlator or a Green function with a specified prescription. See Propagators and Kernels and the QFT Bridge.

Common mistake: assuming every object called a propagator is the same mathematical object. Write its operator or correlator definition before comparing formulas.

Quasienergy

A quasienergy is the eigenphase label of a Floquet operator. If

$$U_F\lvert u_a\rangle = e^{-i\varepsilon_aT/\hbar} \lvert u_a\rangle,$$

then $\varepsilon_a$ is defined only modulo the drive quantum:

$$\varepsilon_a \sim \varepsilon_a+m\hbar\Omega, \qquad \Omega=\frac{2\pi}{T}.$$

The invariant datum is the eigenvalue on the unit circle; a real-valued quasienergy requires a Floquet-zone or logarithm-branch choice. See Quasienergies.

Common mistake: interpreting a jump at a reduced-zone boundary as a physical spectral discontinuity. It may be only a branch-cut artifact.

Resolvent

The resolvent of a Hamiltonian is the operator-valued function

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

defined for $z$ in the resolvent set, where the inverse exists as a bounded operator. Its poles, boundary values, and spectral representation encode eigenvalues, projectors, continua, and density of states. For a self-adjoint $H$, every nonreal $z$ lies outside the real spectrum. See Resolvent Operator.

Common mistake: writing $(E-H)^{-1}$ as an ordinary bounded operator at a real spectral value. One generally needs a limiting prescription such as $E\pm i0$ or another generalized interpretation.

Schrödinger Picture

In the Schrödinger picture, states carry the unitary time dependence while operators are fixed except for any explicit time dependence:

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

For a time-independent Hamiltonian,

$$\lvert\psi_S(t)\rangle = e^{-iH(t-t_0)/\hbar} \lvert\psi_S(t_0)\rangle.$$

See Schrödinger Picture.

Common mistake: saying operators never depend on time in this picture. An externally controlled observable or Hamiltonian may have explicit time dependence even though unitary picture evolution is assigned to states.

Stationary Phase

Stationary phase is an asymptotic method for oscillatory integrals. For

$$I(\lambda) = \int dx\, a(x)e^{i\lambda f(x)},$$

large $\lambda$ causes cancellation away from points satisfying

$$f'(x_*)=0.$$

Neighborhoods of stationary points then organize the asymptotic expansion, with phases and prefactors determined by derivatives such as $f''(x_*)$. In path integrals, the analogous stationary-action configurations are classical solutions, but multiple saddles, boundaries, caustics, and zero modes can matter. See Stationary Phase.

Common mistake: concluding that only one classical path contributes exactly. Stationary phase is generally an approximation, and several saddles can interfere.

Stationary State

For a time-independent Hamiltonian, a pure stationary state is an energy eigenstate or any vector within one degenerate energy eigenspace:

$$H\lvert E\rangle = E\lvert E\rangle.$$

Its Schrödinger-picture vector acquires only a phase,

$$\lvert E,t\rangle = e^{-iE(t-t_0)/\hbar} \lvert E,t_0\rangle,$$

so all expectation values of time-independent observables are constant. A density operator is stationary when $[H,\rho]=0$. See Stationary States and Phases.

Common mistake: requiring the state vector itself to be time independent. Its ray and physical statistics are stationary even though the vector carries a global phase.

Time-Evolution Operator

The time-evolution operator maps a state at $t_0$ to the corresponding state at $t$:

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

It satisfies

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

as well as composition and, for closed systems with self-adjoint generators, unitarity. See Time-Evolution Operator.

Common mistake: writing $U(t,t_0)=e^{-iH(t-t_0)/\hbar}$ for a time-dependent Hamiltonian without checking unequal-time commutators.

Time Ordering

Time ordering is the rule that places later-time operators to the left in products:

$$\mathcal T \left[ A(t_1)B(t_2) \right] = \begin{cases} A(t_1)B(t_2), & t_1\gt t_2,\\ B(t_2)A(t_1), & t_2\gt t_1. \end{cases}$$

It is required in the formal exponential for a Hamiltonian that does not commute with itself at unequal times:

$$U(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(s)\,ds \right].$$

See Time Ordering.

Common mistake: treating $\mathcal T$ as decorative notation. It changes operator products whenever unequal-time factors do not commute.

Unitarity

Unitarity means

$$U^\dagger U = UU^\dagger = I.$$

It preserves inner products, norms, orthogonality, and total probability. For a closed system, a self-adjoint Hamiltonian generates unitary evolution under appropriate domain assumptions. Unitarity also implies reversibility:

$$U(t,t_0)^{-1} = U(t,t_0)^\dagger = U(t_0,t).$$

See Unitarity and Conservation of Probability.

Common mistake: using norm preservation as proof that a numerical evolution is accurate. A wrong but unitary approximation still preserves every norm.

Wigner Function

The Wigner function is a real phase-space quasiprobability associated with a density operator. In one common convention,

$$W(x,p) = \frac{1}{2\pi\hbar} \int dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2} \right\rvert \rho \left\lvert x-\frac{y}{2} \right\rangle.$$

Its position and momentum marginals reproduce the corresponding probability densities, but $W$ may be negative. Operator products become star products, and commutator dynamics becomes Moyal-bracket dynamics. See Wigner Function and Phase-Space Conventions.

Common mistake: interpreting negative values as negative outcome probabilities. The Wigner function is a quasiprobability representation, not a joint probability distribution for simultaneously sharp position and momentum.

Common Confusions

• A propagator kernel is a matrix element of time evolution; an energy Green function is a kernel of an inverse such as $(z-H)^{-1}$.
• A stationary state has time-independent physical statistics; the Schrödinger-picture vector can still accumulate a phase.
• A picture transformation redistributes time dependence exactly; an approximation made after the transformation is a separate step.
• Time ordering arranges noncommuting operator products; chronological sorting of scalar functions has no analogous algebraic effect.
• A Wigner function is real and normalized but need not be positive; a classical phase-space probability must be nonnegative.
• A quasienergy is modular because it comes from a unitary eigenphase; an ordinary energy eigenvalue is not made modular unless one deliberately samples a static system stroboscopically.

Self-Checks

1. Propagator or resolvent?

Classify $\langle x\rvert e^{-iH(t-t_0)/\hbar}\lvert x'\rangle$ and $\langle x\rvert(z-H)^{-1}\lvert x'\rangle$.

Solution

The first is a time-domain propagator kernel: a position-basis matrix element of the unitary evolution operator. The second is an energy-domain Green kernel: a matrix element of the resolvent. Fourier or Laplace transforms can relate them under suitable prescriptions, but they are not the same definition.

2. Is the interaction picture approximate?

Explain where approximation enters an interaction-picture perturbation calculation.

Solution

The change of picture defined by the chosen $H_0$ is exact. Approximation enters when the transformed evolution is truncated, for example by keeping only finitely many Dyson terms, applying a rotating-wave approximation, or replacing the exact transformed generator by an effective one.

3. Why is quasienergy modular?

Show that $\varepsilon$ and $\varepsilon+m\hbar\Omega$ define the same Floquet eigenvalue.

Solution

Because $\Omega T=2\pi$,

$$e^{-i(\varepsilon+m\hbar\Omega)T/\hbar} = e^{-i\varepsilon T/\hbar} e^{-i2\pi m} = e^{-i\varepsilon T/\hbar}.$$

The unitary eigenvalue fixes only an equivalence class modulo $\hbar\Omega$.

4. Does unitarity prove accuracy?

Give a unitary evolution rule that can still be numerically wrong.

Solution

A product of exact short-step unitary exponentials in reversed chronological order is unitary, but for a noncommuting time-dependent Hamiltonian it approximates the wrong ordered evolution. Unitarity tests structure preservation, not operator accuracy.

5. What is stationary about a stationary state?

Why can an energy eigenstate be stationary even though its state vector changes with time?

Solution

The vector changes only by a global phase $e^{-iEt/\hbar}$. The ray, probabilities, and expectation values of time-independent observables are unchanged. “Stationary” refers to physical statistics, not literal constancy of a chosen vector representative.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. S. Schulman, Techniques and Applications of Path Integration, Dover, 2005.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution Functions in Physics: Fundamentals,” Physics Reports 106, 121–167 (1984), doi:10.1016/0370-1573(84)90160-1.
• J. H. Shirley, “Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time,” Physical Review 138, B979–B987 (1965), doi:10.1103/PhysRev.138.B979.