Common Pitfalls

Quantum dynamics has a compact notation, which is useful until the notation hides assumptions. This page collects common mistakes that lead to wrong calculations or misleading physical interpretations.

Using Ordinary Exponentials for Noncommuting Hamiltonians

Mistake. Writing

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

for any time-dependent Hamiltonian.

Correction. This expression is valid when $[H(t),H(t')]=0$ for all relevant times. In general, use the time-ordered exponential

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

or the equivalent Dyson expansion.

See: Time Ordering.

Mixing Pictures Inconsistently

Mistake. Evolving the state as if it were Schrödinger-picture while also evolving the operator as if it were Heisenberg-picture, then taking the expectation value as though nothing changed.

Correction. Pick a picture and transform states, operators, and Hamiltonians consistently. The expectation value is invariant only when the whole transformation is applied:

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

See: Translation Table of Formulations.

Treating the Interaction Picture as an Approximation

Mistake. Saying that the interaction picture is approximate because it is used in perturbation theory.

Correction. The interaction picture is an exact change of representation once a split $H=H_0+V$ is chosen and the required evolution operators are well defined. Approximation enters when one truncates the Dyson series or makes assumptions about $V_I(t)$.

See: Interaction Picture.

Confusing the Propagator with the Wavefunction

Mistake. Treating $K(x_f,t_f;x_i,t_i)$ as the wavefunction at $x_f$.

Correction. The propagator is a kernel of the evolution operator. It evolves a wavefunction by integration:

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

The kernel depends on two spacetime endpoints; the wavefunction depends on one endpoint and the chosen initial state.

See: Propagator Kernel.

Ignoring Boundary Conditions in Propagators

Mistake. Reusing the free-particle propagator in a box, on a ring, or near a boundary without modification.

Correction. A propagator is tied to the Hamiltonian and its domain. Boundary conditions determine the allowed states, the spectral resolution, and the kernel. For example, a particle on a ring must respect periodicity, while a particle in a box must satisfy the chosen endpoint boundary conditions.

See: Boundary Conditions and Infinite Square Well.

Confusing Green Functions with Propagator Kernels

Mistake. Using “Green function” and “propagator” as interchangeable words in every context.

Correction. In ordinary quantum mechanics, $K(x_f,t_f;x_i,t_i)$ is a time-domain evolution kernel. The resolvent Green function,

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

is an energy-domain object. They are related by transforms and boundary prescriptions, but they are not the same mathematical object.

See: Formula Sheet.

Forgetting Boundary Prescriptions

Mistake. Dropping the small imaginary term in expressions such as

$$G^\pm(E) = \lim_{\epsilon\to 0^+} \frac{1}{E-H\pm i\epsilon}.$$

Correction. The sign of $i\epsilon$ selects boundary conditions, such as outgoing or incoming behavior in scattering and retarded or advanced response in time-domain language. It is not decoration; it is part of the definition.

See the scattering discussion in Normalization Conventions until the full scattering-theory volume is available.

Treating Real-Time Path-Integral Weights as Probabilities

Mistake. Reading

$$e^{iS[x]/\hbar}$$

as a probability weight.

Correction. The real-time path integral sums amplitudes. The phase is oscillatory and interference is essential. Euclidean path integrals use damping weights of the form $e^{-S_E/\hbar}$, but even there normalization, measure, and analytic-continuation assumptions matter.

See: Why Path Integrals? and Path Integral Conventions.

Assuming Smooth Classical Paths Dominate Automatically

Mistake. Thinking the path integral literally sums mostly over smooth classical trajectories.

Correction. The formal integral includes highly irregular paths. Classical equations emerge through stationary-phase reasoning, coarse graining, decoherence, and limiting procedures. The classical path is a saddle, not a statement that all contributing histories are smooth.

See: From Propagators to Path Integrals and Path Integral Conventions.

Interpreting Wigner Functions as Ordinary Probabilities

Mistake. Treating $W(x,p)$ as a joint probability density for position and momentum.

Correction. A Wigner function is a quasiprobability distribution. Its marginals reproduce the correct position and momentum distributions, but it can become negative and does not represent a classical joint distribution in general.

See: Formula Sheet.

Overreading Ehrenfest’s Theorem

Mistake. Concluding that Ehrenfest’s theorem by itself explains the whole classical limit.

Correction. Ehrenfest’s theorem gives exact equations for expectation values, such as

$$\frac{d}{dt}\langle p\rangle = -\langle V'(x)\rangle.$$

Classical Newtonian motion requires additional assumptions, for example that $\langle V'(x)\rangle\approx V'(\langle x\rangle)$ and that the state remains sufficiently localized.

See: Ehrenfest Theorem.

Forgetting That Quasienergies Are Modular

Mistake. Treating Floquet quasienergies as ordinary energies with an absolute ordering.

Correction. For a drive period $T$ and frequency $\Omega=2\pi/T$, quasienergies are defined modulo $\hbar\Omega$:

$$\varepsilon_\alpha \sim \varepsilon_\alpha+n\hbar\Omega.$$

The physically meaningful object is the one-period unitary and its phase spectrum, not an unrestricted real-valued energy ladder.

See: Floquet Theorem in Quantum Mechanics and Formula Sheet.

Thinking a Picture Is More Real Than the Others

Mistake. Saying that the Schrödinger picture is the “real” one because states move, or that the Heisenberg picture is the “real” one because observables move.

Correction. Pictures are representation choices. Correctly transformed predictions agree. Which picture is best depends on the calculation: the Schrödinger picture is natural for wavefunction evolution, the Heisenberg picture for operator equations and field theory, and the interaction picture for perturbation theory.

See: Which Formulation Should I Use?.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45–57, 1984.

Exercises

1. Let $H(t)=f(t)A+g(t)B$ with $[A,B]\ne 0$. State a sufficient condition on $f$ and $g$ under which the ordinary exponential of $\int H(t)\,dt$ is nevertheless valid.

Solution

It is sufficient that $H(t)$ commute with $H(t')$ for all pairs of times. Since

$$[H(t),H(t')] = \left[f(t)g(t')-g(t)f(t')\right][A,B],$$

one sufficient condition is that $f(t)/g(t)$ be constant wherever both functions are defined. Then all $H(t)$ are proportional to a single fixed operator.

2. A wave packet obeys Ehrenfest’s theorem exactly. Does that alone imply the packet follows a classical trajectory?

Solution

No. Ehrenfest’s theorem gives exact equations for expectation values, but the force term is generally $-\langle V'(x)\rangle$, not $-V'(\langle x\rangle)$. A classical trajectory requires additional conditions, such as localization and controlled spreading, so that the expectation value of the force is well approximated by the force at the expectation value.