Quantum Dynamics and Formulations

Quantum dynamics predicts how a prepared system changes with time. This volume develops several ways to make that prediction: evolving states, evolving operators, propagator kernels, Green functions, path integrals, and phase-space functions. They describe the same closed-system physics when the Hamiltonian, domains, boundary conditions, and conventions agree; each makes different calculations easier.

Required background. The Schrödinger equation states the evolution law. Unitary Time Evolution introduces the evolution operator. The Time-Dependent Schrödinger Equation shows how it acts on wavefunctions.

One evolution, several useful descriptions

For a time-independent self-adjoint Hamiltonian,

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

The exponential is defined for every state; the differential Schrödinger equation holds in norm for initial vectors in the Hamiltonian’s domain. Finite matrices avoid this domain issue and provide good first examples.

Take a two-level system with $H=\hbar\omega\sigma_z/2$ and initial state $|+x\rangle=(|0\rangle+|1\rangle)/\sqrt2$. The two components acquire opposite phases:

$$|\psi(t)\rangle= \frac{e^{-i\omega t/2}|0\rangle+e^{i\omega t/2}|1\rangle}{\sqrt2}, \qquad \langle\sigma_x\rangle_t=\cos(\omega t).$$

The Schrödinger picture finds this by evolving the state. The Heisenberg picture keeps $|+x\rangle$ fixed and evolves the measured operator. The prediction is identical. Spin Precession in Three Pictures develops this comparison and checks the rotation sign.

Find the description that fits your calculation

Question	Starting point
How is evolution defined, including a changing Hamiltonian?	Foundations of Time Evolution
Should I evolve a state or an observable?	Pictures of Quantum Mechanics
How do commutators, density operators, and split evolutions work?	Operator Dynamics
How does a wavepacket propagate between two times?	Propagators and Kernels
How can an inverse operator reveal energies and boundary conditions?	Green Functions and Resolvents
How does the action enter a quantum amplitude?	Path Integral Formulation
How can quantum states be represented on phase space?	Phase-Space Formulation
How are evolution and state geometry related?	Algebraic and Geometric Formulations
What simplifies when a drive repeats?	Periodic and Driven Dynamics
When does a classical trajectory approximate quantum motion?	Classical Limit and Semiclassical Bridges
Which tools carry over to fields?	QFT Bridge

For a first calculation, follow the time-evolution operator through Schrödinger and Heisenberg pictures, then compare them in the spin-precession example. Next calculate a free-particle propagator or oscillator propagator. The translation table connects the objects and conventions across formulations.

Use Notebooks and Worked Examples for computational calculations and Dynamics Reference for formulas and conventions.

Keep exact rewritings separate from approximations

Changing picture is exact. Truncating a Dyson series, replacing a path integral by stationary phase, or truncating a Moyal expansion is an approximation whose error needs a separate argument. A time-dependent Hamiltonian generally requires time ordering. If $[H(t),H(s)]=0$ for all relevant times, the finite-dimensional solution is

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

not an exponential obtained by substituting the final value $H(t)$ into the time-independent formula. Infinite-dimensional time-dependent problems also require conditions ensuring a well-defined propagator.

For controlled approximation methods continue to Approximation and Semiclassical Methods; for amplitudes and cross sections continue to Scattering Theory. Environmental noise, measurement records, and dissipative evolution are developed in Measurement and Open Quantum Systems.

Exercises

In the two-level example above, what is the probability of finding $\sigma_x=+1$ at time $t$? Check $t=0$ and $t=\pi/\omega$ for $\omega>0$.

Solution

Since the outcomes are $\pm1$, $\langle\sigma_x\rangle=p_+-p_-$ and $p_++p_-=1$. Therefore

$$p_+(t)=\frac{1+\cos(\omega t)}2=\cos^2(\omega t/2).$$

The initial $+x$ outcome is certain. At half a precession period the state points along $-x$, and the $+x$ probability vanishes. Norm conservation does not mean that individual measurement probabilities stay constant.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977 — evolution and quantum pictures.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020 — spin dynamics and time-dependent evolution.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981 — propagators and path integrals.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994 — state evolution, pictures, and the classical limit.