What does a spin Hamiltonian do in time?
Once the electronic structure has been reduced to a spin Hamiltonian, the problem changes. We are no longer solving for the electrons in real space; we are propagating amplitudes, populations and coherences in spin space.
Propagate pure states and density operators and connect them to measurable expectation values.
Separate coherent evolution from population relaxation and dephasing.
Recognize when an effective open-system description is required instead of closed Schrödinger dynamics.
Spin-\(\tfrac12\)
The smallest non-trivial spin
Spin is intrinsic angular momentum—not a tiny classical ball rotating in space
What it is: An intrinsic quantum-mechanical angular momentum carried by particles such as electrons and many nuclei.
What it changes: It produces discrete angular-momentum states and, through the particle's magnetic moment, couples to magnetic fields and other spins.
What you observe: Stern–Gerlach-type quantization, EPR/NMR transitions and magnetic moments.
What it is: A two-level angular-momentum representation with two possible projections along any chosen measurement axis.
What it changes: Its state can be a coherent superposition of the two basis projections rather than a classical vector pointing in one fixed direction.
What you observe: Two-level Zeeman splitting, Rabi oscillations and interference of spin amplitudes.
A spin-\(\tfrac12\) system has a two-dimensional Hilbert space. Choosing the \(z\)-axis as quantization axis gives the basis states \(\lvert\alpha\rangle\) and \(\lvert\beta\rangle\). The corresponding spin operators are
These are not three independent classical components. They are non-commuting operators. That non-commutativity is what gives spin dynamics its genuinely quantum character.
Many spins
Hilbert spaces multiply very quickly
For several spins the total Hilbert space is a tensor product. Two electron spins already give four basis states; adding nuclear spins multiplies the dimension again. For \(N\) spin-\(\tfrac12\) particles,
This exponential growth is the basic scaling problem behind large radical-pair and magnetic-resonance simulations. It is also why stochastic trace sampling, sparse representations and carefully chosen propagators become useful.
Coherent dynamics
Start with Larmor precession
Precession is phase evolution generated by an energy splitting
What it is: The angular frequency at which a spin phase advances in a static magnetic field; for a simple electron it is set by \(g\mu_BB/\hbar\).
What it changes: It determines how quickly transverse spin components rotate in the plane perpendicular to the field.
What you observe: The resonance frequency and oscillation period of transverse magnetization or spin populations in driven experiments.
What it is: The off-diagonal phase relationship between basis states in the density matrix.
What it changes: Coherence allows interference and oscillatory population transfer; losing it removes the well-defined relative phase.
What you observe: Oscillations, echoes and interference-sensitive signals rather than a simple static population.
For one approximately isotropic electron spin in a static field, the Zeeman Hamiltonian is enough to generate precession. The frequency is
Electron-spin Larmor precession
At 1 mT an electron with \(g\approx2\) precesses at roughly 28 MHz.
The numerical frequency is physical; the visual rotation rate is slowed down enormously so that you can see it.
Density matrices
Populations and coherences in one object
The density matrix separates 'how much is in each state' from 'how the states interfere'
What it is: A diagonal density-matrix element giving the probability weight of a basis state.
What it changes: Population transfer changes occupation of spin levels or singlet/triplet manifolds.
What you observe: State-selective reaction yields, magnetization components and level populations.
What it is: An off-diagonal density-matrix element carrying the relative amplitude and phase between two basis states.
What it changes: It enables interference and oscillatory transfer; environmental phase noise suppresses it without necessarily changing populations immediately.
What you observe: Quantum beats, Rabi oscillations, free-induction signals and spin echoes.
What it is: A statistical ensemble that cannot be represented by one pure state vector because different members occupy different quantum states.
What it changes: It requires density-matrix language and naturally describes thermal ensembles or partially decohered systems.
What you observe: Reduced polarization and ensemble-averaged signals rather than one deterministic wavefunction trajectory.
A state vector is enough for a pure closed state. For ensembles and open systems, the density operator is more convenient:
Diagonal elements of \(\rho\) encode populations in the chosen basis; off-diagonal elements encode coherences. The closed-system equation of motion is the Liouville–von Neumann equation
Relaxation
\(T_1\), \(T_2\) and pure dephasing
Population relaxation and phase randomization are different physical processes
What it is: The characteristic time for populations to exchange energy with the environment and return toward longitudinal equilibrium.
What it changes: It changes diagonal density-matrix elements and therefore the population difference between spin levels.
What you observe: Longitudinal recovery after saturation or inversion.
What it is: The characteristic decay time of transverse coherence.
What it changes: It destroys phase relationships between spin states, through both population relaxation and additional phase-randomizing processes.
What you observe: Decay of transverse magnetization, echo amplitude and homogeneous resonance linewidth.
What it is: Phase randomization that changes coherence without requiring energy exchange between spin levels.
What it changes: It shortens \(T_2\) while leaving \(T_1\) unchanged in the simple decomposition.
What you observe: Broader homogeneous lines or faster coherence decay than can be explained from \(T_1\) alone.
Real spin systems are not isolated. Longitudinal relaxation changes populations, while transverse relaxation destroys phase coherence. A useful relation is
\(T_\phi\) is the pure-dephasing time. So \(T_2\) is not simply “the same relaxation as \(T_1\)”. Even if pure dephasing vanished completely, the largest possible value would be \(T_2=2T_1\).
Bloch-type relaxation explorer
Population recovery and coherence decay occur on different timescales.
This is the phenomenological Bloch picture: \(M_z(t)=1-e^{-t/T_1}\) after saturation and \(|M_{xy}(t)|=e^{-t/T_2}\). Microscopic relaxation theory asks where those rates come from.
Open quantum systems
Where do relaxation rates come from?
At the microscopic level, relaxation comes from fluctuating interactions. Molecular rotation, vibrations, conformational motion and solvent dynamics modulate the spin Hamiltonian. Different theories make different assumptions about those fluctuations.
A perturbative, usually Markovian treatment that connects correlation functions and spectral densities to relaxation.
A projection-operator framework that retains memory through a time-nonlocal kernel and is useful when Markovian assumptions become questionable.
Represents open-system evolution through ensembles of stochastic state-vector trajectories rather than propagating the full density matrix directly.
Use \(H(t)\) obtained from molecular motion when the fluctuating interactions themselves are available along a trajectory.
Selected reading
Examples from my work
Modeling spin relaxation in complex radical systems using MolSpin
Open-system density-matrix dynamics for complex radical systems.
J. Comput. Chem. (2023) →Spin Dynamics of Radical Pairs Using the Stochastic Schrödinger Equation in MolSpin
Stochastic state-vector propagation for large radical-pair spin systems.
J. Chem. Theory Comput. (2024) →Multiscale modeling approaches in biomolecular physics
Connecting atomistic motion, electronic structure and quantum observables.
Advances in Physics: X (2026) →Key external literature
Where to read next
These are deliberately selected from outside my own work: foundational papers or reviews that are especially useful for this topic.
On the Theory of Relaxation Processes
A. G. Redfield · IBM Journal of Research and Development (1957). The foundational density-matrix treatment behind Redfield relaxation theory.
Open DOI →On the Generators of Quantum Dynamical Semigroups
G. Lindblad · Communications in Mathematical Physics (1976). The canonical structure of completely positive Markovian quantum dynamics.
Open DOI →