Two-Level Quantum Systems: Landau-Zener, Density Matrix, and Hydrogen Scaling | MIT OpenCourseWare

In this MIT OpenCourseWare lecture, the instructor expands on the Landau-Zener problem and coherent population transfer through resonance, then introduces the density matrix formalism for a two-level system. The talk covers open system dynamics, decoherence, and relaxation via T1 and T2, culminating in a Bloch vector picture and the Bloch equations. A segment on hydrogen atom scaling provides intuition for length and energy scales, Bohr radius, and the impact of nuclear charge on electronic structure.

Introduction and Recap

The lecture builds on the Landau-Zener transition and avoided crossing, emphasizing full coherence during population transfer and the idea that the effective transfer time is set by a window where detuning is small. The discussion frames this transfer as the coherent evolution described by Schrödinger dynamics, with a focus on the conditions under which detuning can be neglected during the short crossing time.

Density Matrix Formalism

The instructor introduces the density operator as a tool to describe quantum systems beyond pure states. This formalism enables treatment of decoherence and particle losses, which a pure state wave function cannot capture. The density matrix combines quantum state averages with ensemble averages, making it possible to describe open systems where the environment induces non-unitary evolution. The diagonal elements are populations, while the off-diagonal elements are coherences, and these quantities evolve under a generalized equation of motion that reduces to Schrödinger evolution for pure states.

Two-Level Systems and Bloch Representation

For arbitrary two-level systems, the most general Hamiltonian can be written as a dot product of a three-vector Ω with the Pauli matrices. The corresponding density matrix can also be expanded in the Pauli basis, yielding a Bloch vector A, whose components describe populations and coherences. The density matrix dynamics are isomorphic to a classical magnetic moment precessing in a time-dependent magnetic field, a powerful conceptual link that extends to any two-level system.

Relaxation and Bloch Equations

Open system dynamics introduce relaxation processes with two characteristic times, T1 and T2. T1 describes population relaxation, typically due to energy exchange with the environment, while T2 describes coherence decay due to dephasing. The Bloch equations encode both the coherent precession and the damping terms, illustrating how coherence can survive or vanish depending on the environment and the presence of inhomogeneities. The discussion also mentions the Weisskopf-Wigner theory of spontaneous emission and clarifies how ground state population and coherence evolve when spontaneous emission channels are present.

Hydrogen Atom Scaling

The lecturer shifts to hydrogen-like atoms to illustrate scaling concepts. The energy levels scale as Rydberg energy proportional to 1/n^2, and the inverse radius scales differently from the radius itself due to the Virial theorem. Length scales, such as the Bohr radius, depend on Z, the nuclear charge, leading to a simple Z scaling for many properties. The talk highlights how certain quantities, like the probability of electron-nucleus overlap, scale with Z and principal quantum number n, and discusses implications for hyperfine structure and field interactions. The explanation emphasizes the distinction between energy scales and spatial scales, and the special behavior of the n-dependent radial exponential part of hydrogenic wave functions.

Outlook

The session sets the stage for further topics including hyperfine structure, external field effects, and a deeper dive into two-level dynamics in real atomic systems. The instructor also previews the next major chapter on atomic structure, including hydrogen, helium, and the origin of energy level separations in atoms.