Two-Level and Finite-Dimensional Canonical Systems

Two-level models describe coherent exchange between two basis states: opposite spin components, two sites, two localized wells, or selected internal levels. Their matrices are small enough to solve exactly, but the physical approximation still needs checking whenever higher levels or environmental effects can matter.

A coupling and a detuning

In a specified orthonormal basis $\lvert1\rangle,\lvert2\rangle$, consider

$$H=\epsilon\sigma_z+K\sigma_x =\begin{pmatrix}\epsilon&K\\K&-\epsilon\end{pmatrix}, \qquad \epsilon,K\in\mathbb R.$$

The basis-state energy difference is $2\epsilon$, while $K$ couples them. The eigenvalues are $\pm\sqrt{\epsilon^2+K^2}$. For nonzero $K$, sweeping $\epsilon$ through zero gives an avoided crossing with minimum gap $2|K|$.

For $\epsilon^2+K^2>0$, starting in $\lvert1\rangle$, the probability of finding $\lvert2\rangle$ is

$$P_{1\to2}(t)=\frac{K^2}{\epsilon^2+K^2} \sin^2\!\left(\frac{\sqrt{\epsilon^2+K^2}}{\hbar}t\right).$$

At zero detuning and nonzero $K$, complete transfer first occurs at $t=\pi\hbar/(2|K|)$. Away from zero detuning, the maximum is smaller. The population oscillation has angular frequency $2\sqrt{\epsilon^2+K^2}/\hbar$; the argument of the sine is half that frequency times time. If both parameters vanish, $H=0$ and the initial state stays fixed.

This static model is distinct from a driven transition. Rabi formulas require a specified drive and, usually, a rotating-frame approximation. Landau–Zener formulas describe a sweep with defined initial and final branches and slope conventions.

Reading sequence

Page	What you will learn
Two-Level Systems	Physical definition, state space, examples, and truncation cautions
Two-State Hamiltonians	Pauli decomposition, mixing, energy projectors, and exact static dynamics
Bloch Sphere: Wave-Mechanics Perspective	Pure-state geometry, measurement axes, overlaps, and precession
Coupled Wells and Avoided Crossings	Localized wavefunctions, tunneling matrix elements, and level repulsion
Tight-Binding Dimer	Two-site hopping, bonding states, and the lattice-model bridge
Landau–Zener Problem: First Encounter	Linear sweep, diabatic and adiabatic labels, and transition probability
Rabi Oscillations: First Encounter	Near-resonant drive, rotating-frame dynamics, and pulse times
Spin-1/2 as a Canonical System: First Encounter	Minimal spin dictionary and magnetic-field realization

Choose a physical realization

Begin with two-state Hamiltonians and the Pauli decomposition, then use the Bloch sphere to picture a state and its evolution. Coupled wells and the tight-binding dimer interpret the same matrix spatially. The spin example explains a magnetic-field realization.

For an effective two-level model, identify omitted levels and the couplings that can populate them. Spectral separation alone is insufficient under a resonant drive, a broad pulse, or a long evolution. Effective Hamiltonians develops that approximation more fully.

Exercises

What is the largest transition probability when $\epsilon=K\ne0$, and why does a common energy shift leave it unchanged?

Solution

The maximum is $K^2/(K^2+K^2)=1/2$. Adding $cI$ to this isolated two-level Hamiltonian multiplies the entire evolving state by $e^{-ict/\hbar}$, which cancels from probabilities. The detuning and level gap do not change.

References

• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• 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.