Two-Level System Hamiltonian

A closed two-level system has a two-dimensional Hilbert space and a self-adjoint $2\times2$ Hamiltonian. In a chosen orthonormal basis $\{\lvert1\rangle,\lvert2\rangle\}$, the most general form is

$$\hat H = \begin{pmatrix} \epsilon_1 & \kappa\\ \kappa^* & \epsilon_2 \end{pmatrix},$$

where $\epsilon_1,\epsilon_2\in\mathbb R$ and $\kappa\in\mathbb C$. Equivalently,

$$\hat H=c_0I+\mathbf b\cdot\boldsymbol\sigma,$$

with real $c_0$ and $\mathbf b\in\mathbb R^3$. This decomposition separates the common energy offset from the level splitting, eigenbasis, and Bloch-sphere rotation axis.

The canonical physical overview is Two-Level Systems, and the complete static diagonalization is Two-State Hamiltonians. This page is the compact Hamiltonian and convention card.

Quick Reference

Property	General closed two-level system
Hilbert space	$\mathbb C^2$
Hamiltonian	$\hat H=c_0I+\mathbf b\cdot\boldsymbol\sigma$
Parameters	$c_0,b_x,b_y,b_z\in\mathbb R$
Eigenvalues	$E_\pm=c_0\pm\lvert\mathbf b\rvert$
Energy gap	$\Delta E=2\lvert\mathbf b\rvert$
Spectral projectors	$P_\pm=(I\pm\hat{\mathbf b}\cdot\boldsymbol\sigma)/2$
Static propagator	Elementary sine-cosine form
Bloch rotation axis	$\hat{\mathbf b}=\mathbf b/\lvert\mathbf b\rvert$
Bloch angular frequency	$2\lvert\mathbf b\rvert/\hbar$
Degeneracy	Only when $\mathbf b=0$
Domain issue	None: every finite matrix is bounded and everywhere defined

This algebra applies to a literal spin-$1/2$, a pair of atomic levels, two localized orbitals, polarization modes, a qubit subspace, or any other two-dimensional sector. Those systems share mathematics but not necessarily physical observables.

Matrix and Pauli Dictionaries

Define

$$c_0=\frac{\epsilon_1+\epsilon_2}{2}, \qquad \delta=\frac{\epsilon_1-\epsilon_2}{2}.$$

With the standard Pauli matrices,

$$\boldsymbol\sigma=(\sigma_x,\sigma_y,\sigma_z),$$

the Pauli vector is

$$\mathbf b = \left( \operatorname{Re}\kappa, -\operatorname{Im}\kappa, \delta \right).$$

The inverse dictionary is

$$\begin{aligned} \epsilon_1&=c_0+b_z,\\ \epsilon_2&=c_0-b_z,\\ \kappa&=b_x-ib_y. \end{aligned}$$

Thus

$$\hat H = \begin{pmatrix} c_0+b_z & b_x-ib_y\\ b_x+ib_y & c_0-b_z \end{pmatrix}.$$

The sign in $b_y=-\operatorname{Im}\kappa$ follows from the standard convention

$$\sigma_y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}.$$

Check this sign whenever importing formulas from a source with a different basis ordering or Pauli convention. The matrices themselves are tabulated at Pauli Matrices.

Parameters and Units

In the energy-vector convention used above, every component of $\mathbf b$ has units of energy.

Quantity	Meaning	Units
$c_0$	Mean of the two eigenenergies	Energy
$b_z=\delta$	Half the bare-basis energy difference	Energy
$b_x,b_y$	Real and phase-quadrature coupling components	Energy
$\lvert\mathbf b\rvert$	Half the exact level splitting	Energy
$\kappa$	Off-diagonal coupling in the chosen basis	Energy

Many atomic, magnetic-resonance, and qubit sources instead use an angular-frequency vector:

$$\hat H =\frac{\hbar}{2} \left( \Omega_0I+\boldsymbol\Omega\cdot\boldsymbol\sigma \right).$$

The conversion is

$$c_0=\frac{\hbar\Omega_0}{2}, \qquad \mathbf b=\frac{\hbar\boldsymbol\Omega}{2}.$$

In this convention the energy gap is $\hbar\lvert\boldsymbol\Omega\rvert$, and the Bloch vector rotates at angular frequency $\lvert\boldsymbol\Omega\rvert$. A factor-of-two error usually comes from combining the energy-vector and frequency-vector conventions.

Spectrum and Projectors

The Pauli identity

$$(\mathbf b\cdot\boldsymbol\sigma)^2 =\lvert\mathbf b\rvert^2I$$

implies

$$E_\pm=c_0\pm\lvert\mathbf b\rvert.$$

For $\mathbf b\neq0$, define

$$\hat{\mathbf b} =\frac{\mathbf b}{\lvert\mathbf b\rvert}.$$

The spectral projectors are

$$P_\pm =\frac12 \left( I\pm\hat{\mathbf b}\cdot\boldsymbol\sigma \right),$$

so

$$\hat H=E_+P_++E_-P_-.$$

The upper eigenstate has Bloch vector $+\hat{\mathbf b}$, and the lower eigenstate has Bloch vector $-\hat{\mathbf b}$. When $\mathbf b=0$, the Hamiltonian is $c_0I$ and every state is an energy eigenstate; $\hat{\mathbf b}$ and the individual rank-one projectors are then undefined.

The gap closes only at the three simultaneous conditions

$$b_x=b_y=b_z=0.$$

This codimension-three fact underlies the generic appearance of avoided crossings when only one control parameter is varied.

Basis Dependence and Invariants

The entries of the matrix are basis coordinates, not invariant physical quantities. Under a passive unitary basis change $U$,

$$\hat H'=U^\dagger\hat H U.$$

The scalar coefficient $c_0$ and length $\lvert\mathbf b\rvert$ are invariant, while the components of $\mathbf b$ rotate by the associated three-dimensional rotation.

A phase redefinition

$$\lvert1\rangle' =e^{i\alpha}\lvert1\rangle, \qquad \lvert2\rangle' =e^{i\beta}\lvert2\rangle$$

changes the off-diagonal element to

$$\kappa' =e^{i(\beta-\alpha)}\kappa.$$

For one static Hamiltonian, a phase choice can make a nonzero $\kappa$ real and nonnegative. That choice may not simultaneously simplify other observables, several couplings, or a time-dependent family of Hamiltonians. Transition probabilities between specified physical preparations and measurements remain invariant.

The basis states are energy eigenstates only when $\kappa=0$, equivalently $b_x=b_y=0$. Calling $\epsilon_1$ and $\epsilon_2$ the energies when $\kappa\neq0$ confuses bare-basis diagonal entries with exact eigenvalues.

Static Time Evolution

For constant $\hat H$ and $\tau=t-t_0$,

$$U(t,t_0) =e^{-ic_0\tau/\hbar} \left[ \cos\left( \frac{\lvert\mathbf b\rvert\tau}{\hbar} \right)I -i\sin\left( \frac{\lvert\mathbf b\rvert\tau}{\hbar} \right) \hat{\mathbf b}\cdot\boldsymbol\sigma \right].$$

The factor involving $c_0$ is a global phase for an isolated fixed two-level sector. It does not alter transition probabilities, but it should not be discarded when comparing absolute energies, thermodynamic weights, or phases relative to another sector.

For the matrix-basis initial state $\lvert1\rangle$, the probability of finding $\lvert2\rangle$ is

$$P_{1\to2}(t) = \frac{\lvert\kappa\rvert^2} {\delta^2+\lvert\kappa\rvert^2} \sin^2\left[ \frac{t}{\hbar} \sqrt{ \delta^2+\lvert\kappa\rvert^2 } \right].$$

The maximum transfer is unity only at zero detuning $\delta=0$. Far from resonance, $\lvert\delta\rvert\gg\lvert\kappa\rvert$, the coupling produces only a small transition amplitude.

Bloch-Vector Dynamics

Any two-level density operator can be written as

$$\rho=\frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right), \qquad \lvert\mathbf r\rvert\leq1.$$

Unitary evolution under the static Hamiltonian gives

$$\frac{d\mathbf r}{dt} =\frac{2}{\hbar}\mathbf b\times\mathbf r.$$

Thus $\mathbf r$ rotates about $\hat{\mathbf b}$ at angular frequency

$$\omega_{\mathrm B} =\frac{2\lvert\mathbf b\rvert}{\hbar}.$$

Pure states lie on the Bloch sphere, while mixed states lie inside it. The geometry is developed at Bloch Sphere: Wave-Mechanics Perspective.

Unitary Hamiltonian evolution preserves $\lvert\mathbf r\rvert$. Relaxation or dephasing that changes its length requires open-system dynamics, not merely another Hermitian $2\times2$ Hamiltonian.

Time-Dependent Hamiltonians

For

$$\hat H(t)=c_0(t)I+\mathbf b(t)\cdot\boldsymbol\sigma,$$

the instantaneous eigenvalues remain

$$E_\pm(t) =c_0(t)\pm\lvert\mathbf b(t)\rvert.$$

They do not by themselves determine the evolution. In general,

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

At two times,

$$[\hat H(t),\hat H(t')] =2i \left[ \mathbf b(t)\times\mathbf b(t') \right] \cdot\boldsymbol\sigma.$$

The ordinary exponential of the time integral is valid when these commutators vanish, for example when all nonzero $\mathbf b(t)$ point along one fixed axis. A changing axis requires time ordering and can produce nonadiabatic transitions.

• Rabi Oscillations: First Encounter treats periodic driving and rotating-frame conventions.
• Landau–Zener Problem: First Encounter treats a sweep through an avoided crossing.

Physical Dictionaries

Realization	Typical Hamiltonian	Meaning of basis
Spin-$1/2$ in a field	$-\gamma\hbar\,\mathbf B\cdot\boldsymbol\sigma/2$	Eigenstates of a chosen spin component
Coupled localized states	$\bar\epsilon I+\delta\sigma_z-J\sigma_x$	Left and right site or well
Driven transition in a rotating frame	$\hbar(\Delta\sigma_z+\Omega\sigma_x)/2$	Lower and upper internal states
Avoided crossing	$vt\,\sigma_z/2+\Delta_0\sigma_x$	Diabatic states
Polarization mode pair	$c_0I+\mathbf b\cdot\boldsymbol\sigma$	Two orthogonal polarization basis states

For a real spin, $\boldsymbol\sigma$ is related to the physical spin operator by $\hat{\mathbf S}=\hbar\boldsymbol\sigma/2$. For a coupled-well or atomic-level model, the Pauli matrices act on an effective two-state label and are not literal spin observables.

Relevant canonical applications include Coupled Wells and Avoided Crossings, Tight-Binding Dimer, and Spin-$1/2$ as a Canonical System: First Encounter.

Validity of a Two-Level Truncation

Some systems are fundamentally two-dimensional, such as an ideal spin-$1/2$. Others are reduced from a larger Hilbert space. A two-level effective Hamiltonian is reliable only when the neglected states remain weakly populated.

Check:

1. Spectral isolation: the retained pair is separated from omitted levels by a gap large compared with relevant couplings.
2. Drive bandwidth: a pulse does not spectrally address leakage transitions.
3. Matrix elements: the perturbation couples only weakly to excluded states.
4. Time scale: small off-resonant amplitudes do not accumulate into significant leakage over the experiment.
5. Effective corrections: virtual excursions may shift $c_0$, detuning, and coupling even when real leakage is negligible.
6. Dissipation: relaxation to states outside the pair cannot be represented by a closed two-level Hamiltonian.

Projecting the exact Hamiltonian as $P\hat H P$ is generally only the first approximation. Eliminating remote states can generate energy-dependent or perturbative corrections. The physical derivation of the effective model belongs at the canonical system or approximation page.

Open-System Distinction

A Hermitian $2\times2$ Hamiltonian generates only unitary dynamics. A dissipative two-level model evolves $\rho$ with a superoperator, commonly

$$\frac{d\rho}{dt} =-\frac{i}{\hbar}[\hat H,\rho] +\sum_\mu \left( L_\mu\rho L_\mu^\dagger -\frac12 \left\lbrace L_\mu^\dagger L_\mu,\rho \right\rbrace \right).$$

The jump operators and rates are additional model data. The canonical treatment is Lindblad–GKSL Equation, and the compact generator card is Lindblad Generator.

An effective non-Hermitian $2\times2$ matrix may describe conditional no-jump evolution or resonances, but it is not by itself a trace-preserving open-system dynamics.

Common Mistakes

• Treating the displayed basis as the energy basis when $\kappa\neq0$.
• Calling $\epsilon_1$ and $\epsilon_2$ exact energies in the presence of mixing.
• Forgetting the minus sign in $b_y=-\operatorname{Im}\kappa$ for the standard $\sigma_y$.
• Mixing the energy-vector convention $\mathbf b$ with the angular-frequency convention $\boldsymbol\Omega$.
• Confusing the spinor phase frequency $\lvert\mathbf b\rvert/\hbar$ with the Bloch-vector rotation frequency $2\lvert\mathbf b\rvert/\hbar$.
• Treating the phase of one isolated coupling as basis invariant.
• Dropping $c_0I$ in a calculation where relative phase between sectors or thermodynamic energy matters.
• Replacing a time-ordered exponential by an ordinary exponential when the effective-field direction changes.
• Calling every two-level degree of freedom a physical spin.
• Assuming a projected pair remains closed under a strong or broadband drive.
• Modeling relaxation or dephasing with a Hermitian Hamiltonian alone.

Exercises

1. Detuned coherent transfer

Let

$$\hat H =\frac{\hbar}{2} \left( \Delta\sigma_z+\Omega\sigma_x \right),$$

with real constants $\Delta$ and $\Omega$. Find the eigenvalues and the probability that an initial $+1$ eigenstate of $\sigma_z$ is later found in the $-1$ eigenstate.

Solution

Here

$$\mathbf b =\frac{\hbar}{2} (\Omega,0,\Delta),$$

so the eigenvalues are

$$E_\pm =\pm\frac{\hbar}{2} \sqrt{\Omega^2+\Delta^2}.$$

The off-diagonal coupling is $\kappa=\hbar\Omega/2$, and the half-difference of the diagonal entries is $\delta=\hbar\Delta/2$. Therefore

$$P_{+\to-}(t) = \frac{\Omega^2}{\Omega^2+\Delta^2} \sin^2\left[ \frac{t}{2} \sqrt{\Omega^2+\Delta^2} \right].$$

On resonance, $\Delta=0$, complete transfer occurs. Detuning reduces the maximum probability to $\Omega^2/(\Omega^2+\Delta^2)$.

2. Basis-phase transformation

Under $\lvert1\rangle'=e^{i\alpha}\lvert1\rangle$ and $\lvert2\rangle'=e^{i\beta}\lvert2\rangle$, find the transformed coupling and show that the spectrum is unchanged.

Solution

The transformed off-diagonal element is

$$\kappa' =\langle1'\rvert\hat H\lvert2'\rangle =e^{i(\beta-\alpha)}\kappa.$$

Its magnitude is unchanged:

$$\lvert\kappa'\rvert=\lvert\kappa\rvert.$$

The diagonal entries are also unchanged, so

$$E_\pm =c_0 \pm \sqrt{\delta^2+\lvert\kappa\rvert^2}$$

is invariant. The coupling phase is a coordinate choice for one isolated static Hamiltonian; the gap is physical.

3. Why pulse order matters

Consider two constant Hamiltonians $\hat H_x=b\sigma_x$ and $\hat H_z=b\sigma_z$. Show that applying them for equal durations in opposite orders generally gives different evolution operators.

Solution

Their commutator is

$$[\hat H_z,\hat H_x] =b^2[\sigma_z,\sigma_x] =2ib^2\sigma_y,$$

which is nonzero. Therefore

$$e^{-i\hat H_z\tau/\hbar} e^{-i\hat H_x\tau/\hbar} \neq e^{-i\hat H_x\tau/\hbar} e^{-i\hat H_z\tau/\hbar}$$

for generic $\tau$. On the Bloch sphere, the two products are rotations about different axes in different orders. This is the finite-pulse version of the time-ordering requirement for a changing $\mathbf b(t)$.

Canonical Links

• Two-Level Systems defines the model class and its physical realizations.
• Two-State Hamiltonians derives mixing, coherent oscillations, and avoided crossings.
• Pauli-Matrix Hamiltonians develops the Pauli-vector geometry.
• Bloch Sphere: Wave-Mechanics Perspective visualizes unitary dynamics.
• Rabi Oscillations: First Encounter introduces periodic driving.
• Landau–Zener Problem: First Encounter introduces a time-dependent avoided crossing.
• Spin in a Magnetic Field gives the literal spin dictionary.
• Finite-Dimensional Hilbert Spaces supplies the linear-algebra setting.

References

• 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.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions, Wiley, 1992.
• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987.