Pauli-Matrix Hamiltonians

Pauli-matrix Hamiltonians are the natural form of two-level dynamics. Any Hermitian operator on a two-dimensional Hilbert space can be written as

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

where $c_0$ is a real average energy, $\mathbf b=(b_x,b_y,b_z)$ is a real vector, and

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

is the Pauli-vector notation.

This form separates what is physically invariant from what depends on the chosen basis. The scalar $c_0$ shifts both levels equally. The vector $\mathbf b$ determines the splitting, eigenstates, and rotation axis of two-level dynamics.

Pauli Basis

The Pauli matrices are

$$\sigma_x= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad \sigma_y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad \sigma_z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

Together with $I$, they form a basis for Hermitian two-by-two matrices. Expanding

$$H=c_0I+b_x\sigma_x+b_y\sigma_y+b_z\sigma_z$$

gives

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

Thus the Pauli-vector coefficients are directly readable from the matrix entries.

Dictionary From Matrix Entries

Start with the general Hermitian two-state Hamiltonian

$$H = \begin{pmatrix} E_1&\Delta\\ \Delta^*&E_2 \end{pmatrix}.$$

Then

$$c_0=\frac{E_1+E_2}{2}, \qquad b_z=\frac{E_1-E_2}{2},$$

and

$$b_x=\operatorname{Re}\Delta, \qquad b_y=-\operatorname{Im}\Delta.$$

Conversely,

$$E_1=c_0+b_z, \qquad E_2=c_0-b_z, \qquad \Delta=b_x-ib_y.$$

The minus sign in $b_y=-\operatorname{Im}\Delta$ comes from the standard convention for $\sigma_y$. If a different convention is chosen, the dictionary must be changed consistently.

Spectrum

The Pauli identity

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

implies that the traceless part has eigenvalues $\pm\lvert\mathbf b\rvert$. Therefore

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

The energy splitting is

$$\Delta E=E_+-E_-=2\lvert\mathbf b\rvert.$$

When $\mathbf b=0$, the Hamiltonian is proportional to the identity and every state is an energy eigenstate. When $\mathbf b\ne0$, the eigenstates are the two states aligned and anti-aligned with $\mathbf b$ in Bloch-vector language.

Projectors Onto Energy Eigenstates

For $\mathbf b\ne0$, define the unit vector

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

The projectors onto the upper and lower energy eigenspaces are

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

and

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

These formulas are often the fastest way to compute measurement probabilities, expectation values, and spectral decompositions.

Effective Field Interpretation

The vector $\mathbf b$ is often called an effective field. The phrase is literal for a spin-$1/2$ magnetic moment in a magnetic field, where the Hamiltonian has the form

$$H = -\boldsymbol\mu\cdot\mathbf B = -\gamma\mathbf S\cdot\mathbf B = -\frac{\gamma\hbar}{2} \mathbf B\cdot\boldsymbol\sigma.$$

In that example,

$$\mathbf b=-\frac{\gamma\hbar}{2}\mathbf B.$$

For a double well, atomic two-level subspace, or tight-binding dimer, $\mathbf b$ is usually not a literal magnetic field. It is a compact parameter vector encoding detuning, coupling magnitude, and coupling phase.

Time Evolution

The average term $c_0I$ contributes only an overall phase. The traceless part generates

$$U_{\mathrm{rel}}(t) = \exp\left[ -\frac{i}{\hbar} (\mathbf b\cdot\boldsymbol\sigma)t \right].$$

Using $(\hat{\mathbf b}\cdot\boldsymbol\sigma)^2=I$, one obtains

$$U_{\mathrm{rel}}(t) = \cos\left(\frac{\lvert\mathbf b\rvert t}{\hbar}\right)I -i \sin\left(\frac{\lvert\mathbf b\rvert t}{\hbar}\right) \hat{\mathbf b}\cdot\boldsymbol\sigma.$$

On the Bloch sphere, the corresponding Bloch vector rotates about the axis $\hat{\mathbf b}$ by angle

$$\phi(t)=\frac{2\lvert\mathbf b\rvert t}{\hbar}.$$

The factor of $2$ appears because spinors use half-angles relative to the associated Bloch-vector rotation.

Standard Special Cases

Hamiltonian	Pauli vector	Interpretation
$H=c_0I+b_z\sigma_z$	$(0,0,b_z)$	diagonal detuning in the chosen basis
$H=K\sigma_x$	$(K,0,0)$	real symmetric coupling between basis states
$H=K\sigma_y$	$(0,K,0)$	imaginary antisymmetric coupling
$H=\epsilon\sigma_z+\Delta\sigma_x$	$(\Delta,0,\epsilon)$	real avoided-crossing model
$H=-(\gamma\hbar/2)\mathbf B\cdot\boldsymbol\sigma$	$-(\gamma\hbar/2)\mathbf B$	spin-$1/2$ in a magnetic field

These examples look physically different, but their spectra and rotations all follow from the same Pauli-vector algebra.

Basis Dependence And Invariants

The components of $\mathbf b$ depend on the chosen basis. A change of basis rotates the Pauli-coordinate description. A phase redefinition of one basis state can move a complex coupling between the $\sigma_x$ and $\sigma_y$ components.

The invariant quantities include:

• the average energy $c_0$;
• the splitting $2\lvert\mathbf b\rvert$;
• transition probabilities between specified physical preparation and measurement states;
• whether $\mathbf b=0$, where the two levels are exactly degenerate.

This is why the Pauli form is useful: it makes basis-dependent coordinates and basis-independent physics easy to separate.

Common Mistakes

• Forgetting the identity term $c_0I$ when comparing absolute energies.
• Keeping $c_0I$ when computing transition probabilities where it contributes only a global phase.
• Using $b_y=\operatorname{Im}\Delta$ with the standard $\sigma_y$ convention.
• Confusing the effective field $\mathbf b$ with a literal magnetic field in non-spin examples.
• Forgetting that the Bloch-vector rotation frequency is $2\lvert\mathbf b\rvert/\hbar$.
• Assuming the displayed basis is the energy basis when $b_x$ or $b_y$ is nonzero.

Where This Is Used

• Two-Level Systems introduces the physical model and examples.
• Two-State Hamiltonians derives eigenvectors, mixing, coherent oscillations, and avoided crossings.
• Pauli Matrices gives the algebraic identities used here.
• Bloch Sphere: Wave-Mechanics Perspective gives the pure-state visualization of the effective-field motion.
• Coupled Wells and Avoided Crossings uses $H=\bar E I+\epsilon\sigma_z-K\sigma_x$ as a concrete wave-mechanics model.
• Tight-Binding Dimer uses $H=\bar\epsilon I+\delta\sigma_z-J\sigma_x$ as the two-site lattice model.
• Landau–Zener Problem: First Encounter uses $\mathbf b(t)=(\Delta,0,vt/2)$ for a swept avoided crossing.
• Rabi Oscillations: First Encounter uses a rotating-frame effective field $(\Omega,0,\Delta)$.
• Spin-1/2 as a Canonical System: First Encounter uses $\mathbf b=-(\gamma\hbar/2)\mathbf B$ for the magnetic-field model.
• Bloch Sphere gives the spinor geometry behind the effective-field picture.
• Bloch Sphere Geometry gives the mathematical coordinate picture.
• Two-Level System Hamiltonian is the compact reference card.

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.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. Decompose

$$H= \begin{pmatrix} 3&1-i\\ 1+i&1 \end{pmatrix}$$

as $c_0I+\mathbf b\cdot\boldsymbol\sigma$.

Solution

Here $E_1=3$, $E_2=1$, and $\Delta=1-i$. Therefore

$$c_0=\frac{3+1}{2}=2, \qquad b_z=\frac{3-1}{2}=1.$$

Also,

$$b_x=\operatorname{Re}\Delta=1, \qquad b_y=-\operatorname{Im}\Delta=1.$$

Thus

$$H=2I+\sigma_x+\sigma_y+\sigma_z.$$

2. Find the spectrum of $H=c_0I+\mathbf b\cdot\boldsymbol\sigma$.

Solution

Use

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

Thus the traceless part has eigenvalues $\pm\lvert\mathbf b\rvert$. Adding the scalar part gives

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

3. Show that $P_+=(I+\hat{\mathbf b}\cdot\boldsymbol\sigma)/2$ is a projector.

Solution

Since $\hat{\mathbf b}$ is a unit vector,

$$(\hat{\mathbf b}\cdot\boldsymbol\sigma)^2=I.$$

Then

$$P_+^2 = \frac14 \left( I+\hat{\mathbf b}\cdot\boldsymbol\sigma \right)^2 = \frac14 \left( 2I+2\hat{\mathbf b}\cdot\boldsymbol\sigma \right) =P_+.$$

It is also Hermitian, so it is an orthogonal projector.

4. Compute the relative time-evolution operator for $H=K\sigma_x$.

Solution

Here $\mathbf b=(K,0,0)$, so $\lvert\mathbf b\rvert=K$ and $\hat{\mathbf b}\cdot\boldsymbol\sigma=\sigma_x$. Therefore

$$U(t) = \cos\left(\frac{Kt}{\hbar}\right)I -i\sin\left(\frac{Kt}{\hbar}\right)\sigma_x.$$