Two-State Hamiltonians

A two-state Hamiltonian is a Hermitian two-by-two matrix acting on a two-dimensional Hilbert space. Its diagonalization is the reusable calculation behind level splitting, coherent oscillation, avoided crossings, spin precession, and many qubit models.

In a chosen orthonormal basis $\{\lvert1\rangle,\lvert2\rangle\}$, the general closed-system Hamiltonian is

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

Here $E_1$ and $E_2$ are real, while $\Delta$ may be complex. The off-diagonal term mixes the basis states.

Average Energy And Detuning

Separate the average energy

$$c_0=\frac{E_1+E_2}{2}$$

from the traceless part. Define

$$\epsilon=\frac{E_1-E_2}{2}.$$

Then

$$H=c_0I+ \begin{pmatrix} \epsilon & \Delta\\ \Delta^* & -\epsilon \end{pmatrix}.$$

The term $c_0I$ shifts both energies by the same amount. It contributes an overall phase to time evolution and usually does not affect transition probabilities.

Pauli-Matrix Form

Using Pauli matrices,

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

with

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

This representation turns a two-state Hamiltonian into the geometry of an effective vector $\mathbf b$. Its magnitude is

$$\Omega=\lvert\mathbf b\rvert =\sqrt{\epsilon^2+\lvert\Delta\rvert^2}.$$

The energies are

$$E_\pm=c_0\pm\Omega.$$

Thus the energy splitting is

$$E_+-E_-=2\Omega.$$

Eigenvectors And Mixing Angle

If $\Delta$ is complex, a basis phase choice can often make it real and nonnegative for a single static two-level Hamiltonian. Assume that has been done, so $\Delta\ge0$.

Define a mixing angle $\theta$ by

$$\tan(2\theta)=\frac{\Delta}{\epsilon}.$$

Equivalently,

$$\cos(2\theta)=\frac{\epsilon}{\Omega}, \qquad \sin(2\theta)=\frac{\Delta}{\Omega}.$$

One convenient choice of normalized eigenstates is

$$\lvert+\rangle =\cos\theta\,\lvert1\rangle +\sin\theta\,\lvert2\rangle,$$

and

$$\lvert-\rangle =-\sin\theta\,\lvert1\rangle +\cos\theta\,\lvert2\rangle.$$

When $\Delta=0$, the basis states are already energy eigenstates. When $\epsilon=0$, the mixing is maximal:

$$\theta=\frac{\pi}{4}.$$

Coherent Oscillation

For time-independent $H$, suppose the system starts in $\lvert1\rangle$. After removing the physically irrelevant overall phase from $c_0I$, the probability to find the system in $\lvert2\rangle$ is

$$P_{1\to2}(t) =\frac{\lvert\Delta\rvert^2}{\Omega^2} \sin^2\left(\frac{\Omega t}{\hbar}\right).$$

This formula contains two important effects:

• the oscillation frequency is set by $\Omega/\hbar$;
• detuning suppresses the maximum transition probability.

On resonance, $\epsilon=0$, so $\Omega=\lvert\Delta\rvert$ and the oscillation can reach unit probability:

$$P_{1\to2}(t) =\sin^2\left(\frac{\lvert\Delta\rvert t}{\hbar}\right).$$

Far off resonance, $\lvert\epsilon\rvert\gg\lvert\Delta\rvert$, the maximum transition probability is small.

Avoided Crossings

Suppose a control parameter changes the detuning $\epsilon$, while $\Delta$ stays nonzero. The two energies are

$$E_\pm(\epsilon) =c_0\pm\sqrt{\epsilon^2+\lvert\Delta\rvert^2}.$$

At $\epsilon=0$, the minimum gap is

$$E_+-E_-=2\lvert\Delta\rvert.$$

If $\Delta=0$, the levels cross. If $\Delta\ne0$, they repel and form an avoided crossing. This is the standard two-level explanation of why coupling turns a crossing of bare basis energies into a nonzero spectral gap.

Time-dependent passage through an avoided crossing is introduced in Landau–Zener Problem: First Encounter, with the advanced treatment in Landau–Zener Transition.

Relation To Bloch-Sphere Motion

The traceless Hamiltonian $\mathbf b\cdot\boldsymbol\sigma$ generates rotations of the state vector on the Bloch sphere. The direction of $\mathbf b$ sets the rotation axis, and its magnitude sets the angular frequency.

The wave-mechanics visualization is developed in Bloch Sphere: Wave-Mechanics Perspective. The spinor interpretation is developed in Bloch Sphere. This page keeps the Hamiltonian algebra as the canonical wave-mechanics first encounter.

Common Mistakes

• Forgetting that the off-diagonal element can be complex while the Hamiltonian remains Hermitian.
• Treating $E_1$ and $E_2$ as exact energies when $\Delta\ne0$.
• Keeping the common shift $c_0I$ in transition-probability calculations where it cancels.
• Confusing detuning $\epsilon$ with the full energy splitting $2\Omega$.
• Assuming an avoided crossing occurs without coupling.
• Forgetting that phase conventions can change the apparent phase of $\Delta$.

Where This Is Used

• Two-Level Systems explains the physical model and examples.
• Pauli-Matrix Hamiltonians gives the effective-field and Pauli-projector form of the same algebra.
• Bloch Sphere: Wave-Mechanics Perspective visualizes the same dynamics as rotations of pure two-level states.
• Coupled Wells and Avoided Crossings applies the same diagonalization to biased localized wells.
• Stark Shift in a Two-Level Approximation applies it to static dipole mixing and derives the quadratic-to-linear crossover.
• Tight-Binding Dimer applies the same matrix to two-site hopping.
• H₂⁺ Ion derives the symmetric two-state model from a one-electron, two-center Coulomb problem and identifies the parity splitting with the coherent-transfer scale.
• Landau–Zener Problem: First Encounter adds a time-dependent sweep through an avoided crossing.
• Rabi Oscillations: First Encounter uses the same diagonalization in a rotating-frame driven system.
• Spin-1/2 as a Canonical System: First Encounter gives the spin magnetic-field dictionary.
• Double-Well Potential uses the symmetric two-state Hamiltonian for tunneling splitting.
• Pauli Matrices gives the matrix basis.
• Matrix Diagonalization gives computational tools for larger finite systems.
• Two-Level System Hamiltonian is the quick 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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. Derive the eigenvalues of the general two-state Hamiltonian

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

Solution

Write

$$H=c_0I+ \begin{pmatrix} \epsilon & \Delta\\ \Delta^* & -\epsilon \end{pmatrix},$$

where $c_0=(E_1+E_2)/2$ and $\epsilon=(E_1-E_2)/2$. The traceless part has determinant

$$-\epsilon^2-\lvert\Delta\rvert^2,$$

and trace zero. Its eigenvalues are therefore

$$\pm\sqrt{\epsilon^2+\lvert\Delta\rvert^2}.$$

Adding back $c_0$ gives

$$E_\pm=c_0\pm\sqrt{\epsilon^2+\lvert\Delta\rvert^2}.$$

2. For $\epsilon=0$, show that an initial $\lvert1\rangle$ state reaches $\lvert2\rangle$ with unit probability.

Solution

When $\epsilon=0$,

$$\Omega=\lvert\Delta\rvert.$$

The transition probability formula becomes

$$P_{1\to2}(t) =\sin^2\left(\frac{\lvert\Delta\rvert t}{\hbar}\right).$$

At

$$t=\frac{\pi\hbar}{2\lvert\Delta\rvert},$$

the sine equals $1$, so $P_{1\to2}=1$.

3. What is the minimum gap in an avoided crossing with energies $E_\pm=c_0\pm\sqrt{\epsilon^2+\lvert\Delta\rvert^2}$?

Solution

The gap is

$$E_+-E_- =2\sqrt{\epsilon^2+\lvert\Delta\rvert^2}.$$

This is minimized at $\epsilon=0$, giving

$$\Delta E_{\mathrm{min}}=2\lvert\Delta\rvert.$$