The Hamiltonian is the observable of energy, and it is also what drives time evolution. Its eigenvalues are the energy levels of the system. Two identical states that the system can tunnel between — a nitrogen atom above or below the plane of an ammonia molecule — are the simplest example: the coupling between them splits one energy into two.

Two identical pendulums joined by a weak spring no longer swing at one frequency but at two, one for swinging together and one for swinging opposite. Coupling two quantum states does the same to their energy.

Without tunnelling, the two positions of ammonia's nitrogen share one energy $E_{0}$. The coupling $A$ splits it into $E_{0}-A$, the even combination, and $E_{0}+A$, the odd one. In ammonia the gap $2A$ is about $10^{-4}$ eV, and the photons it emits are the 24 GHz radiation of the first maser.

The Hamiltonian $\hat{H}$ is the Hermitian operator whose eigenvalues are the possible energies of a system, and it generates the system's evolution in time. For a two-state system with equal energies $E_{0}$ and a coupling $-A$ between the states, it is a symmetric matrix.

• The eigenvectors are the even and odd combinations: $\ket{g}=\tfrac{1}{\sqrt2}(\ket{L}+\ket{R})$ with and with , for .

Write the characteristic equation of the $2\times2$ Hamiltonian. It is a difference of two squares, whose roots lie symmetrically about $E_{0}$ at distance $A$. Putting each root back picks out the even and the odd combination of the two states.

The determinant of the $2\times2$ matrix.

Take square roots.

Each row of the matrix acting on $(1,1)$ gives $E_{0}-A$.

And on $(1,-1)$ it gives $E_{0}+A$.

The Hamiltonian is the observable whose eigenvalues are the energies the system can have. The same operator appears in the Schrödinger equation.

Two states of equal energy coupled by an amount $A$ split into two levels, one $A$ below and one $A$ above the original energy.

With a negative coupling between the two states, the even combination has the lower energy and the odd one the higher.

Adding the same constant to every energy multiplies every state by a common phase in time, which no measurement can see.

Great work! You now know how to:

• read the Hamiltonian as the energy observable
• find the split levels of two coupled states
• say why only energy differences are measurable