Put a system in a superposition of two energies and each part turns its phase at its own rate. The relative phase between them turns at the difference of the two rates, and the probabilities swing back and forth at that frequency — the Bohr frequency. An ammonia molecule started with its nitrogen on one side tunnels to the other and back again, over and over.

Two tuning forks of slightly different pitch, sounded together, swell and fade at a slow beat equal to the difference of their frequencies. Two energies in a superposition beat in exactly the same way.

The probabilities of finding ammonia's nitrogen on the left and on the right, against $At/\hbar$, for a molecule started on the left: $\cos^{2}(At/\hbar)$ and $\sin^{2}(At/\hbar)$. The nitrogen tunnels across and back at the Bohr frequency $2A/\hbar$.

Expand the initial state in energy eigenstates; each component turns at its own rate. Probabilities and averages then contain cross terms with the relative phases, which oscillate at the Bohr frequencies, the differences of energies over $\hbar$.

• The probabilities of the energies themselves, $|c_{n}|^{2}$, never change.
• An observable that connects two levels, with $\bra{E_{m}}\hat{A}\ket{E_{n}}\neq0$, has an average oscillating at .

Write $\ket{L}$ as the sum of the even and odd energy eigenstates, attach to each its own phase, and take the overlap with $\ket{L}$ again. The two phases, $E_{0}\mp A$, combine into a common phase times a cosine of .

The left state is the sum of the two energy eigenstates.

Each part turns at its own rate.

$\braket{L|g}=\braket{L|e}=\tfrac{1}{\sqrt2}$.

Euler's formula for the cosine.

The common phase drops out of the modulus squared.

In a superposition of two energies the relative phase turns at the difference of the energies divided by $\hbar$, and observables that connect the two levels oscillate at that rate.

Started on the left, ammonia’s nitrogen is certainly on the right when $At/\hbar$ reaches a quarter turn, $\pi/2$.

The probability of each energy is the modulus squared of its coefficient, and the coefficients only gain phases. So the energy distribution never changes.

Started in $\ket{L}$, the probability of $\ket{L}$ at a later time is $\cos^{2}(At/\hbar)$.

Great work! You now know how to:

• evolve a superposition of energy eigenstates
• find the Bohr frequency of its oscillations
• derive the tunnelling of ammonia back and forth