The ammonia maser

Ammonia, NH₃, is a pyramid: three hydrogens form a triangular base, and the nitrogen sits either above or below their plane. The two arrangements are mirror images of each other, and classically both are equally good equilibria. Call them $|1\rangle$ and $|2\rangle$ . Quantum mechanically the nitrogen can tunnel from one side to the other, and what follows from that one fact is close to the simplest nontrivial system in quantum mechanics: two states, coupled by a single number. It is also the system that gave masers, and shortly after lasers, their name.

Two configurations, one Hamiltonian

Restrict attention to $|1\rangle$ and $|2\rangle$ and write the Hamiltonian in that basis. Left alone in either well, the molecule would sit at the same energy $E_0$ , since the two configurations are mirror images and nothing distinguishes them energetically. That fixes the diagonal:

\langle 1|\hat H|1\rangle = \langle 2|\hat H|2\rangle = E_0.

The barrier separating the wells is finite, though, so the nitrogen has a small but nonzero amplitude to tunnel through it rather than stay put. That coupling is the off-diagonal element, real by the same mirror symmetry, written $-A$ :

\langle 1|\hat H|2\rangle = \langle 2|\hat H|1\rangle = -A, \qquad \hat H = \begin{pmatrix} E_0 & -A \\ -A & E_0 \end{pmatrix}.

$A > 0$ measures how easily the molecule tunnels: a taller or wider barrier makes $A$ smaller, and $A = 0$ decouples the wells entirely, leaving $|1\rangle$ and $|2\rangle$ each stationary on its own.

Symmetric and antisymmetric states

$\hat H$ is not diagonal in $\{|1\rangle, |2\rangle\}$ , so those are not the states of definite energy. Diagonalising it gives two,

|+\rangle = \frac{1}{\sqrt2}\big(|1\rangle + |2\rangle\big), \quad E_+ = E_0 - A; \qquad |-\rangle = \frac{1}{\sqrt2}\big(|1\rangle - |2\rangle\big), \quad E_- = E_0 + A.

The sign names the combination, not the energy: $|+\rangle$ is the symmetric state and lies lower, since the coupling enters as $-A$ . (Writing these $|S\rangle$ and $|A\rangle$ is common, but $A$ is already the coupling, and $E_A = E_0 + A$ is asking for trouble.)

Tunneling, invisible on the diagonal, has split a single energy $E_0$ into two, $2A$ apart. The symmetric combination sits lower and the antisymmetric one higher, the same pattern that puts a bonding orbital below an antibonding one. For real ammonia $2A/h \approx 23.87\ \text{GHz}$ , a wavelength of about $1.25\ \text{cm}$ : a photon at that frequency connects $|+\rangle$ and $|-\rangle$ . Cleeton and Williams measured this absorption line in 1934, and it is exactly the transition that Gordon, Zeiger and Townes drove into oscillation around 1953–54 to build the first maser, Microwave Amplification by Stimulated Emission of Radiation, the direct ancestor of the laser.

The molecule flips

Prepare the molecule in a definite configuration, $|\psi(0)\rangle = |1\rangle = \frac{1}{\sqrt2}\big(|+\rangle + |-\rangle\big)$ : an equal superposition, not an eigenstate, so it does not sit still. Each component turns at its own rate, set by $E_+$ and $E_-$ ,

|\psi(t)\rangle = \frac{1}{\sqrt2}\Big(e^{-iE_+ t/\hbar}|+\rangle + e^{-iE_- t/\hbar}|-\rangle\Big),

and projecting back onto the original basis gives

P_1(t) = |\langle 1|\psi(t)\rangle|^2 = \cos^2\!\Big(\frac{At}{\hbar}\Big), \qquad P_2(t) = |\langle 2|\psi(t)\rangle|^2 = \sin^2\!\Big(\frac{At}{\hbar}\Big).

The nitrogen genuinely oscillates from one side of the molecule to the other, at angular frequency $2A/\hbar$ , exactly the gap that separates $E_+$ from $E_-$ . Ask a different question, though: not where the nitrogen is, but what energy a measurement would return. Nothing moves at all. $|c_+(t)|^2 = |c_-(t)|^2 = \tfrac12$ for every $t$ , since $|+\rangle$ and $|-\rangle$ are themselves stationary states and only their phase evolves. The flipping lives entirely in the configuration basis $\{|1\rangle, |2\rangle\}$ ; it is invisible to a measurement made in the energy basis.

\hat{H} = (0 - A - A 0)

ψ (0) = ∣1 ⟩

t

in units of

ℏ/ A

Solving…

t = 0.00·gap 0.00·flip period 0.00

The state is two complex numbers, so it is drawn as two: each arrow is one amplitude,

⟨ 1∣ ψ (t)⟩

and

⟨ 2∣ ψ (t)⟩

, its squared length the probability of finding the nitrogen on that side. Watch one shorten as the other grows: that is the inversion, at

2 A /ℏ

. Below, the same state read two ways. The configuration probabilities oscillate, while

∣ c_{+} ∣^{2}

and

∣ c_{-} ∣^{2}

sit at

1/2

and stay there, stated as bars since there is nothing for them to do over time. Time is in units of

ℏ/ A

, which is all

A

ever sets here: it is the only scale in the problem, so changing it renames the clock and leaves every curve exactly where it is.

From oscillation to amplification

A beam of ammonia by itself just flips back and forth. Turning that into a maser needs one more piece, a way to separate $|+\rangle$ from $|-\rangle$ before they reach a cavity. Each geometric configuration carries an electric dipole moment, pointing opposite ways in $|1\rangle$ and $|2\rangle$ , so an external electric field pushes the two levels apart in energy by an amount that grows with the field and has opposite sign for each. Passing the molecular beam through a strongly inhomogeneous field turns that state-dependent Stark shift into a spatial sort: the upper state $|-\rangle$ is focused into a beam, $|+\rangle$ is steered away. What enters the cavity next is a population living almost entirely in $|-\rangle$ , a population inversion, upside down from thermal equilibrium, feeding a cavity resonant at $2A/\hbar$ . A single photon at that frequency, striking one of those molecules, stimulates the $|-\rangle \to |+\rangle$ transition and comes out accompanied by a second photon identical to the first. Repeated across the whole beam, that is amplification by stimulated emission, the physics behind the name.