Does the nitrogen atom move with in the molecule in an ammonia maser, or does the molecule flip?
Let me expand a little on my earlier comment. If we only consider the inversion of the ammonia molecule (we ignore rotation and vibrations other than the inversion mode), the Schrödinger equation is given by
$$
-\frac{\hbar^2}{2\mu_\text{red}}\frac{\text{d}^2\psi{z}}{\text{d}z^2}+V(z)\psi(z)=E\psi(z),
$$
with $\mu_\text{red}$ the reduced mass and $z=d_\text{NH}\sin{\alpha}$, where $d_\text{NH}$ is the NH bond distance and $\alpha$ the so-called umbrella angle. The potential V(z) is shown in the link given by porphyrin in one of his comments. Although the reduced mass does, in principle, depend on $\alpha$ (see Eq. 4 of this paper), the dependance is weak and we may consider that the hydrogen atoms move simultaneously with respect to the nitrogen atom. In this case
$$
\mu_\text{red}=\frac{3m_\text{H}m_\text{N}}{3m_\text{H}+m_\text{N}}\approx2.5\text{ u}.
$$
The tunneling does not change the center of mass (c.o.m.) of the molecule, so while the nitrogen moves up, the plane of hydrogen atoms moves down to ensure the stationary c.o.m. (and vice versa). I made the following animated gif at some point during my PhD (sorry for the watermark, but I've  had some negative experiences in the past):



The first paper - to my knowledge - that treats the tunneling of ammonia appeared already in 1932. The first application of QM tunneling was done by Gamow appeared 4 years prior to the ammonia work and solved the mystery of the huge range of observed lifetimes of alpha emitters (20 orders of magnitude). 

In the 1932 paper, Dennison and Uhlenbeck also derive the magnitude of the tunneling splitting as a function of the reduced mass
$$
E_\text{inv}=\frac{h\nu_0}{\pi}\exp\left \{-\frac{2\pi}{h}\int_\limits{-z_0}^{z_0}\left [2\mu_\text{red}\left (V(z)-E \right ) \right ]^{1/2} \right \},
$$
where $\pm z_0$ are the equilibrium positions corresponding to the potential minima, $\nu_0$ is the vibration frequency in one of the two wells and $E$ is the energy of the system with respect to the potential minimum. This expression is derived from the WKB approximation and it basically is the product of the tunneling probability with the frequency with which the reduced mass 'hits' the barrier. As can be seen from the formula, the tunneling frequency scales exponentially with the square root of the reduced mass, which is also observed experimentally. The following picture is taken from my thesis and basically a copy of Fig. 2 of this paper. 



If the molecule would simple rotate instead of tunnel, the change in the energy splitting would be inversely proportional to the reduced mass (via the moment of inertia) and this is clearly not observed.