Double Well

This is a typical double well potential (as are Potentials 1,2, and ~ 8). In recent decades four Nobelprizes have been won by clever applications of double well potentials! Their main features are tunneling splitting of each level below the middle cusp which above it leads to a sequence of levels with half the expected distance for a single well. Look at the numbers in the table for the lowest levels n=1,2/3,4/5,6 for the splitting, where the resolution of the plot does not show it. Note the increase of the level-doubling the higher the molecule gets inside its private well. This is a direct consequence of the increasing tunneling probability which depends (inverse) exponentially on the size of the (remaining) potential barrier to the other state. Go back and point your browser to the Particle in a Box and its explanations to understand tunneling (Potentials 0,1,2). Consider the pyramidal NH3 (C3v) molecule which on vibrational excitation can reach a flat NH3 (D3h) structure at the cusp of the potential well and relax into the same or the other well. In this symmetrical double well the splitting is called 'inversion doubling' because the nuclear motions during this umbrella vibration - or the tunneling through the barrier - invert the configuration of the molecule (the structures in the two wells are connected by inversion symmetry). In a substituted NH3, e.g. NHDT, the flip-flop of the N atom through the plane of the ligands (or the other way round) inverts its handedness or chirality. This happens rapidly with most NABC molecules at ambient temperature except for some species with unwieldy ligands A,B,C, e.g. Tröger's base, which V.Prelog was able to resolve into S- and R-enantiomers. In NH3 the energy difference between the flat and the pyramidal forms is only 22.7 kJ/mol (0.235 eV, at 0 K, with HF/6-31g(d) optimized model chemistry, incl. zero point energies, the flat form with imaginary umbrella frequency). An important real life application of the inversion doubling of ammonia is the Ammonia Maser, a molecular microwave laser at 23874 MHz, wavelength 1.256 cm. It has been invented by Charles H. Townes who, with his students J.P.Gordon and H.J.Zeiger, built the first maser 1954 at Columbia University, NY. It operates between the two lowest split levels 1 and 2 which are separated by only 0.0000987 eV = 0.796 cm-1 in the real molecule (which actualy carries only three split levels below the cusp). Townes won a Nobelprize 1964 for this discovery. The theory can be read in the Feynman Lectures on Physics, Vol.III, chapters 8 and 9 where the maser is explained for the ammonia molecule as a two states system. By the way, the next splitting between n=3 and n=4 in the plot, has been discovered in the (far) infrared spectrum of ammonia at 36 cm-1. The splitting is, of course, a quantum mechanical effect and arises through the interaction of the two equivalent states on "either side" of the cusp (they are in the same region of space!) which perturbs the single level into a pair. The lower level n=1 has a symmetrical, the higher level n=2 an antisymmetrical superposition of the two vibrational wavefunctions. This fact was used by Townes to separate the split states by an inhomogeneous electrical field in a molecular ammonia beam. By this trick the upper state molecules are driven into the beam, i.e. they are focused into the beam axis, while the lower state ones diffuse out of the beam and are pumped away. The remaining beam now has a population inversion (i.e.there are more molecules in the upper state than in the lower, contrary to thermal equilibrium, hence at a negative absolute temperature!) compared to the beam without electrical field gradient. It is fed into a resonance cavity - a metallic box with holes for the beam to pass through and an antenna to pump in or draw out of microwave radiation. The maser does not start spontaneously since the probability for spontaneous emission at this low frequency is too small. The cavity is shortly irradiated by an external microwave radiation at 24 GHz which induces some n=2 molecules to emit a microwave photon and relax into the lower state. Their electromagnetic field induces more molecules to radiate, as predicted 1917 by A. Einstein. The rising electromagnetic field is leading to an avalanche of emission. In a few microseconds the maser is born and the external microwave source can be shut off. The maser is self sustaining. It is recharged by the continuous stream of upper state molecules from the running beam and some of the radiation generated within the cavity can even be extracted into an external circuit. The oscillations are at a very precise frequency, and can be used to drive a "molecular clock". Several of the other applied "atomic clocks", hydrogen(molecules), caesium(atoms), rubidium(atoms) clocks, operate similarly although the level schemes used may be more complicated than with ammonia. Finally, it is worth mentioning that ammonia masers and many other molecular masers have been detected since 1965 in the molecular clouds of young or forming stars, where they produce intense beams of radiation, investigated by radio astronomers. So, one of the most recent discoveries of mankind has existed for billions of years. Unfortunately, nobody has been around then to tell us ... (;-)