The heart of quantum mechanics is the wave-particle duality: matter and light possess both wave-like and particle-like attributes. Typically, the wave-like properties are inferred indirectly from the behavior of many electrons or photons, though it's sometimes possible to study them directly. However, there are fundamental limitations to those experiments—namely information about the wave properties of matter that is inherently inaccessible.

And therein lies a loophole: two groups used indirect experiments to reconstruct the wave structure of electrons. A.S. Stodolna and colleagues manipulated hydrogen atoms to measure their electron's wave structure, validating more than 30 years of theoretical work on the phenomenon known as the Stark effect. A second experiment by Daniel Lüftner and collaborators reconstructed the electronic structure of individual organic molecules through repeated scanning, with each step providing a higher resolution. In both cases, the researchers were able to match theoretical predictions to their results, verifying some previously challenging aspects of quantum mechanics.

Neither a wave nor a particle description can describe all experimental results obtained by physicists. Photons interfere with each other and themselves like waves when they pass through openings in a barrier, yet they show up as individual points of light on a phosphorescent screen. Electrons create orbital patterns inside atoms described by three-dimensional waves, yet they undergo collisions as if they were particles. Certain experiments are able to reconstruct the distribution of electric charge inside materials, which appears very wave-like, yet the atoms look like discrete bodies in those same experiments.

Researchers typically deal with this behavior using wave functions. The wave function is a mathematical description of the external attributes of a particle: its position, momentum, and rotational characteristics. Much of quantum mechanics involves calculating wave functions and their evolution using the Schrödinger equation, named for the same guy famous for the cat thought experiment.

The wave function contains two pieces: an absolute piece called the amplitude and a relative component called the phase. When the amplitude is squared, it gives the probability of the outcome of certain measurements, but the phase is not directly accessible. In other words, there's always an aspect of the wave character that cannot be obtained experimentally without resorting to some kind of cleverness. That's a disappointing proposition for those of us interested in direct comparisons between theory and measurement.

However, full knowledge of the wave function is important for understanding chemical reactions and material properties on the atomic or molecular scale. Understanding at that level of detail is especially significant for the next generation of materials and molecular design.

A Stark contrast

Hydrogen is the simplest of atoms, consisting of just one proton and one electron. That means its wave function can be calculated, as it is by innumerable physics and chemistry students at universities every year as class exercises. Since its electron is charged when a hydrogen atom is placed in a uniform electric field (such as exists inside a large capacitor), its wave functions change. That change results in different responses to light, which is known as the Stark effect.

The wave functions in the Stark effect have a peculiar mathematical property, one which Stodolna and colleagues recreated in the lab. They separated individual hydrogen atoms from hydrogen sulfide (H 2 S) molecules, then subjected them to a series of laser pulses to induce specific energy transitions inside the atoms. By measuring the ways the light scattered, the researchers were able to recreate the predicted wave functions—the first time this has been accomplished.

The authors also argued that this method, known as photoionization microscopy, could be used to reconstruct wavefunctions for other atoms and molecules. Since the Stark effect is a general response to external influences, the technique would be very handy for studying atoms' responses to other electric and magnetic fields—essential for understanding the behavior of materials under a wide variety of conditions.

Just a phase

Lüftner and colleagues took a different approach, examining the wave functions of organic molecules chemically attached (adsorbed) on a silver surface. Specifically, they looked at pentacene (C 22 H 14 ) and the easy-to-remember compound perylene-3,4,9,10-tetracboxylic dianhydride (or PTCDA, C 24 H 8 O 6 ). Unlike hydrogen, the wave functions for these molecules cannot be calculated exactly. They usually require using "ab initio" computer models.

The researchers were particularly interested in finding the phase, that bit of the wave function that can't be measured directly. They determined that they could reconstruct it by using the particular way the molecules bonded to the surface, which enhanced their response to photons of a specific wavelength. The experiment involved taking successive iterative measurements by exciting the molecules using light, then measuring the angles at which the photons were scattered away.

Reconstructing the phase of the wave function required exploiting the particular mathematical form it took in this system. Specifically, the waves had a relatively sharp edge, allowing the researchers to make an initial guess and then refine the value as they took successive measurements. Even with this sophisticated process, they were only able to determine the phase to an arbitrary precision—something entirely to be expected from fundamental quantum principles. However, they were able to experimentally reconstruct the entire wave function of a molecule. There was previously no way to check whether our calculated wave functions were accurate or not.

Quantum physics of solace

When we discuss quantum physics, the weirdness of the theory is often emphasized. However, quantum mechanics is the basis of most of modern technology, and these experiments highlight how much we actually understand about it. The wave functions generated by these experiments are exact matches to theoretical predictions. The physics works as expected.

In both the molecular and hydrogen cases, the method used to reconstruct the wave functions could be applied to other systems. As researchers work to understand chemical reactions and material properties on the molecular and atomic levels, such techniques would be very powerful, perhaps leading to new insights about how to control them.

Physical Review Letters, 2013. DOI: 10.1103/PhysRevLett.110.213001 and

PNAS, 2013. DOI: 10.1073/pnas.1315716110 (About DOIs).