# The Humble Hydrogen Atom

Back in May 2013, scientists announced that they had managed to capture a photo of an electron’s whizzing orbit within a hydrogen atom, using a unique new technology of ‘quantum’ microscopy.  Ladies and gentlemen, let’s take a short trip into the infinitesimally small!  Here is the first photograph of a hydrogen atom!

According to NASA’s Astrophysics Dictionary, atomic hydrogen H (“monatomic” hydrogen) constitutes about 75% of the elemental mass of the Universe.  (Although it is worth noting that most of the Universe’s mass is not actually in the form of chemical elements, or “baryonic” matter.  Food for thought, and another story to be told!)

On Earth, it is extremely rare to come across isolated hydrogen atoms outside experimental settings.  Hydrogen usually combines with other atoms into compounds, or with itself to form ordinary (diatomic) hydrogen gas, H2

The hydrogen atom H contains a single positively charged proton p and a single negatively charged electron e, bound to the nucleus by the Coulomb force.  It is electrically neutral.  The hydrogen atom is unique because it has only one electron.

And the diameter of a hydrogen atom is no bigger than about twice…

As early as 1913, Niels Bohr proposed what is now called the Bohr model of the atom, and suggested that electrons could only have certain classical motions.

The model describes the atom as a small, positively charged nucleus, surrounded by electrons travelling in circular orbits around it.  In a way, the concept is similar in structure to the Solar system, but with attraction provided by electrostatic forces, rather than by gravityAlthough the Bohr model is now obsolete, the quantum theory at the heart of it is still regarded as valid.

The Bohr radius for the hydrogen atom remains an important physical constantThe Bohr radius $a_0 = 5.29 \times 10^{-11}m$ corresponds to the radius of the lowest energy electron orbit predicted by the Bohr model of the atom.

The radius of an atom is over 10,000 times the radius of its nucleus, and less than 1/1000 th  of the wavelength of visible light.  The Bohr model only applies to atoms and ions with a single electron, such as singly ionized helium He II, positronium Ps, and of course hydrogen H.

So, the size of a hydrogen atom in its ‘ground state’ is of order $2a_0 \approx 10^{-10}metre$.

That’s MIGHTY small !!

# The Trouble with Mighty Small Things

Observing the tiniest building blocks of matter has always been tricky.  Not merely because of the infinitesimal size of an atom…

You see, mighty small things operate in mighty strange ways!  At the atomic scale, Nature’s behaviour seems so absurd that particles interactions can only be explained by a special branch of physics.  Electrons have neither definite orbits, nor sharply defined ranges.  Instead, their positions must be described by probability distributions that taper off gradually as one moves away from the region of the nucleus, without any sharp cut-off.

### Mighty small things operate in mighty strange ways!

The development of  Quantum Mechanics in the early part of the 20th century has had a profound influence on the way that scientists now understand the world.  At the centre of it, is the concept of a wave function that satisfies the time-dependent Schrödinger equation.

Here we encounter another very real difficulty.  Things get even weirder.  The basic act of observing such infinitesimal particles seems to be affecting their very existence!

Getting around such a reality-bending concept as the Uncertainty Principle, scientists have relied upon quantum theory to define the behaviour of particles in time and space, with complex equations that predict the probabilities of finding electrons, at any particular moment OR in any particular locations of their orbit around an atom’s densely packed nucleus.

The Schrödinger equation governs the atomic structure, describing it as a wave function.  But so far, the actual observation of that structure has seemed to inevitably destroy it…

The new “quantum microscope” invented by Aneta Stodolna and her colleagues at the FOM Institute for Atomic and Molecular Physics (AMOLF) in the Netherlands, uses the process of photoionization and an electrostatic magnifying lens to observe directly the electron orbital paths of an excited hydrogen atom.

## Atomic Energy Levels and Transitions… of Mighty Small Atoms

Unlike classical particles which can have any energy, a quantum mechanical system, or ‘bound’ particle, can only take on certain discrete values of energy.  These discrete values are called energy levels.  The term is used in the context of the energy levels of electrons in atoms or molecules, bound by the electric field of the atomic nucleus.  The energy spectrum of a system with such discrete energy levels is said to be ‘quantized‘.

#### Energy is always conserved.

This implies that if an atom absorbs a photon with a given energy, the energy of that particular atom must inevitably increase by the exact same amount of energy.  In the same token, if an atom emits a photon, the energy of an atom must decrease by a fixed amount of energy (or ‘quanta’), corresponding to that of the emitted photon.

So that the energy of the photon $E_{ph} = h \nu$ equals the change $\Delta$ in the energy of the atom $E_{atom}$:

$E_{ph} = \Delta E_{atom}$

# A New Look at the Hydrogen Atom Wave Function

As described in the journal Physical Review Letters, Stodolna et al. 2013’s experiment imaged the wave function of a hydrogen atom.  Hydrogen is uniquely suited for the new photography technique because the first element in the periodic chart contains just a single electron.

Initially proposed over 30 years ago, the experiment provides a unique look at one of the few atomic systems that has an analytical solution to the Schrödinger equation.

The hydrogen atom is zapped with laser pulses, thereby forcing the ionised electron to escape from the hydrogen atom along direct and indirect trajectories.

Stodolna and her team first fired two lasers at hydrogen atoms inside a special chamber, thus ejecting electrons from the atoms at speeds and directions dependent on their underlying wave functions.  A strong electric field inside the chamber guided the electrons through a lens and onto a detector, which displayed the electron distribution as light and dark rings on a phosphorescent screen.

The phase difference between the trajectories leads to an interference pattern, which Stodolna et al. 2013 magnified with an electrostatic lens and managed to capture.

This interference pattern was photographed using a high-resolution digital camera.  This is the first ever such photo taken.