# Molecules at a double slit

The double slit experiment is one of the best known experiments in physics. It has been realized with photons, electrons, neutrons, atoms and large molecules.

A source emits objects that traverse two neighboring slits on the way towards the detector.

Try to predict what the distribution of particles on the detector could look like?

When an ensemble of particles of high mass or high velocity traverses the double slit, you’ll find a random pattern of single particles straight behind the two openings. This looks like…

For waves we would expect an interference pattern as a superposition of two diffracted partial waves behind the two slits.

A quantum interference pattern reveals both properties, of waves and particles in a single image. Like for classical particles we can identify individual objects, for instance using fluorescence or surface probe microscopy. But even when the individual particles travel independently from each other through the apparatus, the ensemble image is that of a wave.

Extra: What does the wave function signify?

For more than 80 years scientists have debated the real meaning of the wavefunction. Is the delocalized object at different positions at once? Or is the wavefunction rather a mathematical construct, similarly to imaginary numbers? Why is it then that it determines the dynamics of particles even when they are well separated from each other?

Delocalized wavefunctions determine the structure of atoms and molecules (orbitals), their chemical bonds (aromaticity) and indirectly also biology and life. While electrons in atoms are only delocalized over ca. $$0,000~000 1~\mathrm{mm}$$, in chemistry occasionally hundred times further, modern atom interferometers have confirmed that atoms can be delocalized over centimeters. The research presented here illustrates how to delocalize molecules that may even be composed of about 1000 atoms.

The delocalized quantum wave nature is verified using matter-wave interference.

The following applet shows the intensity distribution of a wave behind two neighboring slits in a screen. You can open or close either slit A or slit B or both and monitor the intensity distribution. Vary the wavelength to see its influence.

 Wavelength $$\lambda$$ - + Distance A-B - + Slit opening - + Slit A Slit B
Extra: Diffraction image behind the double slit.

## Locating the interference minima

The interference intensity behind the double slit reaches a minimum when the two wavelets are out of phase, i.e. when a wave crest encounters a wave trough. This is given when the path length difference $$\Delta s$$ equals an odd multiple of half a wavelength, i.e. $$\Delta s = \left(\pm \frac{1\cdot\lambda}{2},\,\pm \frac{3\cdot\lambda}{2},\,\pm \frac{5\cdot\lambda}{2},\,\dots \right)$$.

## Shape of the interference curve

The distribution of particles follows the probability given by the wave equation. A formal similarity in the equations of electrodynamics and quantum physics explains the similarity in the diffraction patterns of light and matter:

$$I(\theta) \propto \left( \frac{\sin\left(\frac{\pi}{\lambda} b \sin\theta\right)}{\frac{\pi}{\lambda} b \sin\theta} \right)^{\!2} \cdot \cos^2\left(\frac{\pi}{\lambda} a \sin\theta\right)$$ $$I(\theta)$$ is the intensity function of the angle to the normal onto the line between the two slits. The slit width is $$b$$ and the distance between the slits $$a$$.