The basic principle of matter-wave optics is that particles such as neutrons, atoms, and molecules behave like a wave. As discussed before, this is quite contrary to our notion of a particle. So how can we achieve this feat and act on a particles in such a way that it gets “delocalized”?

**Heisenberg’s uncertainty relation**

The way to go to delocalize a particle is to make use of Heisenberg’s uncertainty relation. This relation states that it is fundamentally impossible to measure two connected entities with arbitrarily high precision. The more exactly you measure one, the less you know about the other. Time and energy are such a pair and position and momentum is another one. However, the level of precision where these effects play a role is quite high, so it’s not something that you usually come across in daily life. For now we focus on the pair of position *x* and the momentum in along this axis *p _{x }*=

*M*×

*v*. Here

_{x}*M*is the mass of the particle and

*v*is its velocity along

_{x}*x*.

Δ*x* × Δ*p _{x}* = Δ

*x*×

*M*Δv

_{x}≥ /2 (1)

The symbol Δ indicates how well we know a certain value. For instance, if Δ*x* gets very small, Δ*p _{x}* must increase accordingly. However, the value of is 1·10

^{-34}Joules × second, which is 34 zeros before the 1, an extremely small number. To illustrate when this effect plays a role, we do a little thought experiment.

**Thought experiment: Heisenberg’s slit**

In this thought experiment we consider a beam of gas atoms, say helium, with a velocity of 500 m/s. We let them fly through a slit, and detect the size of the resulting stripe 1 m behind the slit at a detector. The width of the slit measures the position of the atoms in *x*-direction (Δ*x*), as the atoms have to pass it to reach the detector. In classical physics the beam at the detector is exactly as wide as the slit if we require that all atoms fly only in *z*-direction and not along *x*.

We will then start to reduce the slit width and measure the position *x* of the atoms more and more precisely. At the same time we take a look at the width of the atomic beam at the detector.

We begin with a slit width of 1 m. When we rearrange equation (1) to

Δ*v _{x}* = /(2

*M*Δ

*x*) (2)

we see that the velocity along *x* is not longer exactly 0, but 0.00000008 m/s. This is a tiny velocity and it broadens the signal at the detector by the diameter of an atom – clearly too small a change to detect.

At a slit width of 1 mm, the signal at the detector is 1.00000003 mm in size, still a negligibly small change. When we reduce the slit width by another factor of 1000, we get to 1 µm, 1/80 of the thickness of a piece of paper. Sending the helium atoms through such a tiny slit leads to a signal which is 32 times wider than the slit itself! Intuitively, we would expect the signal at the detector to be the same as the slit – 1 µm. However, on these scales Heisenberg’s uncertainty relation starts to have an effect and the increase in width at the detector is clearly visible.

This little though experiment also shows why we do not come across this effect in daily life – its is simply too small to observe without special equipment.

**An observable effect**

This effect does not only occur in theory, but can also be observed in real experiments. Nairz *et al.,* for instance, have shown this for the molecules C_{60} [1].

The peculiar point is that it is fundamentally impossible to know by how much the velocity component *v*_{x} of a single atoms is changed in advance. It may take any velocity between the boundaries of +Δ*v _{x}* and -Δ

*v*; it may even stay unchanged. As we don’t know exactly in which direction the atom is flying, we also cannot ascribe a determined position to the atom anymore as soon as it leaves the slit –

_{x}*it got delocalized*.

In our experiment we use the same principle that we just discussed to prepare delocalized molecules. We focus the laser down to a spot size of ≈1.5 µm to thermally evaporate molecules from a glass window. The tiny spot of the laser measures the position of the molecules as only molecules hit by the laser are hot enough to leave the surface. After 1.5 m of free flight the molecules are sufficiently delocalized that we can probe their wave properties.

**Make it cold**

However, this is not the only way to do it. In atom optics experiments researcher often use a cloud of strongly cooled atoms. During the cooling process the velocity (and thus the momentum) of the atoms is continuously decreased. From Heisenberg we know that this must have an effect on the position of the atoms as the momentum and the position are connected by the uncertainty relation. Indeed, when the atoms are cold enough their position expands over the whole extend of the atomic cloud, leading to a *Bose-Einstein condensate* [2]. When such a cloud of cold atoms is dropped on a grating, it also shows diffraction.

Now that we know how to delocalize atoms and molecules, we will focus on how to diffract them. In the next post we will take a closer look at the possibilities how to do this.

[1] O. Nairz, M. Arndt, and A. Zeilinger, *Experimental verification of the Heisenberg uncertainty principle for fullerene molecules*, Phys. Rev. A **65**, 032109 (2002)

[2] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, *Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor*, Science **269**, 198 (1995)

01. Waves

02. The de Broglie wavelength

03. Delocalizing matter