How We Came to Know the Cosmos:
Light & Matter

by Dr Helen Klus

# Chapter 16. Heisenberg’s Uncertainty Principle

## 16.1 Heisenberg’s microscope

In 1925-1926, German physicist Werner Heisenberg, Danish physicist Niels Bohr’s assistant at the University of Copenhagen, showed that if electrons orbit the nucleus of atoms as waves, then you cannot measure their position and momentum at the same time.[1]

A microscope, for example, could be used to measure the position of an electron wave by illuminating it with light. The microscope is more accurate the smaller the wavelength, but when the wavelength gets too small, Compton scattering means that the electron’s momentum is changed by an unknown amount; the smaller the wavelength, the larger the change.

Heisenberg showed that it is physically impossible to design an experiment that allows you to measure both of these properties at once, and that there’s a similar relationship between energy and time. He proved this using a new mathematical method called matrix mechanics.

## 16.2 Matrix mechanics

In mathematics, the term ‘matrix’ refers to numbers or symbols that are arranged in rows and columns. A matrix for position, for example, consists of a grid of position values. The plural of matrix is ‘matrices’, and matrix mechanics interprets the physical properties of particles as matrices that evolve in time.

Matrix mechanics was developed in 1925 by Heisenberg and fellow German physicists Pascual Jordan and Max Born[2,3] - who coined the term ‘quantum mechanics’.[4,5] They did this in order to describe how electrons ‘jump’ between atomic shells, creating spectral lines (discussed in Chapter 10).

Heisenberg knew how to multiply matrices, but his equations were not giving him the correct results. With the help of Born, he realised that if you multiply position matrices, Q, by momentum matrices, P, you get a different result than if you multiply momentum matrices P by position matrices Q.

Born found that the matrices are related by:

 QP = PQ+iħ (16.1)

Here, ħ = h/2π, where h is Planck’s constant, and i is the imaginary unit (discussed in Chapter 17), which is also equal to a constant number. This means that the order in which you measure position (x) and momentum (p) matters, and both cannot be measured simultaneously. Energy (E) and time (t) are also related in a similar way:

 ΔxΔp ≥ ħ/2 and ΔEΔt ≥ ħ/2 (16.2)

Here, Δ should be read as ‘change in’.

The position-momentum uncertainty relation confirms the fact that the lowest energy state of an electron in an atom is not zero and the energy-time uncertainty relation determines the width of spectral lines, where the line width is a measure of the uncertainty of the energy.

## 16.3 Quantum tunnelling

The energy-time uncertainty relation also allows electrons - and anything else that obeys the laws of quantum mechanics - to sometimes ‘borrow’ enough energy to overcome forces they wouldn’t be able to classically, and so they can exist in classically forbidden regions. This effect is known as quantum tunnelling.

If a stream of electrons, for example, is fired at an impenetrable wall, some will have enough energy to tunnel through and appear on the other side. If the time needed to overcome the force is Δt, then they can borrow an energy of ΔE. The probability of an electron tunnelling through a barrier gets exponentially lower as the barrier gets wider.

Quantum tunnelling is responsible for alpha radiation, which occurs when alpha particles (helium nuclei) tunnel out of an unstable nucleus, overcoming the strong nuclear force, which is related to the nuclear binding energy[6] (discussed in Chapter 14). The quantum nature of this event means that you cannot predict when a particle will be emitted, only assign a probability to different times.

Quantum tunnelling is also responsible for covalent bonding - the form of chemical bonding that occurs when pairs of electrons are shared between atoms[7] - and nuclear fusion - which occurs when nuclei tunnel through the Coulomb barrier to form a strongly bound state.[8]

### 16.3.1 Electron microscopes

Quantum tunnelling is utilised in scanning tunnelling electron microscopes (STM), which were developed by German physicist Gerd Binnig and Swiss physicist Heinrich Rohrer while working for IBM in Switzerland in 1981.[9]

Electrons at the tip of an STM tunnel through the air to a surface under the influence of an electric field. The probability of transmission depends on how far the tip is from the surface, and so they can be used to map the topology of microscopic surfaces.

In the early 1990s, it was shown that STMs can be used to move individual atoms.[10] This is because the tip exerts a small force on each atom, which can be increased by increasing the voltage. By 1993, single atoms were arranged into closed structures known as quantum corrals.[11] Mike Crommie, Chris Lutz, and Don Eigler first achieved this effect while working for IBM in California. Figure 16.1 shows cobalt atoms - the raised points - on a copper surface. The ripples on the surface are electron waves.

 Figure 16.1Image credit Quantum corral made from cobalt atoms (the raised points) on copper, the ripples on the surface are electron waves.

Heisenberg’s theory was extremely successful, and could be used to calculate where almost all known spectral lines would occur. Yet within a year, Austrian physicist Erwin Schrödinger had developed a new equation that used wave mechanics to make the same predictions[12] (discussed in Chapter 17). Schrödinger’s paper was endorsed by German-Swiss-American physicist Albert Einstein, who saw it as more intuitive than Heisenberg’s matrix mechanics.[13]

## 16.4 References

1. Heisenberg, W., Zeitschrift für Physik 1925, 33, 879–893.

2. Born, M., Jordan, P., Zeitschrift für Physik 1925, 34, 858–888.

3. Born, M., Heisenberg, W., Jordan, P., Zeitschrift für Physik 1926, 35, 557–615.

4. Born, M., Zeitschrift für Physik 1924, 26, 379–395.

5. Fedak, W. A., Prentis, J. J., American Journal of Physics 2009, 77, 128–139.

6. Gamow, G., Zeitschrift für Physik 1928, 51, 204–212.

7. Basdevant, J. L., Lectures on Quantum Mechanics, Springer Science & Business Media, 2007.

8. D’E Atkinson, R., Houtermans, F. G., Nature 1929, 123, 567–568.

9. Binnig, G., Rohrer, H., Gerber, C., Weibel, E., Applied Physics Letters 1982, 40, 178–180.

10. Frankel, F., American Scientist 2005, 93, 261.

11. Crommie, M. F., Lutz, C. P., Eigler, D. M., Science 1993, 262, 218–220.

12. Schrödinger, E., Physical Review 1926, 28, 1049–1070.

13. Dongen, J. van, Einstein’s Unification, Cambridge University Press, 2010.