The strange world of quantum tunnelling Adam Reid The simple phrase `quantum tunnelling' evokes all of the mystery and wonder of quantum mechanics, yet this ubiquitous process can often be quite poorly understood by many otherwise scientically-literate people. It is exploited to great eect in electron microscopes, and other examples of its occurrence include the nuclear fusion reaction occurring on a vast scale within the heart of the Sun (in which hydrogen nuclei fuse into one another to form helium), and also many biochemical processes in our bodies. Thus, we all have an intense personal interest in the phenomenon! What exactly do we mean by `tunnelling' though? In order to answer this question, we rst need to review some basic concepts of energy. Kinetic energy and potential energy One of the most fundamental laws of physics is the conservation of energy: it cannot be created or destroyed, but can only be inter-converted between dierent forms. The forms of relevance to our discussion are kinetic energy (possessed by an object by virtue of its motion) and potential energy (possessed by an object by virtue of its position). Consider a woman skiing in the mountains, as depicted in Figure 1. 1 In this idealized scenario we will assume that there is no friction, air resistance or other source of energy loss. When she has been raised to the top of a slope by a ski lift, to point (a), she possesses gravitational potential energy by virtue of her height above ground level (i.e. her position); suppose that this equals two arbitrary units. While stationary at the top, she possesses zero units of kinetic energy, as she is not in motion. Her total energy is thus two units see Figure 2. Once our skier sets o down the slope, her potential energy reduces as she moves towards ground level, but her kinetic energy increases as she picks up speed; her original two units of potential energy are progressively converted into kinetic energy. Halfway down the slope, at (b), she has one unit of potential energy and one unit of kinetic energy. At the bottom of the slope, at (c), the skier is at ground level and hence possesses zero units of potential energy; all of her energy is now in kinetic form, and she is moving with her maximum speed. This process is then exactly mirrored as the skier climbs the opposite slope. The two units of kinetic energy possessed at (c) are progressively converted into potential energy as she moves further from ground 1 An illustration inspired by a talk on free energy landscapes by Prof Daan Frenkel at the Department of Chemistry. 1
(a) (e) (b) (d) (c) Figure 1: An idealized skiing scene. Energy 2 = potential energy = kinetic energy fast 1 0 (a) (b) (c) (d) Position of skier Figure 2: The variation of potential and kinetic energies as the skier moves from (a) to (e) (see Figure 1). (e) 2
level. The skier ends up at (e), at exactly the same height whence she began on the opposite side of the valley. 2 Potential energy barriers Now consider Figure 3 with a skier starting from rest at (f), descending a steep slope and then encountering a small intermediate peak at (h). Figure 4 shows that while our skier s down as she approaches (h), she will still have sucient kinetic energy remaining to surmount the peak. As her starting energy exceeds the potential energy of point (h), the skier is energetically `permitted' to ski to any location between points (f) and (j). Now imagine that our skier commences her run at a height lower than the small central peak, as depicted in Figures 5 and 6. She starts at (k) and her kinetic energy reduces to zero when she arrives at (l), and hence she is unable to attain the top of the peak. An exactly analogous scenario unfolds if she begins at (m) and skis down into the right-hand valley: (n) is the maximum height she can reach on the right-hand valley wall. The skier is only energetically `permitted' to ski to locations between (k) and (l) or between (m) and (n); other regions, for example on the central peak between (l) and (m), are `forbidden'. We refer to peaks like this as `potential energy barriers'. Tunnelling: a microscopic phenomenon The skier is a macroscopic 3 object which we observe to obey the laws of `classical mechanics', as rst elucidated by Isaac Newton. As explained above, such objects can pass over potential energy barriers (if they have sucient energy), but are otherwise trapped by them. In contrast however, microscopic objects (such as electrons or protons) are observed to pass from one side of a barrier to the other, even if they have insucient energy to do so; they have, in a sense, `tunnelled' through it! To understand this, we must recognise that classical mechanics breaks down when attempting to describe how matter behaves on very small scales. 2 This is true in our idealized scenario, but in reality the skier would not make it all the way to (e) without propelling herself along using her ski poles, as she would lose energy because of friction, air resistance, etc. 3 From the ancient Greek µακρo, meaning `long' or `large' (Oxford English Dictionary). 3
(f) (j) (h) (g) (i) Figure 3: A skier with sucient energy to surmount the central peak. Energy = potential energy = kinetic energy fast fast (ish) 0 (f) (g) (h) (i) (j) Position of skier Figure 4: The energy plot for the skier in Figure 3. The total energy (as shown by the dashed horizontal line) exceeds the height of the central barrier at point (h). 4
(k) (l) (m) (n) Figure 5: In contrast, this skier has insucient energy to surmount the central peak. Energy = potential energy = kinetic energy fast fast 0 (k) (l) (m) (n) Position of skier Figure 6: The energy plot for the skier in Figure 5. The total energy (as shown by the dashed horizontal line) is lower than the height of the central barrier. The forbidden regions are indicated by the polka-dot pattern. 5
Matter as waves: the double-slit experiment By the turn of the twentieth century, matter was considered to be made up of particles (i.e. tiny discrete units) and light was held to be a wave (i.e. an oscillation in a medium). However, this neat dichotomy was to be short-lived, and its inadequacy is demonstrated by the famous double-slit experiment. 4 Most variants thereof consist of two screens; the rst contains two slits, which can be opened and closed, and the second is placed parallel to the rst, attached to a detector. Figures 7-9 show schematic depictions of two versions of the experiment, the rst with tennis balls and the second with a beam of electrons (both of which, being matter, would have been considered to be particles). 5 The detector signals shown in Figures 7 and 8 appear entirely unremarkable. The tennis balls or electrons are red through either one slit or the other (only one is open in each gure), and the peaks of the detector signals are located opposite the corresponding slits. Now consider what happens when both slits are open. We expect each particle (either a tennis ball or an electron in the beam) to pass through either the top slit (50% of the time) or the bottom slit (the other 50% of the time), and hence for the overall detector signal simply to be the sum of the two single-slit signals shown in Figures 7 and 8. Indeed, exactly this observation is made for the tennis balls see Figure 9(a). However, something very unexpected happens with the electrons, as shown in Figure 9(b). Instead of a simple sum, we observe a pattern of peaks and troughs exactly like that observed when light waves pass through the same apparatus see Figure 10. 6 This pattern seems to imply that electrons in the beam passing through the top slit interfere with electrons in the beam passing through the bottom slit i.e. the electrons behave like waves rather than particles. The mystery deepens, however, if we down the ring rate such that electrons can be seen arriving one by one at the detector, thus incontrovertibly establishing their particle-like behaviour. We might now expect the interference pattern to disappear, as there are no longer two separate beams (one through the top slit and one through the bottom slit) to interfere with one another only a single electron passes through at any one time. Intriguingly however, the interference pattern is retained! 7 The electrons are thus behaving both as waves (by interfering) and as particles (by arriving one by one), at the same time. The phenomenon is called `wave-particle duality' and it is universally applicable: all microscopic objects are observed to behave like both a particle and a wave, concurrently. Even light does the same thing: the intensity of a source can be reduced to such a degree that individual packets of light are emitted. These are called photons, and their behaviour in the double-slit experiment is analogous to that of electrons. 4 For background reading on this experiment, see http://en.wikipedia.org/wiki/double-slit_experiment 5 Partly inspired by Feynman & Hibbs, Quantum Mechanics and P ath Integrals : Emended Edition (Dover, 2010), 4-5 6 You can also watch a highly-informative video on interference here: http://youtu.be/j_xd9huz2ay 7 See the experiment in action here: http://youtu.be/zj-0pbruthc 6
Figure 7: Two dierent double-slit experiments: (a) has tennis balls being red (by a racquet) through the top slit in a wall, while (b) has an electron beam being red (by an electron gun) through the top slit in a screen; both show the outcome after many objects have been red through. The curve on the right of (a) shows the signal recorded by the tennis ball detector, while the curve on the right of (b) shows the signal recorded by the electron detector. Figure 8: As Figure 7, but with the bottom slits open instead. Figure 9: As Figures 7 and 8, but now with both slits open. In (a), the single-slit signals are shown by the two pale lines and their sum (which is observed in the experiment) is shown by the darker line. 7
Figure 10: Interference of light waves passing through a double-slit apparatus. Constructive interference creates peaks in the detector signal, while destructive interference creates troughs. Compare this to Figure 9(b). Explaining tunnelling using quantum mechanics In order to understand counterintuitive behaviour like this, physicists were forced to abandon classical physics and develop a new theory quantum mechanics. One of the cornerstores of this theory is Erwin Schrödinger's famous wave equation, which mathematically describes the wave-like properties of matter. In order to work out how a microscopic object behaves, we solve the Schrödinger Equation to obtain a `wavefunction' for the object. We then square this wavefunction and interpret the resulting value as indicating how likely we are to nd the object at a particular position in space. 8 Thus, the area under a specic section of the squared wavefunction curve provides the probability of nding the object in that particular region. Consider a proton being passed between two water molecules (Figure 11). Although its potential energy derives from electrostatic interactions with the two water molecules, rather than gravity, the resulting curve is still very similar to that in Figures 3-6. If we solve the Schrödinger Equation for the proton and then square the resulting wavefunction, we obtain the result shown in Figure 12. The largest areas beneath the squared wavefunction are located in the two `valleys' on either side of the central `peak', indicating that the proton is highly likely to be found in these regions (exactly as expected from the perspective of classical mechanics). However, even though the total energy of the proton (around 0.28 arbitrary units) is lower than the barrier 8 Warning: other interpretations are available! 8
Figure 11: As the proton moves horizontally its potential energy varies and can be read from the graph (dotted line). The proton has its lowest potential energy when located at either x 0 or x 0, which is directly analogous to the skier being located at (g) or (i) in Figure 3. = potential energy = squared wavefunction Figure 12: The dashed horizontal line is the proton energy (which is lower than the barrier). The polka-dot patterned regions were previously described as forbidden (see Figure 6), and the shaded areas below the squared wavefunction curve give the probability of nding the proton in these regions. 9
Figure 13: (a)-(c) show progressively more schematic `matter waves' being overlaid, while (d) shows what happens when the waves in (c) are summed. (around 0.35), there is still a small area in the so-called `forbidden' barrier region (approximately between x = 1 and x = 1). This area equals the probability of nding the proton within the barrier! We thus see that there is a small (but non-zero) chance of the particle tunnelling through the barrier and emerging on the other side. Can humans tunnel? In principle, macroscopic objects like the skier also obey the laws of quantum mechanics and hence we might expect to see her tunnel through the mountainside! Why is this not observed? The skier is a highly complex organism made up of an astronomically large number of particles. Even though each particle behaves like a wave, when combined together these `matter waves' will interfere, resulting in a cancelling-out of the peaks and troughs see Figure 13(d). This decoherence is what prevents large objects from displaying observable wave-like behaviour, including tunnelling. 9 Many apologies to any would-be tunnellers! 9 Shin Takagi, Macroscopic Quantum T unneling (Cambridge University Press, 2002), 4-5 10