* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Uncertainty in the classroom
Atomic orbital wikipedia , lookup
Path integral formulation wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Quantum teleportation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Hydrogen atom wikipedia , lookup
Canonical quantization wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum state wikipedia , lookup
Coherent states wikipedia , lookup
Renormalization group wikipedia , lookup
Werner Heisenberg wikipedia , lookup
Hidden variable theory wikipedia , lookup
Renormalization wikipedia , lookup
Identical particles wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
EPR paradox wikipedia , lookup
Double-slit experiment wikipedia , lookup
Elementary particle wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Atomic theory wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Particle in a box wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
FEATURES www.iop.org/journals/physed Uncertainty in the classroom—teaching quantum physics K E Johansson and D Milstead Fysikum, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden Abstract The teaching of the Heisenberg uncertainty principle provides one of those rare moments when science appears to contradict everyday life experiences, sparking the curiosity of the interested student. Written at a level appropriate for an able high school student, this article provides ideas for introducing the uncertainty principle and showing how it can be used to elucidate many topics in modern physics. Introduction Physics syllabuses for pupils in the final few years of their high school studies, e.g. A-levels in the UK system, include the physics of waves and provide an introduction to many topics in quantum physics. The students are thus equipped to understand at a basic level one of the most thought-provoking results in modern physics: the Heisenberg uncertainty principle. One form of the Heisenberg uncertainty principle asserts that, contrary to our expectations from classical physics, it is impossible to simultaneously know the position and momentum of a particle to arbitrary accuracy, whilst another form relates uncertainties in energy with time. The Heisenberg uncertainty principle therefore has profound scientific and philosophical implications. There already exist a number of studies discussing the appropriate way to teach quantum mechanics topics at high school (see, for example, [1–4]). As a complement to these works, we provide here a short, self-contained article specifically on the uncertainty principle. We show how it can be demonstrated and how it can be used to gain a deeper understanding of various physical phenomena such as radioactivity and exchange forces. It is hoped that this article could be of interest both for physics teachers and high school students and could be a resource for schemes such as the UK’s Gifted and Talented program. Getting to the uncertainty principle There are a number of ways to arrive at the Heisenberg uncertainty principle. One approach is to use a gedanken (thought) experiment method to demonstrate that even with ‘perfect’ apparatus it is impossible in principle to simultaneously determine a particle’s position and momentum to arbitrary accuracy [5]. Other authors adopt a more rigorous approach, demonstrating how fundamental uncertainties arise from dispersions in a particle’s wavefunction [6]. Always mindful that the longer the explanation, the fewer attentive students, we feel that it is best to start with a situation with which the students are already very familiar. Therefore, we use single-slit diffraction to demonstrate the effects of the Heisenberg uncertainty principle and show that it is nothing more than wave–particle duality cast in a different light. Furthermore, with this approach (outlined below) there is little new mathematics to consider. 0031-9120/08/020173+07$30.00 © 2008 IOP Publishing Ltd PHYSICS EDUCATION 43 (2) 173 K E Johansson and D Milstead _ e _ e _ e Δx py _ e _ e B θ px A not to scale Figure 1. Single slit diffraction with an electron beam. A broad beam of electrons with momentum p y moves in the y -direction towards a wall containing a small slit of width l , as shown in figure 1. Far behind the slit is a detector which can detect a single electron. We know that any electrons which are detected must have passed through the slit. When we look at the locations of the scattered electrons in the electron detector, we observe that they are spread out in a diffraction pattern with maxima and minima. Of course, we are only witnessing the consequences of wave–particle duality, something with which most students are already familiar. An electron has a de Broglie wavelength λ = phy related to the momentum p y it possessed as it entered the slit. As long as the wavelength is of the appropriate size, we obtain an observable single-slit diffraction pattern. The experiment can be rerun with photons, protons, pions, or indeed any sub-atomic particle and similar results are obtained. However, as a complement to the wave picture, the diffraction experiment can be understood as a consequence of the fundamental limit nature imposes on how well we can know the position of any particle x and its momentum p x .1 When the electron is within the slit, x is simply the slit width. It is impossible to say exactly where it is but it must be somewhere between the slit walls. The Heisenberg uncertainty principle states that if nature allows us to gain knowledge about the position of a particle we must lose information on its momentum, a consequence which is contrary 1 Heisenberg uncertainty principle relations can also be written for the position and momentum in the y and z directions although, for simplicity, only uncertainty in one dimension is considered throughout this article. 174 PHYSICS EDUCATION to our expectations from classical physics. However, therefore, there must be a spread of values of momentum in the x -direction p x , explaining why the electron paths are spread. We are allowed to know roughly by how much the momentum can change and we can see that some values (and hence electron paths) are more probable than others. For example, an electron is more likely to be found in the region around point A than point B in figure 1. However, we can never know in advance which value of momentum an individual electron will take. Thus, nature will never allow us to simultaneously know the position and momentum of a particle to arbitrary precision. This is a handwaving mathematics-free way of introducing the Heisenberg uncertainty principle. To write the uncertainty principle mathematically can be easily done using our electron experiment. We can see that the momentum spread occurs in the x -direction and is mostly restricted such that the electrons tend to fall near to the central peak (the maximum of the distribution). The study of diffraction tells us that the scattering angle of λ the first minimum can be approximated as θ ≈ x , where λ is the de Broglie wavelength of the incoming electron, given by λ = phy ; here p y is the incoming electron’s momentum. Taking the angle between the maximum and the first minimum as λ the angular uncertainty, we find that θ ≈ x . The angle θ is related to the momentum of the x electron as it leaves the slit: θ ≈ p , where py px is the magnitude of the scattered electron’s momentum in the x -direction, which is required if it were to come close to the first minimum. Thereλ x fore, we find that p ≈ x = p yhx . The unpy certainty in momentum and position can thus be written xp x ≈ h . This confirms our earlier assertion that nature will not let us simultaneously know the exact values of a particle’s position and momentum. To achieve high accuracy in the position we need to narrow the slit, but at a price of a greater uncertainty in the momentum. The Heisenberg uncertainty principle cannot, of course, be used to predict all the features of the diffraction pattern, for example, the locations and intensities of all of the maxima and minima. However, it can explain the fundamental observation that particles will deviate from their path, and it will also give an approximate estimate of the location of the first minimum. March 2008 Uncertainty in the classroom—teaching quantum physics A formal derivation of the Heisenberg uncertainty principle using the full machinery of quantum mechanics gives a more refined and general formula than that which we obtained for single-slit diffraction xpx h . 4π Applications of the Heisenberg uncertainty principle The Heisenberg uncertainty principle is valid for any particle in any situation, not only for a specific experiment such as single-slit diffraction of electrons. If we have any knowledge of the uncertainty in a position then the Heisenberg uncertainty principle provides information on the uncertainty in momentum and vice versa. For example, an electron bound within a hydrogen atom is known to be confined across a distance that is about the size of the atomic radius. The Heisenberg uncertainty principle allows us to estimate the minimum momentum of such an electron. The Heisenberg uncertainty principle can also elucidate the motion of particles at low temperature. It is popularly and wrongly supposed that at 0 K all motion stops and that even the electrons in atoms are somehow stationary. The Heisenberg uncertainty principle can easily show this statement to be fallacious. As the above example showed, if a particle is confined within a finite distance x then it must have a minimum spread in momentum p 4πhx , which cannot be zero. Hence, there must be some motion even at 0 K. The lowest possible energy of a particle is known as the zero-point energy, and according to quantum mechanics this cannot be zero. Does a particle possess a definite position and momentum? It is tempting to suppose that the Heisenberg uncertainty principle must be wrong, and that it must be possible to simultaneously know exactly the position and momentum of a particle. For 80 years physicists have attempted to design an experiment which would breach the Heisenberg uncertainty principle. Many ingenious gedanken experiments were dreamed up by the early quantum pioneers, including Einstein, Heisenberg, March 2008 and Bohr [7, 8]. Even assuming ‘perfect’ apparatus they found it was impossible to design an experiment which would measure the values of the position and momentum of a particle exactly. Since we can neither predict nor measure them, is it even worth worrying about whether or not a particle has definite values of its position and momentum? It was Heisenberg’s view that physics should only concern itself with measurable quantities. Einstein disagreed and thought that the failure of the Heisenberg uncertainty principle and quantum mechanics to predict definite values of position and momentum shows that quantum mechanics is somehow an incomplete theory. However, in the most common view of quantum mechanics, the so-called Copenhagen interpretation, a particle simply does not possess specific values of position and momentum, but the act of measurement forces it to take a value. This issue remains an open and hotly debated question in quantum mechanics. Niels Bohr preferred to call the Heisenberg uncertainty principle the indeterminacy principle, to stress that it is really impossible to determine the quantities to an arbitrary precision, independent of the experimental set-up. Another uncertainty principle: energy and time An experiment at a particle collider measures the mass of a Z particle. The text books report that the Z mass is measured to several decimal places: 91.1876 GeV/c2 . Hence, if the experiment measures the mass of a single Z boson and finds it to be 88 GeV/c2 , do we assume that there is something wrong with the detector? Actually, no. If we measure the masses of a large number of Z particles we see a broad distribution which peaks at 91.1876 GeV/c2 (as shown in figure 2) but which has a large population even at 88 and 94 GeV/c2 .2 This spread of mass values is due to the very short lifetime of the Z particle (of the order 10−25 s). A particle with such a short lifetime simply does not have a single, well-defined mass; as for position and momentum, there is a spread of possible values M one can obtain when making a measurement. We cannot predict in advance with which mass value an individual Z particle will be 2 The Hands on CERN education program [9] contains a large number of particle collisions from the LEP electron–positron collider, which can be used to further explore the Z0 particle. PHYSICS EDUCATION 175 K E Johansson and D Milstead A No. Z bosons A B C B ΔM time A B Figure 3. An interaction taking place via a particle exchange. 88 92 90 Z mass (GeV/c2) 94 Figure 2. Measurement of the Z-mass made at the LEP electron–positron collider. produced but, as figure 2 shows, some values are more likely than others. Einstein’s equation relates the energy of a particle to its mass via E = mc2 . We can use the Z mass experiment to demonstrate another form of the Heisenberg uncertainty principle: Et ≈ h . Here, E = mc2 is effectively the 4π uncertainty in the mass (around 2 GeV/c2 for the Z particle) and t is the particle’s mean lifetime and not an uncertainty in time, which is anyway a meaningless concept in non-relativistic physics. Quantum mechanics allows us to write a general expression for the energy–time uncertainty principle: Et 4hπ , which relates the minimum uncertainty in energy E of any particle system undergoing a change during a time interval t . This naı̈vely implies that nature can allow extremely large energy changes as long as E 4πht . In practice, as we saw with the measurements of the Z, nature prefers to keep energy shifts to a minimum (Et ≈ 4hπ ). We therefore use the minimum uncertainty relation to discuss (below) radioactivity and differences between the fundamental forces. Understanding fundamental forces with the Heisenberg uncertainty principle The energy–time relation can also help us understand forces between particles. The traditional picture, included in high school syllabuses, is that particles exchange another particle in order to transmit a force, as shown in figure 3, where particles A and B undergo 176 PHYSICS EDUCATION repulsion via the exchange of particle C. Such exchanged particles are examples of virtual particles, so called since they are never directly observed and only exist for an unmeasurably short time. However, our naı̈ve picture of particle exchange has several problems. One difficulty is that if particle A is at rest and emits the exchange particle then energy conservation can be violated: the mass of A < mass of B + mass of exchange particle C.3 This leads many authors to assert that nature allows energy E to be ‘borrowed’ (the mass of the exchanged particle) for a short time t whilst the interaction takes place. However, since we have already abandoned much of our familiar classical physics as we enter the sub-atomic world 3 Another flaw is that our picture can only be easily used to visualize repulsive forces: particle A recoils after emitting the exchange particle which moves to kick particle B further away. Attractive forces can, however, be accommodated by considering the exchange particle to be emitted from A in a direction away from B, thereby pushing A towards B. Furthermore, owing to the uncertainty in the position of the exchange particle (the Heisenberg uncertainty principle again), it may well ‘reappear’ to the right of B and push it towards A. March 2008 Uncertainty in the classroom—teaching quantum physics it is comforting to know that conservation of energy still holds. As we saw with the Z decay, particle states which exist for a short time are allowed to take various values of mass. The shortlived virtual particles can possess a range of mass values as long as energy is always conserved. We shall use the ‘borrowing of energy’ technique here when discussing the forces since it is simple and convenient. However, it should be remembered that it is nothing more than a shorthand way of hiding the mathematics involved with calculating the masses of the various virtual particles. The strong nuclear force is traditionally viewed as the exchange of pions between nucleons (protons and neutrons). A proton emits a pion which is later absorbed by a neutron. The energy ‘borrowed’ must be the pion rest mass energy E = m π c2 = 0.14 GeV, and this regulates how far the pion can travel: ct ≈ 4πhcE = force4 , which keeps the nucleus together. A typical energy of an α -particle is 4 MeV, which is far less than the energy required to overcome that force field and escape. Therefore, α -decay should not happen. However, we observe α -decay, and people die because of cancer caused by α -particles ejected from nuclei. How does this happen? The Heisenberg uncertainty principle allows an α -particle to occasionally ‘tunnel’ through the energy barrier and into the region in which only the electric repulsion force is relevant. Brushing under the carpet the complicated interactions of the α particle with the force field, it is possible to view the Heisenberg uncertainty principle as allowing the α -particle to ‘borrow’ sufficient energy E for the short time t needed to escape. The more energy needed to be borrowed, E ≈ 4πht , the shorter the time available for borrowing and the less likely it is to escape. 6.6×10−34 ×(3×108 ) 4×3.14×0.14×109×1.6×10−19 = 7 × 10−16 ≈ 10−15 m, which is consistent with inner nuclear distances. If nature had chosen the pion mass to be smaller we could be dealing with nuclei of far greater sizes! We can use similar reasoning to understand the range of the weak and electromagnetic forces. As mentioned earlier, the Z boson has a mass of 91 GeV, implying that it can be exchanged over tiny distances of ct ≈ 4πhcE = 6.6×10−34 ×(3×108) 4×3.14×91×109×1.602×10−19 ≈ 10−18 m. This shows the very short range of the weak force, which also explains why weak processes are so unlikely at low energies, where the ‘borrowed’ energy is so large. The electromagnetic force, unlike the weak force, is mediated by photons which are massless, and the range of this force is hence infinite. Barrier penetration In our classical world, a ball rolling along a surface is unable to overcome a barrier (see figure 4) if its kinetic energy at the foot of the barrier 12 mv 2 is less than the gravitational potential energy mgh it would possess at the top. This can also be regarded as the ball not possessing sufficient energy to pass through a gravitational force field. An analogous situation happens in a nucleus, but this time there is a different ending to the story. Approximately 1022 times per second, an α -particle, a bound state of two protons and two neutrons, bounces off the walls of a force field created by the strong nuclear March 2008 Consequences of the Heisenberg uncertainty principle A deterministic view of the world holds that if only we knew all of the laws of physics and could measure positions and properties of each particle then we could ultimately predict how the universe, and presumably human beings, would behave with time (assuming we had the necessary computing power!). The uncertainty principle conflicts with this, since it forbids us to know the exact values of the position and momentum of any particle. If we cannot know the present, then we cannot predict the future! Quantum mechanics gives us uncertainties and probabilities for various outcomes and nothing more. Thus, if quantum mechanics is the correct theory for the sub-atomic domain then we must accept that our model of nature is not a like wound clock but instead more like a game of dice. Although the game of dice is played out at small distances it could affect our macroscopic world as well. For example, as discussed earlier, the α -decay of a nucleus is permitted by the uncertainty principle, but we cannot predict when this will happen for any individual nucleus. Should 4 If it is not obvious why the α -particle should have any motion at all then recall the discussion on zero-point energy. An α particle is known to be localized within a finite distance (the size of a nucleus) and therefore must have non-zero momentum according to the position–momentum form of the Heisenberg uncertainty principle. PHYSICS EDUCATION 177 K E Johansson and D Milstead Table 1. Summary of physical phenomena which can be understood/calculated using the Heisenberg uncertainty principle. Physical phenomenon Single-slit diffraction Zero-point energy Exchange particles α -decay Explanation with the Heisenberg uncertainty principle xp 4hπ , a particle position is determined to an accuracy of the slit-width x = l , implying a spread in momentum xp 4hπ , x = 0 leading to non-zero energy Et 4hπ , short-lived particles can possess a range of mass values. Can also be viewed as nature ‘borrowing energy’ E for a time t to allow exchange processes to happen which would otherwise violate energy conservation. This also explains the range of various forces Et 4hπ an α -particle borrows energy for a short time to escape a force field a person be unfortunate enough to be in the vicinity of a radioactive source when it emits a burst of α -particles, then this may cause a fatal cancer to develop. The Heisenberg uncertainty principle in the classroom In the classroom the scattering of electrons not only demonstrates the wave nature of the electron, that is the wave–particle duality related to quantum mechanics, but also the Heisenberg uncertainty principle relating to position and momentum at work. Diffraction of light, easily accessible at school, can also be interpreted as a consequence of the Heisenberg uncertainty principle. The mass width of particles, particularly striking for the W and Z particles [9], is a manifestation of the Heisenberg uncertainty principle relating energy and time. The effect is present also in nuclear and atomic physics, but as the lifetime of the particle states are normally much longer, the effect is smaller. In addition, the Heisenberg uncertainty principle can help in understanding nuclear decays and barrier penetration and also particle decays and interactions as particle exchange mechanism. However, for a detailed description of particle interactions, the dynamics of the standard model of particles is needed. Discussion Many authors and popularizers of science assign different meanings to the terms x, p x , E, t . 178 PHYSICS EDUCATION Many texts prefer to introduce the position– momentum form of the Heisenberg uncertainty principle using a gedanken approach in which x is clearly an experimental resolution and p x a momentum ‘kick’ received by a particle. Others prefer more rigorous derivations in which x and px represent the dispersion of the particle wavefunction. We feel that the most suitable approach is simply to refer to x and p x as the limits nature will impose on our knowledge of position and momentum. We used the paradigm of the already familiar single-slit diffraction and stressed that there is no physical situation, for example an experiment or a bound atomic electron, for which the Heisenberg uncertainty principle is not valid. It is not necessary to introduce the concept of a particle wavefunction and the consequent discussion of superposition of states. Such an approach does not so easily provide the final expression of the Heisenberg uncertainty principle and is best left to a rigorous undergraduate course. However, it is important to be aware of the limitations of using the Heisenberg uncertainty principle blindly. As observed in one of the more erudite texts on particle physics, if a physicist uses the uncertainty principle to put forward an argument the listener should ‘keep his hands on his wallet’ [10]. The same argument which was used to explain the infinite range of the electromagnetic force fails when applied, for example, to explain why exchanges of gluons only take place within hadrons, as the role of the strong charge, the colour charge, also affects the interaction range. March 2008 Uncertainty in the classroom—teaching quantum physics Such occasional inconsistencies are to be expected owing to the impossibility of simplifying quantum field theory to such a basic level. Summary The Heisenberg uncertainty principle (or indeterminacy principle) is an important and thoughtprovoking part of quantum mechanics. It is also a very powerful tool; two short inequalities can be used to elucidate a number of physical phenomena, as summarized in table 1. Used with care, it enables approximate calculations to be made of otherwise intractable situations. Several of the consequences of the Heisenberg uncertainty principle can be demonstrated in the classroom, and it is also an excellent starting point for a discussion on the bizarre nature of the quantum world. Received 4 September 2007, in final form 24 October 2007 doi:10.1088/0031-9120/43/2/006 References [1] Ireson G 1999 A brief history of quantum phenomena Phys. Educ. 35 381 [2] Hadzidaki P, Kalkanis G and Stavrou D 1999 Quantum mechanics: a systemic component of the modern physics paradigm Phys. Educ. 35 386 [3] Popiesch G 1999 Uncertainty and complementarity: the heart of quantum physics Phys. Educ. 35 393 [4] Bergström L, Johansson K E and Nilsson Ch 2001 The physics of ‘Copenhagen’ for students and the general public Phys. Educ. 36 388 [5] Young H and Freedman R (eds) 2003 University Physics with Modern Physics (London: Addison-Wesley) March 2008 [6] Griffiths D 1994 Introduction to Quantum Mechanics (Englewood Cliffs, NJ: Prentice-Hall) [7] Cassidy D C 1992 Uncertainty, the Life and Science of Werner Heisenberg (New York: Freeman) [8] Pais A 1991 Niels Bohr’s Times in Physics Philosophy and Politics (Oxford: Oxford University Press) [9] Johansson K E 2006 Hands on CERN—a well-used physics education project Phys. Educ. 41 250–4 http://hands-on-cern.physto.se [10] Griffiths D 1987 Introduction to Elementary Particles (New York: Wiley) Erik Johansson is a Professor at Stockholm University and Director emeritus of Stockholm House of Science. He is the co-coordinator of the Education and Outreach project of the ATLAS experiment at CERN. He shared the first European particle physics outreach award in 2001, and received a Webby award in the science category in 2005 for the Hands on CERN particle physics education project. D Milstead is a lecturer at Stockholm University Physics Department and is, by trade, a particle physicist. He has done research at the HERA electron–proton collider and currently works at the ATLAS experiment at the LHC at CERN. In 2003 he won an international Science Communication prize given by the Association of Friends and Sponsors of the Deutsches Elektronen SYnchrotron Labor (DESY). This was for a television documentary (on BBC2) about a new type of particle, known as a magnetic monopole. PHYSICS EDUCATION 179