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The Uncertainty Principle, Virtual Particles and Real Forces

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The Uncertainty Principle, Virtual Particles and Real Forces SPECIAL FEATURE: TEACHING QUANTUM PHYSICS www.iop.org/journals/physed The uncertainty principle, virtual particles and real forces Goronwy Tudor Jones School of Education and School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Abstract This article provides a simple practical introduction to wave–particle duality, including the energy–time version of the Heisenberg Uncertainty Principle. It has been successful in leading students to an intuitive appreciation of virtual particles and the role they play in describing the way ordinary particles, like electrons and protons, exert forces on each other. This resource article is based on experience of Classical physics input teaching this topic at an introductory level to a • Kinetic energy and momentum. It will variety of audiences. Several approaches have be assumed in this paper that the concepts been tried; this has been the most successful of kinetic energy E and momentum p have because the crucial argument—that the better the been discussed. (It takes typically a two- frequency of a wave is to be measured, the more hour session to get non-scientists to feel time is needed—is invariably produced by class comfortable with these ideas.) members, and seems intuitively reasonable to For the more mathematically inclined, it is them. worth pointing out that the three equations The material is divided into several sections: (1) Comments on preliminary ideas from classi- p2 E = 1 mv2 p = mv E = cal physics. 2 2m (2) A practical introduction to E = hf and p = h/λ (wave–particle duality). show that if we know any two of the quantities (3) The energy–time uncertainty relation as a E, p and v (speed) for an object, we can consequence of (2). calculate its mass m. (In passing, the same (4) An argument to motivate considering the is also true for relativistic particles [1].) possibility of processes that violate energy • Light is a wave. It is also assumed that conservation, based on the energy–time Young’s two-slit experiment demonstrating uncertainty relation (the ‘magic loophole’) that light is a wave has been performed. (5) Using this loophole to speculate about such Non-scientists are generally delighted to be processes, eventually arriving at the modern able to measure something as small as the way of describing a force between two real wavelength of light (less than one thousandth particles in terms of the passage of virtual of a millimetre) by measuring just two particles from one to the other. quantities—the distance from a He–Ne laser (6) Emphasizing the power of what has been done to a screen and the separation of two regions by applying this picture to some real forces. of high intensity—and then using the method (7) Drawing attention to the dangers of taking an of similar triangles! (To get here from scratch oversimplified approach too literally. with non-scientists takes 1–2 hours.) 0031-9120/02/030223+11$30.00 © 2002 IOP Publishing Ltd P HYSICS E DUCATION 37 (3) 223 G T Jones Wave–particle duality: The equation p = h/λ is known as the a practical approach de Broglie relation after the Frenchman who Assuming that light is a wave and that electrons are predicted wave–particle duality. particles, one needs two experiments to establish wave–particle duality. Two of the simplest are the An aside: Why do particle physicists need following: high-energy accelerators? (1) The photoelectric effect. When light Particle physicists are microscopists—they study of frequency f is shone onto a metal surface, the structure of neutrons and protons. It is a electrons can be knocked out. Increasing the rule of microscopy that you cannot see anything intensity of the light does not affect the energy smaller than the wavelength of the radiation you with which individual electrons emerge, a fact are shining on it; one way to come to terms with that cannot be described in terms of classical this idea is by means of the following analogy: physics, which would predict electrons of greater you cannot sense the details of Braille with a energy. Einstein was awarded his Nobel prize blackboard eraser, but you can with the point of for his contributions to mathematical physics, and a pencil. especially for his discovery of the law of the To get the very small wavelengths needed photoelectric effect. to study the interiors of neutrons and protons, Einstein’s picture is the following: the light particle physicists use particle beams of very high striking the metal surface should be thought of momentum (λ = h/p). So, a particle physics photons, as a stream of particles called each laboratory like CERN is a huge microscope! having a kinetic energy E = hf , where h is a constant of nature known as Planck’s constant. We now have the two basic formulae of quantum The more intense the light, the greater the number mechanics: E = hf and p = h/λ, relating the of photons striking the surface per second and the particle properties E and p to the wave properties number of electrons ejected. The energy with f and λ. which they are ejected does not vary with intensity The waves concerned are, however, mysteri- because photons of a given colour have the same ous ones: they determine the relative probabilities energy whether they are in a weak beam or an of where an electron (that has gone through a two- intense one. (The chance of more than one photon slit experiment in our case) will be found. We do striking the same electron is ignored.) not understand these waves in the way we under- (2) The two-slit experiment with electrons. stand other waves such as sound waves, and it is In 1974, Jonsson¨ [2], overcoming considerable amazing to remember this when one thinks of the technical difficulties, succeeded in firing a beam of power that quantum mechanics has given us. A electrons at two slits and observing the interference good discussion of the mystery of quantum me- pattern. This experiment is clearly described with chanics is to be found in The Quantum: Illusion many helpful illustrations in an excellent book or Reality by A Rae [5]. called The Quantum Universe by Hey and Walters Here we take a different approach and try to [3]. come to terms with these quantum waves (without From the separation of the maxima of the mathematics1). interference pattern, the wavelength λ of the The approach to be taken in the next section electron wave can be determined by Young’s to show how the Heisenberg Uncertainty Principle method. By varying the momentum of the electron 1 By without mathematics we mean for those who claim not beam and measuring the corresponding wave- to be able to do mathematics, typically graduates in non- length, one can show that pλ = h; Planck’s con- scientific disciplines who, although they may have distant school memories of mathematics, seem to have lost confidence stant again! This relationship tells us that the in their abilities. Such people are very often capable of higher the momentum p of a particle, the smaller following a mathematical argument, provided every step is is its de Broglie wavelength. performed in detail, and appreciate the experience. People are very happy to attend classes on the appreciation of music Perhaps it is also worth pointing out that here or poetry without expecting to be able to play an instrument or is an experiment that is simple to describe and write a poem afterwards. Why should the same not be true of gives a method of determining Planck’s constant. mathematics, which shares aesthetic qualities with both?! 224 P HYSICS E DUCATION May 2002 The uncertainty principle, virtual particles and real forces Worked example for students Problem. Derive an expression for the momentum p of an electron of mass m and charge e that has been accelerated through a voltage V . In his experiment, Jonsson¨ accelerated electrons through 50 kV. What de Broglie wavelength would these electrons have? (Ignore relativistic effects that are beginning to become significant at such voltages; see [4].) Solution. A stone of mass m dropped from a height h2 to a height h1 loses a potential energy of m × (the gravitational potential difference) = m × (gh2 − gh1). Likewise, an electron of charge e dropped through a potential difference of V volts loses a potential energy e × V . Here we come across a little problem with units: an electron accelerated through 1 volt acquires an energy of 1 electron-volt (eV) and 1 eV = 1.602 × 10−19 J. The potential energy lost appears as kinetic energy gained. Thus: p2 √ = eV ⇒ p = 2meV . 2m The de Broglie wavelength is then given by h λ = √ . 2meV The relevant numbers: h = 6.63 × 10−34 Js,e = 1.602 × 10−19 C, m = 9.11 × 10−31 kg. Using these and the conversion factor from electron-volts to joules: 6.63 × 10−34 λ = . 2 × 9.11 × 10−31 × (50 × 103) × 1.602 × 10−19 These give λ = 5.49 × 10−12 m, which is much smaller than the diameter of an atom, about ∼ 10−10 m. is a consequence of wave–particle duality is the Now, confined waves have definite frequencies, so following: although we do not understand we can say that an electron in a hydrogen atom has these quantum waves as we would like, we will definite frequencies: f1, f2, f3,... etc. It is no assume that, whatever properties other waves more than a convention (based on the knowledge share, the quantum waves have them too. from E = hf that h times a frequency has units of energy) to say that the electron (or electron wave An aside in the H-atom) has a set of allowed values of hf : hf , hf , hf ,..
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