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Quantum Tunneling Applications Quantum Tunneling Applications Can the effects of quantum tunneling be seen on the macroscopic scale? Maxence Caron Philippe Chatigny Emile Demers Contemporary Physics 203-BNL-LW Gwendoline Simon Champlain St-Lawrence College April 15th 2016 Table of Content 1 Research Question ..................................................................................................................................... 2 2 Introduction to Quantum Tunneling .......................................................................................................... 2 2.1 The Wave-Particle Duality of Matter ................................................................................................. 2 2.2 Heisenberg's uncertainty principle ..................................................................................................... 3 2.3 Schrödinger’s Equation and 1-dimensional tunneling ........................................................................ 5 3 Applications .............................................................................................................................................. 7 3.1 Nuclear Physics .................................................................................................................................. 7 3.1.1 Alpha Decay ................................................................................................................................ 7 3.1.2 Stellar Nucleosynthesis ............................................................................................................. 10 3.2 Technology ....................................................................................................................................... 11 3.2.1 Scanning Tunneling Microscope ............................................................................................... 11 3.2.2 Tunnel Diodes ........................................................................................................................... 13 3.3 Biology ............................................................................................................................................. 15 3.3.1 Photosynthesis ........................................................................................................................... 15 3.3.2 Vibration Theory of Olfaction ................................................................................................... 17 4 Conclusion .............................................................................................................................................. 18 Bibliography ............................................................................................................................................... 19 1 1 Research Question In contemporary physics, there still exists a clash between the classical and quantum approach to the prediction of the behavior of the universe. In the macroscopic realm, classical physics seems to prevail as it can model our observable universe. However, at the atomic scale, it is quantum physics which appears to accurately describe the behavior of particles. If a process which is impossible according to classical physics can have repercussions in our everyday life, the importance of adapting the current model to incorporate such predictions would be proven. An example of such classically impossible occurrence is quantum tunneling. The question is whether or not the effects of quantum tunneling manifest themselves in the macroscopic scale. If they do, it would prove the need for a reconciliation between the two models. In order to answer this question, the process and applications of quantum tunneling must be investigated. 2 Introduction to Quantum Tunneling 2.1 The Wave-Particle Duality of Matter Fundamentally, every elementary particle and entity that can be found in its minimal amount (quantum state) exhibits both particle and wave properties. This was demonstrated in 1801 by Thomas Young in his “Double-slit experiment”, in which light is displayed as simultaneously behaving as a classical wave and particle. His experiment also displayed the fundamentally probabilistic nature of quantum mechanics (Encyclopaedia Britannica). However, this duality does not simply apply to quantum entities. Indeed, as proposed by Louis de Broglie, all matter constituted from these particles and entities also behave according to this wave-particle duality. The “de Broglie wavelength” of a macroscopic object can be calculated using the following formula, where λ is the wavelength of the object, p its momentum and h the Planck constant (Cohen-Tannoudji et al.). 2 Equation 1 ℎ � = � Thus, for a macroscopic particle, the mass is so large that the momentum is always sufficiently large to make the de Broglie wavelength smaller than the range of experimental detection (R. Eisberg and R. Resnick). This explains why, in the domain of the macroscopic, the intuitive classical mechanics seem to describe the universe accurately. There are, of course, exceptions to this rule as quantum mechanics can indeed interfere with the larger scale, such as in the quantum tunnelling effect (Stanford Encyclopedia of Philosophy). 2.2 Heisenberg's uncertainty principle In the realm of subatomic particles, the size of the de Broglie wavelength is comparatively large since the particle’s momentum is very small. Because of the small size and momentum of these particles, they more blatantly exhibit another property of quantum mechanics which is Heisenberg's uncertainty principle. This principle, introduced in 1927 by physicist Werner Heisenberg, states that for any particle in the quantum realm, both its position and momentum cannot be known simultaneously (D., Sen). More formally, this relation can be described by the following inequality where �& is the standard deviation of the position, �'is the standard deviation of the momentum and ħ is the reduced Planck constant (h/2π). Equation 2 ħ � � ≥ & ' 2 3 This inequality illustrates a fundamental principle of quantum mechanics. There is minimum uncertainty in nature that makes it impossible to know the exact position and momentum of a particle. Heisenberg’s uncertainty can be illustrated by Heisenberg’s famous thought experiment called “Heisenberg’s microscope”. Imagine you want to know the position and the momentum of a certain quantum particle using a microscope. In order to precisely know the position of this quantum particle a photon must be used. If a photon hits and is then reflected by a quantum object onto the microscope, the position of the object can be known. The precision of this position would then depend on the wavelength of the oscillating photon. For instance, if we take as our quantum particle an electron, we can then use as our medium photon a gamma ray since its wavelength is smaller than the size of the photon. Furthermore, we know, from Einstein, the following equation, where E is the energy of the photon, h is Planck’s constant, c the speed of light and � the wavelength: Equation 3 ℎ� � = � Thus, if our medium photon has a very small wavelength, which is required in order to have an uncertainty smaller than the size of the observed particle, it also must have a very large energy. Because this additional energy must be converted into kinetic energy, the medium photon will significantly affect the momentum of the observed particle. This will result in a larger uncertainty for the momentum of the quantum particle. Hence, Heisenberg’s uncertainty is coherent with de Broglie’s wavelength since the instruments used to measure the position and the momentum of a macroscopic object are not precise enough for their uncertainty to be observed. 4 The uncertainty of the instruments, in this case, would be greater than the standard deviation of both the position and the momentum of the object. 2.3 Schrödinger’s Equation and 1-dimensional tunneling Schrodinger’s equation is quantum physics’ analogue to Newton’s second law of motion as they both are used to observe the effect of time on a system. Where Schrodinger’s differs is that, as previously illustrated, quantum physics can only predict via possibilities, that is to say, only to probabilistically predict the effect of time on a system. Indeed, as illustrated by Heisenberg’s uncertainty, there is a fundamental uncertainty attributed to the momentum of a quantum object before it is observed. Momentum is only one of the possible quantum state vectors. Thus, Schrodinger’s equation must describe many potential outcomes in the form of wave functions derived from a single quantum state (Schrödinger, E). Schrodinger’s equations can be used to demonstrate the effect of 1-dimensional quantum tunneling through a potential barrier, as shown in Graph 1. Graph 1 (Graph 1 was drawn using paint.net) 5 Indeed, from a classical standpoint, to overcome the potential barrier V0 of length a, a particle must have enough kinetic energy, E. This means that if � < �/, the particle will not be able to cross the potential barrier. On the other hand, if � > �/, the particle will assuredly cross the barrier. Yet quantum mechanics portray the situation quite differently. From Schrodinger’s equation, we can obtain the equation transmission probability, T, which represents the probability that the particle will cross a certain potential barrier. This relation is represented by Equation 4, where V0 is the potential barrier, E the energy of the particle, m the mass of the object, a the length of the barrier and ħ the reduced plank’s constant (Sharma,
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