5.4 The scanning tunnelling microscope The scanning tunnelling microscope (STM) is a device of such extraordinary sensitivity that it can reveal the distribution of individual atoms on the surface of a sample. It can also be used to manipulate atoms and even to promote chemical reactions between specific atoms. The first STM was developed in 1981 at the IBM Laboratories in Zurich by Gerd Binnig and Heinrich Rohrer. Their achievement was recognised by the award of the 1986 Nobel prize for physics. In an STM the sample
5.3 Stellar astrophysics If tunnelling out of nuclei is possible then so is tunnelling in! As a consequence it is possible to trigger nuclear reactions with protons of much lower energy than would be needed to climb over the full height of the Coulomb barrier. This was the principle used by J.D. Cockcroft and E.T.S. Walton in 1932 when they caused lithium-7 nuclei to split into pairs of alpha particles by bombarding them with high-energy protons. Their achievement won them the 1951 Nobel prize for physics. The same p
5.2 Alpha decay You have probably met the law of radioactive decay, which says that, given a sample of N0 similar nuclei at time t = 0, the number remaining at time t is N(t) = N0e−λt, where λ, the decay constant for a particular kind of nucleus, determines the rate at which the nuclei decay. The half-life is the time needed for half of any sufficiently large sample to decay. It is related to the de
5.1 Overview The discovery that quantum mechanics permits the tunnelling of particles was of great significance. It has deep implications for our understanding of the physical world and many practical applications, particularly in electronics and the developing field of nanotechnology. This section introduces some of these implications and applications. Applications naturally involve the three dimensions of the real world, and realistic potential energy functions are never perfectly square. Despite these
4.4 Stationary states and tunnelling in one dimension We will now use the stationary-state approach to analyse the tunnelling of particles of energy E0 through a finite square barrier of width L and height V0 when E0 < V0 (see Figure 21). 4.1 Overview One of the most surprising aspects of quantum physics is the ability of particles to pass through regions that they are classically forbidden from entering. This is the phenomenon of quantum-mechanical tunnelling that was mentioned in Session 1. In Session 4 we first demonstrate the phenomenon of tunnelling with the ai 3.6 Scattering in three dimensions Sophisticated methods have been developed to analyse scattering in three-dimensions. The complexity of these methods makes them unsuitable for inclusion in this unit but it is appropriate to say something about the basic quantities involved. In three dimensions, we are obliged to think in terms of scattering at a given angle, rather than in terms of one-dimensional reflection or transmission. We distinguish between the incident particles (some of which may be unaffected by the target) a 3.5 Scattering from finite square wells and barriers The procedure used to analyse scattering from a finite square step can also be applied to scattering from finite square wells or barriers, or indeed to any combination of finite square steps, wells and barriers. The general procedure is as follows: Divide the x-axis into the minimum possible number of regions of constant potential energy. Write down the general solution of the relevant time-independent Schrödinger equation in 3.4 Probability currents The expressions we have derived for reflection and transmission coefficients were based on the assumption that the intensity of a beam is the product of the speed of its particles and their linear number density. This assumption seems very natural from the viewpoint of classical physics, but we should always be wary about carrying over classical ideas into quantum physics. In this section we shall establish a general quantum-mechanical formula for the beam intensity. The formula will be consi 3.3 Scattering from a finite square step The kind of one-dimensional scattering target we shall be concerned with in this section is called a finite square step. It can be represented by the potential energy function 2.1 Overview Session 2 discusses the scattering of a particle using wave packets. We shall restrict attention to one dimension and suppose that the incident particle is initially free, described by a wave packet of the form This is a superposition of de Broglie waves, with the function Acknowledgements The content acknowledged below is Proprietary (see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence The author of this unit is Peter Sheldon. Grateful acknowledgement is made to the following sources for permission to reproduce material 4.4 Other Wenlock Limestone fossils Among the other fossils common in the Wenlock Limestone are brachiopods (Figure 12a and b), gastropods (Figure 12c) and bryozoans (Figure 12d). You may need to reread Section 1.3 to remind yourself about various aspects of these groups. Figure 13 (the unit image) is a reconstruction of a typical scene from a Wenlock Limestone environment. See 4.1 Trilobites As we've seen, the Cambrian explosion left the seas teeming with a huge variety of animals. In the following activity you will study some of the marine life at one particular time in the Palaeozoic Era – the middle part of the Silurian Period, 430 Ma ago. You'll look in detail at some fossils which come from a deposit in the UK called the Wenlock Limestone, famous for its many beautiful fossils. The Wenlock Limestone crops out mainly around Birmingham and the borders of Wales. Figure 4.5 What can individuals do? Whatever age they are, men, women and children can all do something to try to prevent future cardiovascular diseases in themselves or their families by eating a balanced diet (see Section 4.6), taking more exercise and modifying their lifestyles to reduce any other known risk factors. If cardiovascular diseases are pre-existi 2.10.1 Mean and standard deviation for repeated measurements In everyday terms, everybody is familiar with the word ‘average’, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation exemplified by Table 2, the sort to use is the mean
(or more strictly the ‘arithmetic mean’) For a set of measurements, this is de Introduction Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply ‘joining up the dots’ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of t 4.2 Least upper and greatest lower bounds We have seen that the set [0, 2) has no maximum element. However, [0, 2) has many upper bounds, for example, 2, 3, 3.5 and 157.1. Among all these upper bounds, the number 2 is the least upper bound because any number less than 2 is not an upper bound of [0, 2).
2.3 Inequalities involving modulus signs Now we consider inequalities involving the modulus of a real number. Recall that if a 1.1 Rational numbers The set of natural numbers is
, then its modulus, or abso
the set of integers is
and the set of rational numbers is
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