4.3 Stationary states and barrier penetration The example of tunnelling we have just been examining can be regarded as a special case of scattering; it just happens to have E0 < V0. As long as we keep this energy range in mind, we can apply the same stationary-state methods to the study of tunnelling that we used earlier when studying scattering. As before, we shall start by considering the finite square step, whose potential energy function was defined in Equations
4.2 Wave packets and tunnelling in one dimension
Figure 18 shows a sequence of images captured from a wave packet simulation program. The sequence involves a Gaussian wave packet, with energy expectation value 〈E〉 = E0, incident from the left on a finite square barrier of height V0. The sequence is broadly simil
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 11 Additional resources Bandolier (2005) Statins: when should you take the tablet? British Red Cross (2007) First aid guidelines in the UK Cardiac Risk in the Young (2003) When a young person dies suddenly Clay, R. A. (2001) Research to the heart of the matter Department of Health (2000) National Service Framework for coronary heart disease, Chapter 4 Department of Health (2007) The coronary heart disease National Service Framework: shaping the future: progress report 2006 The Nat 9 Summary Now you will be very familiar with cardiovascular diseases, their development and their diagnosis. You will also know their treatment and many of the cardiovascular disease risk factors – what they are and how they can be influenced positively to minimise cardiovascular diseases. You will understand the overall importance of a balanced diet, regular exercise and weight management (guided by adiposity measurements) throughout life, to maintain cardiac and vascular health. You will also be a 3.2.1 Fats Fats, also known as lipids, are important components of living tissues, and are used by the body for making cell membranes and for storing energy. Fats come in a variety of different biochemical types, which may be obtained from the diet or can be synthesised within the body. Many cells of the body can convert certain types of fat into others, but by preference, fats will be obtained from the diet, if available. The fatty acids that cannot be synthesised by the body and therefore must 1 Developing modelling skills The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Click 'View document' to open the workbook (PDF, 0.2 MB). Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is used under licence. All materials included in this unit are derived from content originated at the Open University. 5.1 Arithmetic with real numbers At the end of Section 1, we discussed the decimals
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).
3 Proving inequalities In this section we show you how to prove inequalities of various types. We use the rules for rearranging inequalities given in Section 2, and also other rules which enable us to deduce ‘new inequalities from old’. We met the first such rule in Author(s): 10 Conclusion This unit has introduced you to some aspects of using a scientific or graphics calculator. However, in many ways, it has only scratched the surface. Hopefully your calculator will be your friend throughout your study of mathematics and beyond. Like any friend, you will get to know it better and appreciate its advantages as you become more familiar with it. Don't expect to know everything at the beginning. You may find the instruction booklet, or other help facility, a bit hard going to begin 9 When to use the calculator Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase ‘garbage in, garbage out’ applies just as much to calculators as to computers. Your calculator is just that – a calculator!< Ease of use Most aspects of the calculator are straightforward to use. Calculations are entered on the screen in the same order as you would write them down. More complicated mathematical functions and features are also reasonably intuitive, and there are ‘escape’ mechanisms, so that you can explore without worrying about how you will get back to where you were. 8.2.2 The screen You can see the calculations that you have entered as well as the answers. This means you can easily check whether you have made any mistakes. 7.2 Square roots Earlier you met the square function and on most calculators the square root is the second function on the same key. Look to see if this is the case for your calculator and check the calculator handbook on how to use this function. In many cases you will need to press the square root key before the number, instead of afterwards, as for the square key. This is the case on the TI-84. Check that you can find the square root of 25 and of 0.49 (you should get 5 and .7 respectively). Now find
and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least Upper Bound Property of Author(s):
