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
Learning outcomes By the end of this unit you should be able to: explain the meanings of the emboldened terms and use them appropriately; describe the behaviour of wave packets when they encounter potential energy steps, barriers and wells; describe how stationary-state solutions of the Schrödinger equation can be used to analyse scattering and tunnelling; for a range of simple potential energy functions, obtain the solution of the time-independent Sc
6.1 A difficult evolutionary transition Before going any further, click on 'View document' below and read pages 78–79 and 82–83 from Douglas Palmer's Atlas of the Prehistoric World. 4.2 Crinoids Figure 7 shows the fossilised remains of a type of echinoderm called a crinoid (‘cry-noyed’). Although crinoids occur today, they were far more common in the Palaeozoic and Mesozoic Eras. Most crinoids feed by bending their umbrella-like arrangement of flexible appendages (called ‘arms’) downstream so as to catch a current, rather as in an umbrella being caught in the wind. Tube feet (multipurpose tentacles) on the arms gather food particles suspended in the water, which are th 4.3 Education, education, education We have already listed some of the main modifiable risk factors, such as smoking and excessive alcohol consumption. Question: Try to recall two ot 1.3.1 Who is affected by cardiovascular diseases? Cardiovascular diseases are the main cause of premature death (before the age of 75) in the UK, across Europe and the USA – indeed, across many parts of the world (Figures 3 and Author(s): 2.5 Expressing probability According to Equation 1, probability is defined as a fraction. However a fraction such as  Complex numbers Numbers Interpreting data: Boxplots and tables Surfaces Introduction In the OpenLearn unit on Developing modelling skills (MSXR209_3), the idea of revising a model was introduced. In this unit you will be taken through the whole modelling process in detail, from creating a first simple model, through evaluating it, to the subsequent revision of the model by changing one of the assumptions. The new aspect here is the emphasis on a revised model, which comes in Section 2. The problem that will be examined is one based on heat transfer. This unit, the four Acknowledgements The content acknowledged below is Proprietary (see terms and conditions) and is available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence All materials included in this unit are derived from content originated at the Open University. Acknowledgements All written material contained within this unit originated at the Open University. The audio extract is taken from M208 © Copyright 2006 The Open University. http://www.flickr.com/photos/re_birf/69485963/ [Details correct as of 9th June 2008] 5.3 Powers Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent). If a > 0, m 5.2 Existence of roots Just as we usually take for granted the basic arithmetical operations with real numbers, so we usually assume that, given any positive real number a, there is a unique positive real number b = 3.4 Further exercises Use the Triangle Inequality to prove that
3.3.1 Post-audio exercises To practise using the techniques described in the audio, we suggest that you now try the following exercises. Use the Binomial Theorem to prove that
3.3 Worked examples The audio provided below illustrates various methods for proving inequalities. In addition to the techniques already described for proving inequalities, we use mathematical induction and the Binomial Theorem, restated below. If x 3.2 Inequalities involving integers In analysis we often need to prove inequalities involving an integer n. It is a common convention in mathematics that the symbol n is used to denote an integer (frequently a natural number). It is often possible to deal with inequalities involving n using the rules of rearrangement given in Section 2. Here
SAQ 1.2
may also be expressed as a decimal number or as a percentage:
You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.Author(s):
This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound Property. First published on Wed, 2
This unit is concerned with two main topics. In Section 1, you will learn about another kind of graphical display, the boxplot. A boxplot is a fairly simple graphic, which displays certain summary statistics of a set of data. Boxplots are particularly useful for assessing quickly the location, dispersion, and symmetry or skewness of a set of data, and for making comparisons of these features in two or more data sets. Boxplots can also be useful for drawing attention to possible outliers in a dat
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the classification theorem of compact surfaces. First published on Thu, 18 Aug 2011 as <
Audio Materials
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Definition
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such that b2Â =Â a. We now discuss the justification
Exercise 21
Exercise 18
Theorem 3.1 Binomial Theorem
