Before you read this, I suggest you read post 23.8. Let’s think about a wave or signal that looks like the picture above. It might be the result of an observation – for example, the electrical signal that results from recording a sound using a microphone. In this example we might want to know the… Continue reading 23.9 Discrete Fourier Transform

Let’s think about a wave, that we can represent by a wave function ψ(t) where t represents time, that has a time-period is T. An example is the time-dependent electrical potential, shown above (see also post 18.14) that has T = 0.25 ms. The Fourier transform of this one-dimensional function is defined by This is… Continue reading 23.8 Frequency analysis

Before you read this, I suggest you read post 21.8. We usually think that when individual animals can breed, to produce fertile offspring, that they belong to the same species. If they can’t – they belong to different species (see post 23.5). So where do individual species come from? The English biologist Charles Darwin (1809-1882)… Continue reading 23.7 Darwin’s theory of evolution

Before you read this, I suggest you read post 21.8. In my previous posts (21.11, 21.14 and 21.23), it may appear that genes are constant sources of information that never change. This idea explains most of what we know about why individuals have a different genotype and, therefore, a different phenotype to their parents. But… Continue reading 23.6 Mutation

Before you read this, I suggest you read post 16.18. I have written this post not because I know a lot about the subject but because I find some aspects of it interesting. “The naming of cats is a difficult matter.” T.S. Elliot (American/British poet, 1888-1965) Old Possum’s Book of Practical Cats. So let’s see… Continue reading 23.5 Biological classification – taxonomy

If you want to understand the background to this post, I suggest you read posts 22.12, 22.14, 22.15, 22.20, 22.21, 22.23, 22.24, 23.1 and 23.3 (route 1). But, for a less mathematical introduction, you can read posts 18.10, 18.16, 22.21 (but not the last paragraph) and 23.2 (route 2). Most of the statements that appear… Continue reading 23.4 X-ray crystallography and molecular structure

Before you read this, I suggest you read posts 22.14 and 22.21. In post 22.21, we saw that an ideal crystal is an arrangement of atoms (that may be bonded together in molecules) or ions in a regularly repeating three-dimensional pattern. And, in post 22.14, we saw that the x-ray diffracted by an object could… Continue reading 23.3 X-ray diffraction by a crystal

Before you read this, I suggest you read posts 22.4 and 22.21. Post 23.1 was about x-ray diffraction by a crystal. If you learnt about this topic at school or university, you were probably taught Bragg’s law. We don’t need to know about this law to understand x-ray diffraction by a crystal; it uses a… Continue reading 23.2 Bragg’s law

This article was first posted on January 9, 2013; it was revised on January 27, 2013. In this post we will think about diffraction of a beam of x-rays, with a single wavelength, λ, as in post 22.14. But here we will consider the special case where the scattering object is an ideal crystal. My… Continue reading 23.1 Observing x-ray diffraction by a crystal

Probability is an important idea in science. An experiment to answer a question like, for example, whether a drug is safe can only be answered with a probability (post 16.1). When we measure anything we only really know that its value lies between certain limits with a given probability (posts 16.7, 16.24, 16.26, 16.28). Probability… Continue reading 22.25 The Monty Hall problem