[...]... review of some of the successes of the theory is given in the next section, with which we conclude this chapter The big question of what the quantum theoretical calculations actually mean is left to Chapter Three 2.5 Other applications of quantum theory In this section we shall outline some of the most important applications of quantum theory to various areas of physics, applications 34 Quantum theory... figure is the magnitude of the wavefunction The rate of oscillation of the real part is proportional to the velocity of the particle part p’s 1 I1 1I , This isthe magnitude -Thisisminus the magnitude Figure 8 A typical wave packet The broken curve indicates the magnitude of the wavefunction and the solid curve gives the ‘real part’ The rate of oscillation is proportional to the average velocity of the particle... 26 Quantum theory We shall see later that it is necessary to have a method of ‘adding’ wavefunctions The method we use can be understood by reference to figure 9 We wish to add the wavefunctions represented by the lines in figures 9(a) and (b).To do this we join the beginning of the first line to the end of the second; then the line joining the beginning of the second to the end of the first is the. .. probability for the particle to be at that point In fact, the relation between the wavefunction and the probability is very simple: the probability is proportional to the square of the magnitude of the wavefunction It does not depend in any way on the angle of the wavefunction The classical notion of a particle’s position is therefore related to the magnitude of the wavefunction What about the classical... more precisely the velocity in the direction of the line between the points 0 and P , only here the relation is the opposite way round: the narrower the peak, the larger the uncertainty In consequence, although there is no limit to the accuracy with which either the position or the velocity can be fixed, the price we have to pay for making one more definite is loss of information on the other This f a... the point labelled P The width of the distribution, shown in the figure as U,, gives some idea of the uncertainty in the true position of the particle There are precise methods of defining this uncertainty but these are not important for our purpose Clearly a very narrow peak corresponds to accurate knowledge of the position of the particle and, conversely, a wide peak to inaccurate knowledge 20 Quantum. .. think of two pebbles being dropped onto the surface of a still pond Ripples will spread out from the points of impact At some positions on the pond the ‘ups’ and the ‘downs’ from the two circular wave patterns will always come at the same time and the wave will therefore be enhanced At others they will be ‘out of phase’, i.e an ‘up’ from one will arrive at the same time as a ‘down’ from the other,... again giving the classical result Before we comment on the implications of these results, it is worth considering a more readily appreciated situation which is in some ways analogous On one of the jetties in the lake of Geneva there is a large fountain, the ‘Jet d’eau’ The water from this tends 10 Reality in the quantum world to fall onto the jetty, in amounts that vary with the direction of the wind On... 1l(c) These two peaks then move away from the barrier in opposite directions, so a little later Quantum theory 28 we have the situation shown in figure 1l(d) Our wavefunction has separated into two peaks, one reflected and one transmitted by the barrier It is a consequence of the Schrodinger equation that, throughout the motion, the total area under the graph of the square of the Pr bility Poslt'lon of. .. true, but they seem unlikely We mention them here to emphasise how completely the results we have discussed in this chapter violate our basic concept of reality, and also because they are, in their complexity, in stark contrast to the Summary 17 elegant simplicity of the quantum theoretical description of these experiments It is this description that forms the topic of the next chapter 1.5 Summary of Chapter . coin, or throw of a dice, or a horse race; in these cases they enter because of our lack of precise knowledge of the orginal state of the system, whereas in quantum theory, even if we. capable of thinking, the world appears to be silly. Indeed the recent upsurge of interest in the topic of this book has arisen from the results of recent experiments; results which, though they. as the continuation of the process, discussed above, whereby we explain the experiences of our senses in terms of the behaviour of external objects. We have learned how to observe the world,