A Primer for Physicists
Aug. 9th, 2012 11:37 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
mtvesselJan 2012
The Quantum Universe: Everything That Can Happen Does Happen - Brian Cox and Jeff Forshaw - Allen Lane, 2011
* * * *
It is famously said of Stephen Hawking’s A Brief History of Time that many people bought it but almost no-one read it. I fear that this may also be true of this book, which is a shame because it is one of the most lucid and substantial explanations of quantum mechanics that I have come across. Given to the right young mind, it might have a similar life-transforming effect to the one that Steven Rose’s Chemistry of Life had on mine.
The book starts, as most texts on quantum mechanics do, with Young's Double Slit experiment, but the first few chapters focus mainly on Richard Feynman's Sum over Histories formulation of particle movement, which I have always liked for its sixties-style "anything goes" vibe, summed up in the subtitle of this book.
Tell you what - as a test of the authors' explicative powers, let's see if I as a non-physicist can describe it. So, space and time are quantised (albeit at a very fine scale) and thus the motion of particles can be modelled as a sequence of hops in a three-dimensional lattice. Let's consider a particle moving along the x axis. At time 0 it's at position A (let's define that as the origin - (0,0,0)) and some time later it's at position B (let's say that's 3,0,0). Well, that's very boring, you might think - at time t=0, the particle is at position (0,0,0), and at time t=1 it's at (1,0,0) and at time t=2 it's at (2,0,0) and so on. But think about the assumption you have made. What constrains the particle to hop to the next door spot with each time increment? Why can't it go to Alpha Centauri and back before ending up at position B?
The short answer in quantum mechanics is - nothing. There is no a priori reason that the particle can't get from A to B via Alpha Centauri, or Betelgeuse, or your left nostril, or any other of the infinite number of routes it could possibly take. If you are going to insist on it taking the straight line approach, or on restricting its paths by restricting its speed (to, say, the speed of light), you are imposing a priori constraints and that makes it a poor model. The fewer constraints that you need to impose to make your model work, the more elegant the model is (Occam's Razor).
So we have a particle at a particular location and time, and we are not going to impose any constraints on how far it can hop in the matrix in successive time periods. This means that after time zero, the particle could end up anywhere in the universe. Not a very realistic model, is it? To make it work, we need to introduce another constraint, and unfortunately it’s a complicated one. Suppose we don't know exactly where the particle is at time t=0 - let's say it could be at (0,0,0) or it could be at a neighbouring position on the x axis ((-1,0,0) or (1,0,0)). This uncertainty is modelled (for reasons that the professors don’t really explain) with a wave whose amplitude represents the probability of finding the particle at a particular location.
To visualise how this probability wave develops with time, Feynman introduced the representation of a particle as an old-fashioned stopwatch whose single hand is wound backwards by a certain amount related to the particle's mass, how far the particle has travelled, and (inversely) to the time*. Now here's the clever bit. To determine the probability that our slightly imprecise particle appears at a location X, place three "ghost" stopwatches at the three positions it could be at time=0, then add up all the amplitudes (the positions of the stopwatch hands) for all the possible paths that they could take to reach X at time t (the "sum over histories" bit). This is easier than it sounds because for all points beyond a certain distance from the origin, the wave amplitudes cancel each other out in an "orgy of interference" and hence the probability that the particle will be found there is essentially zero. Only when the stopwatch goes round less than one turn (because the distance to X is very short or the time is very long) do the contributions from the different possible paths not cancel out. So, if you wait a short time, the only places where you are likely to find the particle are close to its place of origin, but if you wait longer, the sphere of probable locations increases, just as we see in reality.
This wonderfully weird way to model reality - smooth motion appearing from lattice-hopping particles with wave interference - is astonishing, and the fact that it accurately describes observations is extraordinary. The professors do go into the various theories about what it tells us about reality, though they generally seem to be on the "shut up and calculate" side of the argument rather than being enthusiasts for the Copenhagen interpretation, with its dead-alive cats, or the many worlds hypothesis.
But the professors don't stop there. They next introduce wave packets and Fourier analysis and from that explain the structure of atoms as patterns of standing electron waves around a positive nucleus. This segues into a description of the Pauli exclusion principle (and I never realised that it applies not just to electrons in an atom, but to all electrons throughout the universe), semi-conductors, Quantum Electrodynamics, the Standard Model (and of course the (almost certainly) recently discovered Higgs boson), and finally a derivation of the Chandrasekhar Limit for the mass of white dwarf stars, which arises directly from quantum mechanics and for which only a single intriguing counter-example has been found.
Inevitably, with so much material to cover there is a certain amount of hand-waving, but nonetheless this is still a pleasingly intellectually dense read, leavened by the authors' lively and occasionally humorous style. They blithely ignore the supposed rule of publishing that every equation in a popular science book decreases its readership by half, and the basic algebra they employ is easy to follow and makes its point (most maths is straightforward when someone takes the time to explain what all the symbols mean and the reasoning behind the less obvious logical jumps). In fact, I would have liked the authors to have covered less ground in more detail, though this would probably have taken them dangerously close to student textbook territory. There is also a typo in the footnote on page 67 - a microgramme is a billionth of a kilo, not a millionth. But as an introduction to some fascinating ideas, this book is hard to better. Buy it for any aspiring physicists you may know.
* This is apparently related to an attribute of a dynamic system called the Action, as in the Principle of Least Action, which explains why thrown balls curve in the earth's gravitational field in the way that they do. As I recall, this wasn't mentioned in my A-level physics course.
The Quantum Universe: Everything That Can Happen Does Happen - Brian Cox and Jeff Forshaw - Allen Lane, 2011
* * * *
It is famously said of Stephen Hawking’s A Brief History of Time that many people bought it but almost no-one read it. I fear that this may also be true of this book, which is a shame because it is one of the most lucid and substantial explanations of quantum mechanics that I have come across. Given to the right young mind, it might have a similar life-transforming effect to the one that Steven Rose’s Chemistry of Life had on mine.
The book starts, as most texts on quantum mechanics do, with Young's Double Slit experiment, but the first few chapters focus mainly on Richard Feynman's Sum over Histories formulation of particle movement, which I have always liked for its sixties-style "anything goes" vibe, summed up in the subtitle of this book.
Tell you what - as a test of the authors' explicative powers, let's see if I as a non-physicist can describe it. So, space and time are quantised (albeit at a very fine scale) and thus the motion of particles can be modelled as a sequence of hops in a three-dimensional lattice. Let's consider a particle moving along the x axis. At time 0 it's at position A (let's define that as the origin - (0,0,0)) and some time later it's at position B (let's say that's 3,0,0). Well, that's very boring, you might think - at time t=0, the particle is at position (0,0,0), and at time t=1 it's at (1,0,0) and at time t=2 it's at (2,0,0) and so on. But think about the assumption you have made. What constrains the particle to hop to the next door spot with each time increment? Why can't it go to Alpha Centauri and back before ending up at position B?
The short answer in quantum mechanics is - nothing. There is no a priori reason that the particle can't get from A to B via Alpha Centauri, or Betelgeuse, or your left nostril, or any other of the infinite number of routes it could possibly take. If you are going to insist on it taking the straight line approach, or on restricting its paths by restricting its speed (to, say, the speed of light), you are imposing a priori constraints and that makes it a poor model. The fewer constraints that you need to impose to make your model work, the more elegant the model is (Occam's Razor).
So we have a particle at a particular location and time, and we are not going to impose any constraints on how far it can hop in the matrix in successive time periods. This means that after time zero, the particle could end up anywhere in the universe. Not a very realistic model, is it? To make it work, we need to introduce another constraint, and unfortunately it’s a complicated one. Suppose we don't know exactly where the particle is at time t=0 - let's say it could be at (0,0,0) or it could be at a neighbouring position on the x axis ((-1,0,0) or (1,0,0)). This uncertainty is modelled (for reasons that the professors don’t really explain) with a wave whose amplitude represents the probability of finding the particle at a particular location.
To visualise how this probability wave develops with time, Feynman introduced the representation of a particle as an old-fashioned stopwatch whose single hand is wound backwards by a certain amount related to the particle's mass, how far the particle has travelled, and (inversely) to the time*. Now here's the clever bit. To determine the probability that our slightly imprecise particle appears at a location X, place three "ghost" stopwatches at the three positions it could be at time=0, then add up all the amplitudes (the positions of the stopwatch hands) for all the possible paths that they could take to reach X at time t (the "sum over histories" bit). This is easier than it sounds because for all points beyond a certain distance from the origin, the wave amplitudes cancel each other out in an "orgy of interference" and hence the probability that the particle will be found there is essentially zero. Only when the stopwatch goes round less than one turn (because the distance to X is very short or the time is very long) do the contributions from the different possible paths not cancel out. So, if you wait a short time, the only places where you are likely to find the particle are close to its place of origin, but if you wait longer, the sphere of probable locations increases, just as we see in reality.
This wonderfully weird way to model reality - smooth motion appearing from lattice-hopping particles with wave interference - is astonishing, and the fact that it accurately describes observations is extraordinary. The professors do go into the various theories about what it tells us about reality, though they generally seem to be on the "shut up and calculate" side of the argument rather than being enthusiasts for the Copenhagen interpretation, with its dead-alive cats, or the many worlds hypothesis.
But the professors don't stop there. They next introduce wave packets and Fourier analysis and from that explain the structure of atoms as patterns of standing electron waves around a positive nucleus. This segues into a description of the Pauli exclusion principle (and I never realised that it applies not just to electrons in an atom, but to all electrons throughout the universe), semi-conductors, Quantum Electrodynamics, the Standard Model (and of course the (almost certainly) recently discovered Higgs boson), and finally a derivation of the Chandrasekhar Limit for the mass of white dwarf stars, which arises directly from quantum mechanics and for which only a single intriguing counter-example has been found.
Inevitably, with so much material to cover there is a certain amount of hand-waving, but nonetheless this is still a pleasingly intellectually dense read, leavened by the authors' lively and occasionally humorous style. They blithely ignore the supposed rule of publishing that every equation in a popular science book decreases its readership by half, and the basic algebra they employ is easy to follow and makes its point (most maths is straightforward when someone takes the time to explain what all the symbols mean and the reasoning behind the less obvious logical jumps). In fact, I would have liked the authors to have covered less ground in more detail, though this would probably have taken them dangerously close to student textbook territory. There is also a typo in the footnote on page 67 - a microgramme is a billionth of a kilo, not a millionth. But as an introduction to some fascinating ideas, this book is hard to better. Buy it for any aspiring physicists you may know.
* This is apparently related to an attribute of a dynamic system called the Action, as in the Principle of Least Action, which explains why thrown balls curve in the earth's gravitational field in the way that they do. As I recall, this wasn't mentioned in my A-level physics course.