'The Physics of Quantum Mechanics' - Chapter 1 - Problems

I've now worked my way through the problems at the end of Chapter 1 of James Binney and David Skinner's 'The Physics of Quantum Mechanics'. And I found them to be manageable. Much more so than some of the problems in later chapters, by the looks of it. The ones in Chapter 1 are no doubt intended as gentle warm-up exercises.

Useful warm-up exercises. If nothing else, they forced me to remind myself of some complex number calculation rules.

The last problem, about an infinite series of potential wells, was the most interesting one. The first task is to find the probability of an electron being in the nth well. A quantum state vector for the electron's location is specified, with probability amplitudes provided as a function of n. So it's pretty straightforward to calculate the probability of the electron being in the nth well as a function of n.

The next task is to determine the probability of finding the electron in well 0 or anywhere to the right of it. I duly calculated the sum of all those probabilities from well 0 to infinity after looking up the appropriate summation formula on Wikipedia.

Only to realise that I could have spared myself the trouble: I could have found the answer more easily by employing a symmetry argument, since the probabilities are arranged symmetrically around 0 and must add up to 1.

But I don't regret my detour. It's given me a bit more practice in infinite series calculations. And that's of course what these problems are all about: practice, practice, practice.

Indeed, I find it very difficult to just read a physics textbook and understand it all without getting some practice in manipulating the mysterious mathematical symbols those texts tend to be peppered with. While I fear that at least some practice is necessary to achieve understanding in physics, I don't think it's sufficient.

Is the potential well probability amplitude distribution given in the problem discussed above realistic? If so, where do those probabilities come from? How and why do electrons move from one well to another? Is it meaningful to say that an electron is in a particular well before we've measured its location? Is it ever meaningful to say that an electron is in a particular well? I'm sure it's possible to be rather good at problem solving without understanding very well what the problem is about.

To adapt a quote I read somewhere many years ago: Physicists often expect to be regarded as profound thinkers even though you can find the most simple-minded people among them, unable to do anything that requires them to think unless it can be achieved by combining symbols in a way which is more a matter of practice than of deep thought.

Understanding 'The Physics of Quantum Mechanics' - Chapter 1

Working my way through the first chapter of 'The Physics of Quantum Mechanics' by James Binney and David Skinner (2015) has left me very pleased with my choice of textbook. The authors introduce new concepts gradually and gently and illustrate them well with examples of relevant physical processes.

Their text also provides a lot of food for thought, including a salutary reminder, an unfortunate misrepresentation and an intriguing suggestion. Let me look at these three highlights one by one.

A salutary reminder

The authors are clear about the limits to our understanding of the physical world when they state bluntly that "we can offer no justification" for the appearance of probability amplitudes in quantum mechanics "beyond the indisputable fact that they work" (p. 11).

Here's a salutary reminder that understanding in physics, as in other branches of knowledge, can only be pushed so far. Ultimately we have to content ourselves with describing what the world is like, rather than being able to say why it is like that.

It's useful to make that description as simple as possible, however. The concepts of cause and effect allow us to make connections between seemingly disparate phenomena, such as temperature and motion. But there comes a point when we have to say: this is a fundamental fact which we cannot explain any further, it's just what the world is like.

According to Binney and Skinner, the fact that probability amplitudes in quantum mechanics allow us to make correct predictions is just such a fact.

An unfortunate misrepresentation

Sadly, the authors go on to misrepresent standard probability theory just as they try to explain what's special about quantum mechanics. Here's what they say (p. 23):

"What's astonishing about atomic-scale physics is the way we calculate probabilities: we work with complex amplitudes. When an outcome can occur in two ways, we add the amplitudes associated with each way, not the probabilities. This rule yields an equation for the final probabilities that violates a basic principle of standard probability theory: the probability that one of two mutually exclusive events occurs is not the sum of the probabilities of the individual events, but this sum plus an oscillating quantum interference term. Everything that is strange and counter-intuitive about quantum mechanics flows from this violation of classical probability theory."

The problem with this passage is that in quantum mechanics there is no violation of the "basic principle of standard probability theory" cited by the authors. Quantum mechanics adheres to it just as much as any other branch of knowledge. And it couldn't be otherwise because the principle cited by the authors is part of the very concept of probability. In other words, if P(S or T) for mutually exclusive events S and T is not equal to P(S) + P(T), then P is not a probability distribution in the first place.

It's instructive to take a closer look at exactly where the authors go wrong. They develop their argument with reference to a quantum physical situation in which "something can happen by two (mutually exclusive) routes, S or T", such as an electron hitting a photographic plate after going through one slit or the other in a double-slit experiment. They claim that, in that case, P(S or T) equals P(S) plus P(T) plus a "quantum interference" term, which "has no counterpart in standard probability theory" and which "violates the fundamental rule" of probability theory, namely that P(S or T) = P(S) + P(T) for mutually exclusive events (pp. 11-12).

The problem with this argument is that the authors work with different sets of events, and therefore different probability distributions, in one and the same equation. On the one hand, we have the event of an electron hitting the photographic plate in an area A after going through slit S or T, while both S and T are open. (For the sake of the argument, I will accept here that every electron goes through one or the other of the slits, as the authors suggest.) The authors denote this event (S or T), so the probability of such an event occurring is P(S or T).

On the other hand, we have the event of an electron hitting the photographic plate in A after going through slit S, while just slit S is open. The authors denote this event S, so the probability of such an event occurring is P(S). They define the event T and the probability P(T) likewise. Now we have indeed that P(S or T) is not equal to P(S) plus P(T) - but that's only because the events on either side of the equation belong to different sets of events: on the left-hand side, both slits are open, on the right-hand side, only one of the slits is open. The probability distributions are therefore different, too, and we should really be saying that P1(S or T) is not equal to P2(S) plus P3(T). The "fundamental rule" of probability is thus not violated in this case, it simply doesn't apply.

Another way of saying this is that the authors are referencing different chance experiments on either side of the equation: in one case the experiment is carried out with both slits open, in the other it is carried out with just one or the other slit open.

If we take care to perform the same chance experiment with one and the same set of possible events, then the "fundamental rule" of probability theory fully applies. In that case, on the left-hand side of the equation we have again the probability of an electron hitting the photographic plate in an area A after going through slit S or T while both S and T are open. On the right-hand side we have the probabilities of an electron hitting A after going through S while both S and T are open, and of it hitting A after going through T while both S and T are open (all this again on the assumption that every electron goes through either S or T). In this situation, the sets of events and the probability distributions on either side of the equation are the same and we have P(S or T) = P(S) + P(T).

The authors' misrepresentation of probability theory is unfortunate because it's liable to confuse readers and to set them on the wrong track. Perhaps what the authors really meant to say is this: what's astonishing about atomic-scale physics is that what happens at this scale seems to be incompatible with the notion that particles such as electrons are point-like objects that travel in straight lines. And that it's not easy to conceptualize what they are instead if we want to make sense of experiments such as the double-slit experiment.

It should perhaps be noted that, if we drop the assumption that electrons are point-like particles that go through either S or T in the double-slit experiment, we can still apply the "fundamental rule" of probability theory on a properly defined set of events. We can, for example, regard the darkening of the photographic plate in an area Ai as an elementary event. The probability of an electron hitting one of two non-overlapping areas Ai and Aj is then simply the sum of the probabilities of the electron hitting Ai and of it hitting Aj - fully in line with the rule declared by the authors to be invalid in quantum mechanics.

An intriguing suggestion

According to the authors, it may well be that "the probabilistic nature of the outcome" of a system collapsing into a new state as a result of a measurement "is due to our incomplete knowledge of the state of the measuring apparatus that causes the collapse". "This conjecture appears likely, but remains unproven," they say.

I'd always understood that, according to the dominant interpretation, probabilities in quantum mechanics are fundamental and not the result of incomplete knowledge of any kind. Is this an outmoded view? Are the authors challenging it? To be watched in subsequent chapters.

Coming up next: my solutions to the problems at the end of chapter 1.

A new chapter in my inquiries

Today marks a new chapter in my inquiries into special relativity as I think I've reached the limits of what I can learn by just consulting writings on that theory itself and on classical electromagnetism.

 

The journey so far has certainly been interesting. I've discovered that

 

• the constancy of the one-way speed of light is the result of a clock adjustment procedure (Newton's or, equivalently, Einstein's) which is not a synchronization procedure in every inertial frame of reference;
• some authors have drawn spectacular but misguided conclusions on causality and simultaneous existence on the basis of the erroneous belief that this clock adjustment procedure is a synchronization procedure in every such frame of reference;
• length contraction and time dilation by the relativistic factors in a frame of reference Σ in which light propagates in symmetrical conditions in all directions and clocks have been Einstein-adjusted (or Newton-adjusted) is sufficient to derive all of special relativity (including the constancy of the two-way speed of light in all inertial frames of reference);
• time dilation by the relativistic factor in such a frame (at least as measured by light clocks) is a consequence of length contraction by the relativistic factor combined with the no-overtaking rule for electromagnetic signals.

 

All this has left me with an unresolved question: is there a model of electricity in which the no-overtaking rule for electromagnetic signals and (electric) length contraction by the relativistic factor in Σ can be explained simply and plausibly?

 

I feel I've made a promising start with my sphere model of electricity, which I have applied to two charges moving at one and the same constant velocity through Σ. But a suitably extended version of that model should also be able to explain the electric interactions between charges moving at different constant velocities through Σ. It should be able to do so without using the concept of magnetism and solely in terms of time and space coordinates in Σ as these coordinates, together with the size of the charges, provide a complete description of the situation.

 

Offline I've explored a number of avenues to extend the sphere model, which all revolve around the notion of a continuous exchange of information between two moving charges. Some of those avenues look promising. But I feel that, if I want to make further progress, I need to find out a bit more about how the exchange of information between moving charges is treated in the physics literature.

 

I will make a start by working through a textbook on quantum mechanics followed by another one on quantum electrodynamics. My choice of book for quantum mechanics is The Physics of Quantum Mechanics by James Binney and David Skinner. I intend to work through it methodically, page by page, chapter by chapter, and exercise by exercise. This will take some time. I'll keep you updated on my progress.