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Quantum mechanics

Quantum mechanics

Impossible things usually don’t happen

14 May 2000

The Odd Quantum
Sam Treiman
1999 Princeton University Press 272pp £15.95/$24.95hb

Sam Treiman was a distinguished particle theorist. The famous Goldberger-Treiman relation was, at the time of its discovery in 1958, an amazing connection between the strong and weak interactions. Colleagues used to credit him with “Treiman’s theorem” – impossible things usually don’t happen.

Shortly before his untimely death late last year, Treiman wrote a book that, he said, was “aimed at a wide audience of the curious, scientists as well as non-scientists”. The publishers advertise it as “a concise account of quantum mechanics written for general readers”.

Writing a popular book on quantum mechanics is a perilous endeavour. When I was asked to review the book, my predatory instincts led me to expect numerous errors and misconceptions that I would happily criticize. That hope has not been fulfilled: I found only one error, and it is not about quantum mechanics but is instead in the preliminary discussion on classical physics. (The error concerns an incorrect statement that the solution of the twin paradox requires general relativity.)

The price paid for this dearth of errors is that the book is not at all of the popular kind. Its unpopular features include Maxwell’s equations, on which Treiman writes: “The Maxwell equations are too irresistible not to display, if only for aesthetics… .” The reader is not, however, asked to understand or solve the Maxwell equations, only to appreciate their elegance (cf Matthew, 7:6). Readers will also encounter complex numbers, multiple integrals, bra-ket notations, spherical harmonics and casual statements of familiar truths such as “according to the principles of statistical mechanics, the atoms at low temperature are mainly in the ground state”.

It is therefore necessary to redefine the set of readers: they should have a general education in physics and be eager to understand the curious features of the quantum. For this type of reader, Treiman’s book is outstanding. It is an informal textbook on quantum mechanics, without detailed proofs of the theorems, only hints on how to prove them. This is not at all a qualitative pictorial description of the quantum world, but a rigorous formulation of its axioms in a casual language. For example, Treiman carefully avoids saying that a wavefunction describes properties of a physical system. Rather, he says that the wavefunction “tells us all we can know about the system” – that is, all we can predict on the results of potential measurements that we may perform on that system.

The uncertainty principle is correctly explained in terms of dispersions in statistical distributions, not of mutual disturbances, as many textbooks present it. Quotation marks are scrupulously used for the “uncertainty relation”, DE/Dt~ hw. Treiman stresses that this is not a true uncertainty relation, in the above sense. Quantum mechanics does imply limitations involving time and energy differences, but their physical meaning is not that of uncertainties.

It is inevitable that some points are oversimplified. In discussing the hydrogen atom, for example, Treiman replaces the radial wavefunction u(r) by the product rR(r), and asserts that “this product must vanish at the origin, since r vanishes there”. Quantum mechanics does not demand this, but only that wavefunctions be square-integrable and belong to the domain of definition of the Hamiltonian. Going beyond the hydrogen atom, the book discusses various applications of quantum mechanics to atomic structure, nuclei, the solid state and astrophysics.

Chapter 7, entitled “What’s Going On?”, is the highlight of the book. It begins with the following statement: “Quantum mechanics deals with probabilities. Observers deal with facts [but] nothing within quantum mechanics tells us how to convert probabilities into facts.” This is the infamous quantum measurement problem, which lies at the interface of classical and quantum physics. It is not part of the standard physics curriculum, and suffers from a vast and mostly misleading literature.

Indeed, some leading contemporary scientists have promoted wild speculations about the quantum measurement problem – for example, that human consciousness plays a role in it. Treiman, however, is faithful to his no-nonsense approach and is exceedingly careful. As he writes: “The quantum assertion is [that] the state of the system ‘collapses’ into the eigenstate that corresponds to the eigenvalue obtained in the measurement”, and quickly adds, “for the present, let’s stick with the naive proposition enunciated above.”

There is the inevitable Schrödinger’s cat parable, followed by a mundane explanation: “We are actually all of us, daily, in the position of Schrödinger’s cat…To the outside observer, we are superpositions until the observation is made.” One brief paragraph is devoted to Hugh Everett’s many-worlds interpretation. “It is undoubtedly amusing to contemplate,” says Treiman, “[but] it can neither be falsified nor built on.”

The last two chapters present elementary particles and quantum fields. These were the author’s research areas and the reader gets an up-to-date education on these subjects. Cross sections are defined, and then time, charge and parity symmetries. The text also discusses in a qualitative way neutrino oscillations, Feynman diagrams, propagators and renormalization.

But for Treiman being qualitative does not mean being sloppy! With admirable care, he asserts that “the virtual particle concept is actually only a proxy for certain mathematical ingredients…Virtual particles are not real objects. [We may] describe them as corresponding to virtual reality.” Treiman stops short of explaining current algebra, on which he co-authored two other books.

There are also many funny anecdotes on the early history of quantum theory and on its founding fathers. In summary, this book is a wonderful guided tour through quantum mechanics, and I recommend it without hesitation to every physicist.

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