Ballentine is a quantum mechanics (QM) book par excellence, my go-to book for graduate-level QM, but also very opinionated and I don't agree with everBallentine is a quantum mechanics (QM) book par excellence, my go-to book for graduate-level QM, but also very opinionated and I don't agree with everything he wrote. (More on this at the end.) But I still give it 5 stars. That's how good it is.
Here are snapshots of QM according to Ballentine.
Chapter 2 presents two mathematical postulates of observables and states and verifies that QM is a consistent generalization of probability theory. (Actually if one formalizes Ballentine's postulates a bit more, the first postulate implies the second by Gleason's theorem.)
Chapter 3 is a highlight for me. The form of a general quantum mechanical Hamiltonian and canonical commutation relations are derived by projectively represented the Galilei group (of non-relativistic rigid motions) on the Hilbert space without relying on the analogy of the commutators with the Poisson brackets (i.e. canonical quantization). I find that the derivation makes more sense once I learn a little bit about Lie algebras and representation theory. The tensor product is introduced here as a natural way to combine probabilistic states.
A minor complaint on chapter 7: I don't like the double bar notation in the Wigner-Eckart theorem. (Sakurai also uses this notation.)
Chapter 8 on state preparation and determination presents an important part of QM neglected in most QM books. How can we prepare a system in the desired state. And if given a system, how can we determine its state? I like the injection of common sense that the determination of unknown pure states "makes an interesting mathematical problem, but in practice, if you do not know the state you are unlikely to know whether it is pure." Here Ballentine gives the correct statements of the uncertainty relations and the no-cloning theorem.
As for the rest of the book, the part that I have read are logical, crisp, to-the-point, and insightful except some parts that rely on his interpretation of states in QM. Ballentine refutes the notion that ("The Interpretation of a State Vector" p. 234)
1. A pure state is a complete description of an individual system (then a complete description of an individual system in Ballentine's interpretation would require a hidden variable) with an eigenstate-eigenvalue link. 2. There is a reduction (collapse) of the state.
because of the measurement problem. But his argument only refutes these if a state is taken to be an objective description of the system. One could adopt the first part of 1. where states are subjective (epistemic) and use the collapse postulate merely as a calculational tool, as in Bohr's interpretation. The eigenstate-eigenvalue link and 2. comes from Dirac and von Neumann's textbooks, not Bohr's. (You may have heard of them together as part of the Copenhagen interpretation, but in fact there is no single, unitary Copenhagen interpretation.) So Ballentine's refutation does not imply his interpretation that states only describe ensembles of quantum systems and that there is no state reduction.
I have tried several time in the past to understand his thinking but have not yet come to a satisfying conclusion, so all I'll say is that, as I see it, Ballentine's ensemble of statistical interpretation is the frequentist interpretation of probablities + QM. States can be defined operationally as equivalent classes of preparations. (Actually Ballentine mentions this definition without the word "equivalence class" only to immediately dismiss it as too restrictive, but this is the way I make sense of Ballentine's interpretation.) Probabilities in QM then are the (limits of) frequencies of measurement results given a measurement and a state. In this way, Ballentine bypasses the need for the state reduction by sacrificing the meaning of states for an individual system. So the measurement problem is not a problem in this interpretation because it is about the state of a single system. But we can't meaningfully talk about the quantum-to-classical transition and decoherence of an individual system (e.g. a table or a chair) either. Taken to the logical extreme, a measurement on a single classical system can never verify a prediction about the system.
If you're however more interested in the ensemble interpretation than I am, you can look at Home and Whitaker, "Ensemble Interpretations of Quantum Mechanics: A Modern Perspective," Physics Reports 210 223 (1992).
Anyway, this leads Ballentine to claim, on p. 342 "The watched pot paradox", that
"we have been led to the conclusion that a continuously observed system never changes its state! This conclusion is, of course, false... The notion of “reduction of the state vector” during measurement was criticized and rejected in Sec. 9.3... Here we see that it is disproven by the simple empirical fact that continuous observation does not prevent motion. It is sometimes claimed that the rival interpretations of quantum mechanics differ only in philosophy, and cannot be experimentally distinguished. That claim is not always true, as this example proves."
I never understand this part. (If you ignore his reasoning, a more charitable reading of his claim is that whether a collapse occurs depend on the type of the measurement, which is true.) It seems like he claims that Zeno effect, which can be observed in the lab ("Quantum reduction seen in real time" in Do We Really Understand Quantum Mechanics?), doesn't exist. But that can't be what he claims because of this paper of his, which merely supplants a different interpretation of the Zeno effect, the same as that in Quantum Theory: Concepts and Methods which I can accept. But I'm not sure that it doesn't need a state reduction or an additional postulate, something I'll have to figure out in the future....more
The selection of topics are diverse and modern but the presentation is not great. The book is more like a collection of disparate abstract formalismsThe selection of topics are diverse and modern but the presentation is not great. The book is more like a collection of disparate abstract formalisms and results, each of which you can probably find a better presentation somewhere else.
Here is a sample of thoughts that I have while reading this book: why are "POVM" and "Estimate of the wave function" in the section "The measurement problem"? Why is "Quantum tomography" in an entirely separate chapter? Why is there only homodyne measurements of the Wigner function in "Quantum tomography" and no heterodyne measurements or general POVM? Why "Qubits" has its own section in "Quantum information and quantum computation" when they're defined as nothing but two-level systems? (Actually there is a deeper operational definition that can be given using Schumacher's compression.) Why only introduce the notion of entropy in the last chapter when it can be utilized when talking about mixed states/entanglement/reversibility?
With so many topics that this book covers, it has the potential to bring them all together into a unified modern understanding of quantum mechanics. But Auletta et al. misses it completely.
An introductory textbook on quantum mechanics that I had been waiting for. Emphasis on entanglement and open system (decoherence) early on. Will writeAn introductory textbook on quantum mechanics that I had been waiting for. Emphasis on entanglement and open system (decoherence) early on. Will write more....more
This is a natural continuation from introductory "wave mechanics" textbooks like Griffiths to remedy the impression that quantum theory is nothing morThis is a natural continuation from introductory "wave mechanics" textbooks like Griffiths to remedy the impression that quantum theory is nothing more than a collection of calculation tools. Nice discussions on abstract mathematical axioms of quantum mechanics and the classical realist preconception, the problem it faces going into quantum mechanics, and the anti-realist position's own problem....more