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Quantum Mechanics: A Modern Development
Although there are many textbooks that deal with the formal apparatus of quantum mechanics (Qm) and its application to standard problems, none take into account the developments in the foundations of the subject which have taken place in the last few decades. There are specialized treatises on various aspects of the foundations of Qm, but none that integrate those topics with the standard material. This book aims to remove that unfortunate dichotomy, which has divorced the practical aspects of the subject from the interpretation and broader implications of the theory. The book is intended primarily as a graduate level textbook, but it will also be of interest to physicists and philosophers who study the foundations of Qm. Parts of it could be used by senior undergraduates too.
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First published October 1, 1989
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January 3, 2017
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 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.
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.
April 12, 2024
This is one of my top 5 (perhaps even favorite) QM books. Even with reservation on this claim, I assert that this is probably the most underrated QM book out there.
I joke around calling this book “The Sakurai without the phrase: ‘It’s trivial to see…’” Not only does this book cover a lot of ground (going over the addition of angular momentum, discrete symmetries, perturbation methods, scattering, EM quantization, many-body theory, and even Bell’s inequality), but there are a lot of examples, diagrams, and even detailed proofs to accompany the learning process. I guess the charm of this book is in the details, as QM is quite counterintuitive without geometric/algebraic insight.
I highly recommend this for all graduate physics students, and would even recommend this for sharp undergraduates with a good mathematical background! After all, I read this as an undergraduate, so I believe that you will like this too!
I joke around calling this book “The Sakurai without the phrase: ‘It’s trivial to see…’” Not only does this book cover a lot of ground (going over the addition of angular momentum, discrete symmetries, perturbation methods, scattering, EM quantization, many-body theory, and even Bell’s inequality), but there are a lot of examples, diagrams, and even detailed proofs to accompany the learning process. I guess the charm of this book is in the details, as QM is quite counterintuitive without geometric/algebraic insight.
I highly recommend this for all graduate physics students, and would even recommend this for sharp undergraduates with a good mathematical background! After all, I read this as an undergraduate, so I believe that you will like this too!
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