Customer Reviews

Linear Operators for Quantum Mechanics (Dover Books on Physics)

5 star:		(3)
4 star:		(2)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

One cannot do quantum mechanics without a thorough knowledge of the geometry of Hilbert space and the linear transformations on them. This book gives a good introduction to operators on Hilbert space, and could be read by a beginning graduate student of physics.

The theory of operators on Hilbert spaces could be viewed as a generalization of the theory of matrix transformations on finite-dimensional vector spaces. This viewpoint is readily apparent in chapter 1, wherein the author introduces Hilbert spaces as infinite-dimensional vector spaces (over the complex numbers) with an inner product. The author shows how to handle infinite sums of vectors, which requires the notion of convergence, and how to guarantee an infinite sequence of vectors converges to a limit vector that is also in the Hilbert space: the famous Cauchy sequences of vectors. The notion of a linear functional is also introduced, the author proving the one-to-one correspondence between continuous linear functionals and vectors, and connects this with the Dirac bra-ket notation.

Observables in quantum mechanics are represented by operators on (separable) Hilbert spaces, and these are studied in chapter 2. It is straightforward to define a linear operator in finite dimensions, but in infinite dimensions one needs the notion of a continuous linear operator. The author proves that a linear operator is continuous if and only if it is bounded. Unitary operators, so crucial to the calculation of probabilities in quantum theory, are introduced in this chapter also. In addition, the author studies projection operators, which are very important in the measurement process in quantum mechanics. lastly, the author discusses unbounded operators, which are ubiquitous in quantum theory, especially in the theory of angular momenta.

Obtaining measurement results in quantum theory corresponds to obtaining an eigenvalue of a Hermitian linear operator. Thus one must develop a notion of diagonalization (or "spectral resolution") of these operators, and this takes place in chapter 3. In infinite dimensions a Hermitian or unitary operator need not have any eigenvalues or eigenvectors, but the author shows how to obtain a spectral resolution using spectral families of projection operators. He proves that a self-adjoint operator is bounded if and only if its spectrum is bounded, and also Stone's theorem, which gives a representation of a unitary operator as an exponential of a unique self-adjoint operator. Such a representation is expected from the standpoint of how time evolution is characterized in quantum mechanics.

Things become more abstract in chapter 4, wherein the author studies operator algebras. The goal of the chapter is to find conditions under which the functions of a set of noncommuting operators include all bounded operators. This problem motivates the definition of a von Neumann algebra or W*-algebra, this definition depending on the important notion of a weak topology on a set of bounded operators. It is this topology that is the most relevant for the connection of quantum theory with laboratory measurements.

In chapter 5, the author makes clearer the concept of a state in quantum mechanics, this being done using the concept of a density matrix. States specify expectation values of bounded operators, and the author shows how to represent the expectation value of a bounded operator using a unique density matrix. Probabilities in quantum-mechanical calculations are then viewed as expectation values for projection operators, and the author uses Gleason's theorem to justify that projection operators are sufficient to determine the representation of a state. Having set up all this formalism, the author then derives the uncertainty principle for a quantity represented by a Hermitian operator. He then shows that real quantities which are simultaneously measurable with unlimited precision are represented by commuting Hermitian operators. lastly, the author addresses the implicit assumption that every bounded Hermitian operator can represent a measurable quantity. He gives an example of a system that cannot, this occurring because of 'superselection rules'. An operator that commutes with every Hermitian operator which represents a measurable quantity, but is not a multiple of the identity operator is then called a 'superselection operator'. He also discusses, but does not prove in detail, the representation of the expectation value of an element of a von Neumann algebra in terms of a density matrix. When a superselection rule is in place, the density matrix is not always unique. The author then shows how these facts enable one to view a von Neumann algebra alternatively as a collection of bounded operators that commute with all the projection operators.

States of course evolve in time, and so do observables. In chapter 6 the author derives the equations of motion both for the states and the observables. For the states this is the 'Schrodinger picture', and for the observables the 'Heisenberg picture'. Wigner's theorem on unitary and antiunitary operators is used to show that the time evolution of states is linear. The Heisenberg picture is illustrated by an example of a single particle. A more complicated situation though is when the classical system is not integrable, and is still the topic of intense research. The author also includes, atypically for books at this level, a discussion of what happens to the Schrodinger picture when superselection rules are included.
The mathematical tools used in this chapter are used in chapter 7 to study Galilean and Newtonian space-time transformations of states.

I really like this book and find myself returning to it often to clarify issues in the mathematical foundations of QM only superficially treated in many standard QM textbooks. The book is very concise, clear and rigorous. It's jammed packed with theorems and proofs, not physics exposition or examples. The pace is very brisk and there's no hand-holding. So be prepared!

I'm teaching myself quantum mechanics using several standard QM textbooks and have found this book complements them very well. It does, however, assume that elusive quality of "mathematical maturity" (this is really a definition/theorem type book, so be forewarned). In fact, Jordan states the book is the result of first-year graduate QM courses he taught and assumes the reader is at that level. From my experience, I think one should be comfortable with at least standard university level calculus (some analysis helpful), linear algebra, some linear functional analysis and a smidgen of group theory. If you are studying a physics textbook on QM and would like to know more about the mathematical foundations, I highly recommend this book. For the price, I think it would be hard to beat.

Some background in linear functional analysis is necessary to really understand the mathematics of Hilbert spaces underlying quantum mechanics. For those without any background in linear functional analysis, I recommend very highly Rynne and Youngson's Linear Functional Analysis (Springer Undergraduate Mathematics Series) (my personal favorite), or the longer but admirably clear classic Introductory Functional Analysis with Applications by Kreyszig. I also highly recommend Helmberg's marvelous Introduction to Spectral Theory in Hilbert Space (Dover Books on Mathematics).

For those, like me, with aging eyes, be aware the font is very small!

Aside: For those who do not have sufficient mathematical background for Jordan but feel they need a succinct guide focusing on key topics that often trouble beginning students, I also highly recommend Bowman's Essential Quantum Mechanics. It only requires standard calculus and linear algebra but provides a number of very insightful concise explanations of various key topics in basic QM. (Cf. my Amazon review.)

This is a five stars book with a mortal sin.The book is a good introduction to functional analyses with quantum mechanics serving as motivation for definitions and theorems.It is very good for physics students that have already had a good course on quantum mechanics. Since it is intended to be a book to be used as described it is incomplete as a functional analysis book. Now the mortal sin: the author presents spectral theory using projection operators.This is fine. However to neglect the conection with the continuos spectrum will surely repel the intended audience. There are other books that do not do this and therefore are to be prefered.An old one is by Friedman( Lectures on applications oriented mathematics) . A new one is by Zeidler(Applied functional anlysis: applications to mathematical physics)

Physics students learn that Classical Mechanics can be formulated in a variety of different ways at increasing levels of abstraction beginning with Newton's Laws, then progressing through the principle of least action and Lagrangian mechanics, to Hamilton's equations, and finally to Hamilton-Jacobi theory, and that each of these "layers" needs to be learned before the next one can be appreciated.

So it is with Quantum Mechanics too. With QM there are at least three levels of formulation beginning with wave functions, progressing to Dirac's formalism, and then to operator formalism which is the subject of this book.

With that said, if you don't already know -- really know -- the wave function and Dirac formalisms, this book should be avoided until you do.

Although perhaps useful as an introduction to the operator formalism, this book's real intent is to provide a glimpse at the substantial mathematical machinery behind the operator formalism, so the student can set their understanding of it on firmer mathematical ground. Nonetheless, this book is only an introduction to the mathematics behind the operator formalism, and many important results are stated without proof.

There are only seven chapters in this book, and the first four are mathematical preparation. Physics doesn't really enter the picture until chapter five, "States". And it is here that the reader begins to appreciate the elegance of the operator formalism: it handles quantum mechanics and quantum statistical mechanics simultaneously. This is because in the operator formalism all states are represented as density operators. And pure states are simply idempotent density operators. That is, density operators for which A^2 = A. This makes them projection operators. And is another point of elegance: in the Dirac formalism a state vector only defines a state up to a phase factor, but in the operator formalism, this ambiguity is removed.

However, there is still plenty of ambiguity within the choice of operators used to represent measurable quantities.

Regardless, I thought this book did a very good job both in selection of topics and level of coverage for its intended audience and purpose. There are, however, only a few exercise which are all at the end of the book, several of which are quite challenging.

If you are an undergrad, I'd suggest you postpone this book and spend your time learning more quantum mechanics itself. In addition to a solid knowledge of quantum mechanics, the well prepared reader should have a strong background in finite dimensional linear algebra. Knowledge of elementary analysis and even a bit of group theory will be helpful as well.

The language used in this book is easy especially for those who have a physics background. The mathematical proof is also clearly discussed. I suggest to someone who would like to speed the understanding of the theory of linear operator and find the relation towards quantum mechanics.
It's a must have book.