Quantum Theory for Mathematicians (Graduate Texts in Mathematics, Volume 267)
Brian C. Hall
Language: English
Pages: 566
ISBN: B01K0SUJPW
Format: PDF / Kindle (mobi) / ePub
Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics.
The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.
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operators z j (i.e., multiplication by z j ) and acting on the space of holomorphic functions on Fock [9] observed that these operators satisfy the following commutation relations: (14.23) These are essentially the same commutation relations as the raising and lowering operators considered in Sect. 11.2. Specifically, (14.23) are the relations that would be satisfied by the natural higher-dimensional analogs of the operators a and a ∗ in that section if we omitted the factor of in the
Schur’s lemma holds for representations over an arbitrary field, the second part holds only for representations over algebraically closed fields. Proof. It is easy to see that ker Φ is an invariant subspace of V 1. Since V 1 is irreducible, this means that either ker Φ = V 1, in which case Φ = 0, or in which case Φ is injective. Similarly, the range of Φ is invariant, and thus equal to either or V 2. If Φ is not zero, then the range of Φ is not zero, hence all of V 2. Thus, if Φ is not zero, it
where the n, l, m ’s are as in (18.10) and where {e 1, e 2} forms a basis for V 1 ∕ 2. Now, from the point of view of rotational symmetry, the basis n, l, m ⊗ e j is not the most natural one. Rather, we should decompose the eigenspaces into irreducible invariant subspaces for the (projective) action of SO(3), where SO(3) acts on both and V 1 ∕ 2. We have already decomposed the eigenspaces inside into irreducible invariant subspaces, namely the span of n, l, m where n and l are fixed and m
section s of L another section ∇ X (s) of L satisfying the following properties. First, for each smooth function f on N, we have (23.1) for all vector fields X and sections s. Second, for each smooth function f on N, we have the product rule (23.2) for all vector fields X and sections s. Note that for any section s of L and any function f on N, the quantity fs is a section of s. Given a connection ∇ and a vector field X, the operator ∇ X is called the covariant derivative in the direction
for the mathematician. There are few books that attempt to translate quantum theory into terms that mathematicians can understand. This book is intended as an introduction to quantum mechanics for mathematicians with little prior exposure to physics. The twin goals of the book are (1) to explain the physical ideas of quantum mechanics in language mathematicians will be comfortable with, and (2) to develop the necessary mathematical tools to treat those ideas in a rigorous fashion. I have
