Quantum Theory for Mathematicians (Graduate Texts in Mathematics, Volume 267)
Brian C. Hall
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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.
components of the magnetic field. These functions on then evolve in time according to Maxwell’s equation. In quantum field theory, one regards, say, Maxwell’s equations as a sort of infinite-dimensional dynamical system, which we may quantize in something like the way we quantize Newton’s equation to get ordinary nonrelativistic quantum mechanics. In the quantum version of Maxwell’s equations, the energy in each mode of the fields is “quantized,” meaning that one can only add energy to each mode
generally write Proposition 2.25 in a more concise form as where the time derivative is understood as being along some trajectory. Proof. Using the chain rule and Hamilton’s equations, we have as claimed. Observe that Proposition 2.25 includes Hamilton’s equations themselves as special cases, by taking f(x, p) = x j and by taking f(x, p) = p j . Thus, this proposition gives a more coordinate-independent way of expressing the time-evolution. Corollary 2.26 Call a smooth function f on a
Theorem 23.24 that Q pre(f) preserves the space of polarized sections with respect to P, provided that the flow of X f preserves (which equals P, in this case). We now establish that for any such f, the Lie derivative preserves the space of polarized sections of This result will eventually allow us to define a quantum operator Q(f) on the half-form Hilbert space associated to P. Proposition 23.38 Suppose X is a vector field on N that preserves P, in the sense of Definition 23.22, and suppose α
units of grams, distance in units of centimeters, and time in units of seconds, then has the numerical value of 1. 054 ×10− 27. Thus, on “macroscopic” scales of energy and momentum, it is possible for the uncertainties in position and momentum both to be very small. But on the atomic scale, the uncertainty principle puts a substantial limitation on how localized the position and momentum of a particle can be. In Sect. 12.1, we prove a version of the uncertainty principle for any two operators A
of the origin. Thus, A is not the zero operator. Lemma 13.15. If A belongs to and A commutes with X j and P j for all j = 1,…,n, then A = cI for some Proof. We may easily prove by induction that for any polynomial g. Thus, for any multi-index k, we have (13.39) Suppose A is a nonzero element of that commutes with each X j . If deg(A) = M, consider a nonzero term in A of degree M: If M > 0, we can pick some j such that the jth entry of k 0 is nonzero. By (13.39) and our assumption on A,