Mathematical Concepts of Quantum Mechanics (Universitext)

Mathematical Concepts of Quantum Mechanics (Universitext)

Stephen J. Gustafson

Language: English

Pages: 382

ISBN: 3642218652

Format: PDF / Kindle (mobi) / ePub

The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline.

Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content.

It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include many-body systems, modern perturbation theory, path integrals, the theory of resonances, quantum statistics, mean-field theory, second quantization, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group.

With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. The last four chapters could also serve as an introductory course in quantum field theory.

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ψ = −2 Im Aψ, Bψ . (5.2) So assuming ψ is normalized ( ψ = 1), and ψ ∈ D(xj ) ∩ D(pj ), we obtain i 2 1 = ψ, ψ = ψ, [pj , xj ]ψ = − Im pj ψ, xj ψ ≤ 2 | pj ψ, xj ψ | ≤ 2 pj ψ xj ψ = 2 (Δpj )(Δxj ). This does it. What are the states which minimize the uncertainty, i.e. the l.h.s. of (5.1)? Clearly, the states for which − Im pj ψ, xj ψ = pj ψ xj ψ would do this. This equality is satisfied by states obeying the equation pj ψ = iμxj ψ for some 1/4 2 e− μj xj /2 μ > 0. Solving the latter

presently no physically motivated proof of the finiteness of the discrete spectrum for α > 2. The proof presented below uses mathematical ingenuity rather than physical intuition. We now prove finiteness of the number of eigenvalues for α > 2. To simplify the argument slightly, we assume the potential V (x) is non-positive and denote U (x) := −V (x) ≥ 0. Let λ < 0 be an eigenvalue of H with eigenfunction φ. The eigenvalue equation (H − λ)φ = 0 can be re-written as 2 (− 2m Δ − λ)φ = U φ. 2 Since

K(λ)}. (8.17) The next step is to prove that #{λ < 0 | 1 EV K(λ)} = #{ν > 1 | ν EV K(0)}. (8.18) To prove (8.18), we begin by showing that ∂ K(λ) > 0 ∀ λ ≤ 0 ∂λ (8.19) K(λ) → 0 as λ → −∞. (8.20) and Writing 2 Δ − λ)−1 U 1/2 φ 2m and differentiating with respect to λ, we obtain φ, K(λ)φ = U 1/2 φ, (− 2 ∂ φ, K(λ)φ = U 1/2 φ, (− Δ − λ)−2 U 1/2 φ ∂λ 2m 2 = (− 2m Δ − λ)−1 U 1/2 φ 2 >0 which proves (8.19). To establish (8.20), we need to derive the integral kernel 2 of the operator K(λ).

Σ)φk ≤ −(const) 1 q + (const) 2 < 0, k k for some positive constants, if n is sufficiently large. We can now apply the min-max principle (see Section 8.1) to conclude that H possesses infinitely many discrete eigenvalues below the threshold Σ. 12.8 Scattering States Unlike in the one-body case, the many-body evolution ψ = e−iHt/ ψ0 behaves asymptotically as a superposition of several (possibly infinitely many) 12.8 Scattering States 139 free evolutions, corresponding to different scenarios of

convergent and analytic in θ as long as Re(λe−2θ ) > 0. (16.7) We continue it analytically in θ and λ, preserving this condition. In particular, for λ ∈ R− , we should have π/4 < Im(θ) < 3π/4. Now observe that the right hand side of (16.6) is independent of θ. Indeed, it is analytic in θ as long as (16.7) holds, and is independent of Re(θ) since the latter can be changed without changing the integral, by changing the variable of integration (b → e−θ b, θ ∈ R). Thus we have constructed an

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