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A Condensed Course of Quantum Mechanics by Pavel Cejnar

By Pavel Cejnar

This e-book represents a concise precis of nonrelativistic quantum mechanics for physics scholars on the collage point. The textual content covers crucial issues, from common mathematical formalism to express purposes. The formula of quantum conception is defined and supported with illustrations of the overall thoughts of common quantum platforms. as well as conventional themes of nonrelativistic quantum mechanics—including single-particle dynamics, symmetries, semiclassical and perturbative approximations, density-matrix formalism, scattering conception, and the idea of angular momentum—the ebook additionally covers glossy matters, between them quantum entanglement, decoherence, dimension, nonlocality, and quantum details. historic context and chronology of uncomplicated achievements can be defined in explanatory notes. excellent as a complement to school room lectures, the booklet may also function a compact and understandable refresher of basic quantum thought for extra complicated scholars.

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All these analogies justify the following requirement: Dirac quantization assumption Postulate: {A, B} = C ⇒ (classical) ˆ B] ˆ = −i Cˆ [A, (quantum) Equivalent representations of quantum mechanics A fascinating feature of physical description is that it can be cast in infinitely many equivalent ways. In other words, there exists a multitude of mathematical representations yielding the same observable output. In classical mechanics, this feature is anchored in the concept of canonical transformations.

Unfortunately, these are precisely the observables whose QM operators make troubles. Coordinate & momentum eigenfunctions Hilbert space H = L2 (R3 ) & rigged Hilbert space H ⊂ H ⊂ H with H ≡ differentiable functions satisfying |ψ(x)||x|→∞ |x|−m for any m>0 Postulate: δ function & plane wave ≡ eigenstates of position & momentum |p ≡ ei |x ≡ δ(x − x ) p·x Operators of coordinate components x = (x, y, z) ≡ (x1 , x2 , x3 ) Action of operator xˆi ≡ multiplication by variable xi : Def(ˆ xi ) := H Hermiticity: xˆi ψ(x) ≡ xi ψ(x) [ˆ xi ψ](x) ψ (x) ψ1 (x)∗ [xi ψ2 (x)]dx = [xi ψ1 (x)]∗ ψ2 (x)dx Eigenstates: xi δ(x−x ) = xi δ(x−x ) ⇒ continuous spectrum xi ∈ (−∞, +∞) with δ(x−x ) ∈ H Operators of momentum components p = (px , py , pz ) ≡ (p1 , p2 , p3 ) Action of pˆi ∝ derivative by xi : Def(ˆ pi ) := H Hermiticity: ψ1 (x)∗ [−i ∂x∂ i ψ2 (x)]dx = [−i 2π λ ∂ ∂xi ψ(x) [ˆ pi ψ](x) ψ (x) ⇔ pˆ = −i ∇ 0 ∂ ∗ ∂xi ψ1 (x)] ψ2 (x)dx + [ψ1 (x)∗ ψ2 (x)]+∞ −∞ ∂ ip·x/ ∂xi e = pi eip·x/ ⇒ continuous spectrum pi ∈ (−∞, +∞) with eip·x/ ∈ H n with k ≡ wave vector pointing along unit vector n Eigenstates: −i p= k= pˆi ψ(x) ≡ −i We obtain de Broglie relation for the wavelength: λ = 2π p ≡ h p 30 Historical remark 1924: Louis de Broglie associates plane waves with moving particles 1926: Erwin Schr¨odinger applies operators within the wave mechanics 1927: Wolfgang Pauli introduces spin matrices 1930: Paul Dirac introduces explicit momentum and position operators 1940’s-60’s: Rigorous mathematical treatment in terms of the distribution theory (L.

Cˆ [X, 0 ˆ Cˆ [X, 1 ˆ Cˆ [X, 2 ˆ Cˆ [X, 3 1 k! ˆ Cˆ +. . 1 [X, 2! 3! ∞ Yˆ =e + k,l=1 1 k! l! ∞ k=0 1 k! ˆ ˆ eY [X, k ˆ Yˆ l [X, k ˆ ˆ [X, ˆ Yˆ ]] = [Yˆ , [X, ˆ Yˆ ]] = · · · = 0 Special case: [X, ˆ ˆ ˆ 1 ˆ ˆ ⇒ eX+Y = eX eY e 2 [X,Y ] Unitary transformations as “quantum canonical transformations” Unitary operators materialize transitions between alternative QM representations, defined by distinct bases in the system’s Hilbert space (see Sec. 3). , by translations, rotations, or Galilean boosts.

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