Download A Condensed Course of Quantum Mechanics by Pavel Cejnar PDF

By Pavel Cejnar

This e-book represents a concise precis of nonrelativistic quantum mechanics for physics scholars on the college point. The textual content covers crucial subject matters, from basic mathematical formalism to express purposes. The formula of quantum thought is defined and supported with illustrations of the overall suggestions of hassle-free quantum platforms. as well as conventional subject matters of nonrelativistic quantum mechanics—including single-particle dynamics, symmetries, semiclassical and perturbative approximations, density-matrix formalism, scattering thought, and the speculation of angular momentum—the booklet additionally covers smooth matters, between them quantum entanglement, decoherence, dimension, nonlocality, and quantum details. ancient context and chronology of easy achievements is usually defined in explanatory notes. perfect as a complement to lecture room lectures, the booklet may also function a compact and understandable refresher of straight forward quantum idea for extra complicated scholars.

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Group G = set of elements {g} closed under a binary operation ◦ (group multiplication) g1 , g2 ∈ G ⇒ (g1 ◦ g2 ) ∈ G satisfying the following properties: (1) (g1 g2 )g3 = g1 (g2 g3 ) associativity (2) ∃ e ∈ G : ge = eg = g ∀g ∈ G unit element (3) ∀ g ∈ G ∃ g −1 : gg −1 = g −1 g = e inverse elements g1 g2 Note: commutativity not required! If g1 g2 = g2 g1 ∀ g1 , g2 ∈ G, the group is called Abelian Unitary representation of group G: Mapping to unitary operators: g → Uˆg , g1 ◦ g2 → Uˆ2 Uˆ1 Group properties naturally satisfied: (Uˆ2 Uˆ1 )† = Uˆ1−1 Uˆ2−1 = (Uˆ2 Uˆ1 )−1 closure (Uˆ3 Uˆ2 )Uˆ1 = Uˆ3 (Uˆ2 Uˆ1 ) associativity e → Iˆ = Iˆ† = Iˆ−1 unit element ˆ ˆ Uˆ = eiA ⇒ Uˆ −1 = e−iA inverse elements Hilbert space H where Uˆ act ≡ carrier space of G QM works with H ⇒ it provides a direct physical “arena” for group theory Invariant subspace: a subspace HG ⊂H is invariant under G if Uˆ |ψ ∈HG  .

G. the system’s dynamics), the symmetry is therefore associated ˆ under the unitary transformation S → S’. with the invariance of H § H. Weyl: Symmetry, Princeton Univ. Press, 1952 54 Usual consequences of this meaning of symmetry: ˆ ˆ Uˆ |ψ ) = E(Uˆ |ψ ) (a) degeneracy of energy levels: H|ψ = E|ψ ⇒ H( ⇒ if Uˆ |ψ = |ψ , the level E is degenerate (b) conservation laws (∃ integrals of motions, see Sec. 5) “Flight over” the group theory Group theory represents a superb example of “the unreasonable effectiveness of mathematics in the natural sciences”, which was pointed out by Wigner.

D2 dξk2 + (λ − ξk2 ) ψ(ξk ) = 0 where ξk = M ωk xk and λ = 2E Mω 2 Eigenstates: |ψnk ≡ ψnk (xk ) ∝ e−ξk /2 Hnk (ξk ) Hn (ξ) ≡ dn dη n eξ 2 −(ξ−η)2 |η=0 ≡ Hermite polynomials generating function 3D: ˆ1 + H ˆ2 + H ˆ 3 ] |ψn |ψn |ψn = [H 1 2 3 ˆ H ω1 n1 +ω2 n2 +ω3 n3 + 32 |ψn1 |ψn2 |ψn3 |ψn1 n2 n3 |ψn1 n2 n3 E n1 n2 n3 Isotropic case: ωk =ω ⇒ En1 n2 n3 → EN = ω(N + 32 ) N = n1 +n2 +n3 ⇒ degeneracy N =0 d0 =1 N =1 d1 =3 N =2 d2 =6 .. Orbital angular momentum dN = (N +1)(N +2) 2 Before we continue with Hamiltonians and potentials, it is useful to construct angular momentum operators associated with orbital motions of a particle.

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