Get A Condensed Course of Quantum Mechanics PDF

By Pavel Cejnar

ISBN-10: 8024623498

ISBN-13: 9788024623498

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

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Energy with potential energy pˆr ≡ −i ∂ ∂r + 1 r radial momentum Separation of variables The isotropic form of the Hamiltonian enables one to separate radial and angular variables through the wavefunction ansatz: ˆ 2 Ylm (ϑ, ϕ) = L 2 2 2 2 2 unl (r) r l(l + 1)Ylm (ϑ, ϕ) d 2 d The equation for Rnl reads: − 2M r12 dr r dr + d − 2M dr 2 + ψnlm (r, ϑ, ϕ) ≡ Rnl (r) Ylm (ϑ, ϕ) l(l+1) 2M r2 2 l(l+1) 2M r2 + V (r) unl (r) = Enl unl (r) + V (r) Rnl (r) = Enl Rnl (r) radial Schr¨ odinger eq. (l) Veff (r) Unbound-state asymptotics (eigenfunctions of radial momentum) r→∞ For V (r) −−−→ 0 we write down an E > 0 asymptotic radial solution for l = 0: 36 pr r Spherical wave (for r −i ∂ ∂r + 1 r eipr r/ r = pr e ei Rpr (r) ∝ r 0): ipr r/ ≡ plane wave of u(r) r Bound state close to the origin r=0 Approximate equation d2 u dr2 − l(l+1) r2 u ≈ 0 can be solved with u(r) ∝ rk r→0 l+1 −l (nonphysical) k(k−1) = l(l+1) ⇒ k = ⇒ unl (r)|r∼0 ≈ rl+1 −−→ 0 Example (a): finite spherical square well Radial equation: d2 dr2 − l(l+1) r2 + 2M (E−V ) 2 V (r) = unl (r) = 0 with V = −V0 <0 for r 0 for E ≥ 0 κr for r < R 2M (E+V0 ) κ= k = 2M E r→ρ≡ = iκ for E < 0 kr for r ≥ R The eigenfunctions in a general case can expressed through Bessel & Neumann functions, or alternatively through Hankel functions:  Bessel jl (ρ) ∝ρ→0 ρl     Neumann nl (ρ) ∝ρ→0 ρ−(l+1) unl (ρ) Rnl (ρ) = ρ = + ei(ρ−lπ/2) h (ρ) = j (ρ) + in (ρ) ∝ Hankel  l l r→∞ l iρ    functions h− (ρ) = j (ρ) − in (ρ) ∝ e−i(ρ−lπ/2) l l r→∞ l iρ For bound states (E < 0) the r=0 & r→∞ conditions restrict the solution to: Ajl (κr) for r < R Rnl (r) = ( ≡Re) + B hl (iκr) for r ≥ R d dρ jl (κR) jl (κR) = 2 2 Conditions κ +κ = d dρ (1) hl (iκR) h+ l (iκR) 2M V0 ⇒ 2 numerical determination of energy levels Enl 2 Example (b): isotropic harmonic oscillator (revisited) V (r) = M2ω r2 From the solution of the separable problem we know: EN = ω(N + 32 ) where N = n1 +n2 +n3 Solution in spherical coordinates (for the derivation see elsewhere): l+1/2 Rnl (ξ) ∝ ξ l Ln (ξ 2 ) with ξ = Mω r i d i+a −ρ and Lai (ρ) ≡ ρ−a eρ dρ e ) i (ρ ≡ generalized Laguerre polynomial Relation between quantum numbers from both solutions: nr = 0, 1, 2, .

I1 (x1 )ψi2 (x2 )ψi3 (x3 ) ≡ |ψi1 |ψi2 |ψi3 ≡ basis in H = H1 ⊗ H2 ⊗ H3 ik =1,2,... Examples of separable potentials (a) particle in a box V (x) = = V(a1 ,b1 ) (x) + V(a2 ,b2 ) (y) + V(a3 ,b3 ) (z) V(ak ,bk ) (xk ) ≡ 1D infinite square well which has the following solution: 1 2 ˆk 2M p + V(ak ,bk ) |ψnk = ˆk H √π n 2M Lk k Enk |ψnk ≡ ψnk (xk ) ∝ sin nk π Lk (x−ak ) Solution of the 3D problem: ˆ1 + H ˆ2 + H ˆ 3 ] |ψn |ψn |ψn = [H 1 2 3 ˆ H |ψn1 n2 n3 2 0 for xk ∈(ak ,bk ),k=1,2,3 ∞ otherwise |ψnk with nk = 1, 2, 3, .

Aˆ1 ⊗ I), ⇒ ˆ2 ), (Iˆ ⊗ Cˆ2 ) . . , (Iˆ ⊗ Aˆ2 ), (Iˆ ⊗ B n1 +n2 ≡ complete set in H ≡ H1 ⊗ H2 ˆ 1 ⊗ I, ˆ Iˆ ⊗ Yˆ2 ] = 0 ∀X ˆ 1 , Yˆ2 [X (the same eigenvalues as the original sets) ⇒ addition of freedom-degree numbers for composite systems: n = n1 + n2 Incompatible observables ˆ B ˆ A. ˆ Such observables cannot be simultaneously We turn to the case AˆB= diagonalized and exhibit a mutual uncertainty: increasing precision of one observable reduces precision of the other. ˆ B] ˆ = iCˆ Nonzero commutator expressed as: [A, ˆ † = (AˆB ˆ −B ˆ A) ˆ †=B ˆ † Aˆ† − Aˆ† B ˆ † = −[A, ˆ B] ˆ = −iCˆ ⇒ Cˆ = Cˆ † for (iC) ˆ Aˆ† A= ˆ ˆ† B=B Uncertainty relation [ A2 ψ − A 2ψ ] [ B 2 A2 ψ − B 2ψ ] ≥ B2 ψ 1 4 ˆ ψ|C|ψ lower bound of the product of dispersions depends on |ψ 2 ψ Proof: ˆ 2 |ψ = ϕ|ϕ with |ϕ = [Aˆ − A ψ I]|ψ ˆ [ A2 ψ − A 2ψ ] = ψ|[Aˆ − A ψ I] ˆ − B ψ I] ˆ 2 |ψ = χ|χ with |χ = [B ˆ − B ψ I]|ψ ˆ [ B 2 ψ − B 2ψ ] = ψ|[B A2 ψ B2 ψ ˆ − A ψ|AˆB|ψ ≥ 1 4 ˆ ψ|C|ψ ˆ B ˆ − B ψ I]|ψ ˆ = ϕ|ϕ χ|χ ≥ | ϕ|χ |2 = ψ|[Aˆ − A ψ I][ 2 ψ B ψ = ˆˆ ˆ ˆ ψ| AB+2 B A |ψ + ψ| ˆ B ˆ Aˆ AˆB− 2 2 ˆ B ˆ cannot be ⇒ Non-commuting operators A, diagonalized simultaneously: ⇒ Observables A, B are “incompatible” i ˆ 2C 2 |ψ − A ψ B ψ 2 = 42 Analogy with Poisson brackets Although incompatible (“non-commuting”) observables are genuinely quantum invention, there exists a surprising parallel of this behavior in classical mechanics.

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A Condensed Course of Quantum Mechanics by Pavel Cejnar

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