高斯单位制（CGS）的意义： This MKS system is convenient in engineering. For a treatment of the fundamental physics of fields and matter, it has one basic defect. Thus the MKS system, as it has been constructed, tends to obscure either the fundamental electromagnetic symmetry of the vacuum, or the essential asymmetry of the sources. That was one of our reasons for preferring the Gaussian CGS system in this book. . Auxiliary field H v.s. Magnetic field B . Categories for the Working Mathematician . ψa, ψb： It takes 【two variables】 to describe the configuration of such a system, say ψa and ψj,. For example, in the case of a simple pendulum free to swing in any direction, the “moving parts” ψa and ψb, would be the positions of the pendulum in the two per pendicular horizontal directions; in the case of coupled pendulums, the moving parts ψa and ψb would be the positions of the pendulums; in the case of two coupled LC circuits, the “ moving parts” ψa and ψb would be the charges on the two capacitors or the currents in the circuits. . (for two degrees of freedom and for linear equations of motion) the most general motion is a superposition of two independent simple harmonic motions(normal m odes: Mode1 and Mode2). Mode1: ψa(t)=A1*cos(ω1*t+φ1), ψb(t)=B1*cos(ω1*t+φ1), ψa(t)/ψb(t) = const1 Mode2: ψa(t)=A2*cos(ω2*t+φ2), ψb(t)=B2*cos(ω2*t+φ2), ψa(t)/ψb(t) = const2 characteristic frequency: ω1, ω2 characteristic “ configuration”/“shape”: A1/B1=const1, A2/B2=const2 . The most general motion of the system: ψa(t) = Mode1's ψa(t) +Mode2's ψa(t) ψb(t) = Mode1's ψb(t) +Mode2's ψb(t) . 1.5 Beats 假设：ψ1=A*cos(ω1*t), ψ2=A*cos(ω2*t). ωav ≡ (ω1+ω2)/2, ωmod ≡ (ω1-ω2)/2, . ψ = ψ1+ψ2 = [2A*cos(ωmod*t)] * cos(ωav*t) ≡ Amod(t) * cos(ωav*t) This repetition rate of large values of A mod 2 is called the beat frequency : ωbeat = 2*ωmod . Example 13: Beats between the two normal modes of two weakly coupled Identical oscillators。 定义： Mode1(Fig. 1.14(b))：ψa = ψb， ω1^2 = g/l Mode2(Fig. 1.14(c))：ψa = -ψb， ω2^2 = g/l + 2K/M 简化模型：A1=A2=A， ψ1=ψ2=ψ。 得： ψa = ψ1+ ψ2 = A*cos(ω1*t)+A*cos(ω2*t) = 2A*cos(ωmod*t)*cos(ωav*t), ψb = ψ1- ψ2 = A*cos(ω1*t) -A*cos(ω2*t) = 2A*sin(ωmod*t)*sin(ωav*t), Ea = 1/2*M*ωav^2*(2A*cos(ωmod*t))^2, Eb = 1/2*M*ωav^2*(2A*sin(ωmod*t))^2, Ea+Eb = 2*M*A^2*ωav^2 ≡ E, Ea- Eb = (Ea+Eb)*cos(ω1*t – ω2*t), . Then, the beats between the two modes(Any system of two degrees of freedom can exhibit beats, but the system we have chosen is convenient because ...): . In the study of microscopic systems— molecules, elementary particles— one encounters several beautiful examples of systems that are mathematically analogous to our mechanical example of two identical weakly coupled pendulums. One needs quantum mechanics to understand these systems. The “stuff’ that “ flows” back and forth between the two degrees of freedom, in analogy to the energy transfer between two weakly coupled pendulums, is not energy but probability. Then energy is “ quantized” — it cannot “ subdivide” to flow. Either one “ moving p art” or the other has all the energy. W h at “ flows” is the probability to have the excitation energy. Two examples, the ammonia molecule (which is the “clockworks” of the ammonia clock) and the neutral K mesons, are discussed in Supplementary Topic 1. . ---------------------------------- Ch 2 The modes of a continuous system are called 【standing waves, or normal modes, or simply modes】. In practice we are often concerned only with the first few (or few dozen or few thousand) modes. As we shall see, it turns out that the lowest modes behave as if the system were continuous. Fig. 2.1 每一个bead表示一个自由度，所有beads的一个组合成为一个mode . classical wave equation: (1D (transverse)wave equation) ∂^2ψ(z,t)/∂t^2 = T0/ρ0 * ∂^2ψ(z,t)/∂z^2 (ρ0: mass density at equilibrium, T0: string tension at equilibrium,) . sqrt(T0/ρ0): phase velocity for traveling waves, . σ≡1/λ: wavenumber. It is the parameter for oscillations in space analogous to the frequency v for oscillations in time. . angular wavenumber k≡ 2π/ λ . frequency ν= sqrt(T0/ρ0) * σ <==> angular frequency ω= sqrt(T0/ρ0) * k (dispersion relation) (continuous system only) . ------------------------- ¹²³⁴⁵⁶⁷⁸⁹ ᵃᵇᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖ ʳˢᵗᵘᵛʷˣʸᶻ ᵅᵝᵞᵟᵠ ₀₁₂₃₄₅₆₇₈₉ ₐ ₑ ₕᵢⱼₖₗₘₙₒₚ ᵣₛₜᵤᵥ ₓ ᵦ º¹²³⁴ⁿ₁₂₃₄·∶αβγδεζηθ ικ λ μ νξοπρστυφχψω ∽ ⊥ ∠ ⊙ ⊕ ⊗∈∩∪∑∫∞≡≠±≈＄㏒㎡㎥㎎㎏㎜ ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ ∈⊂∂Δ∇∀∃e̅Ζ͏͏͏͏͏͏ Z̅ ▹◃ ∧† ┘˩⌋⎦┙┚┛
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