Simple Nature - Light and Matter

18 bestt = t19 c1 = bestc120 c2 = bestc221 c3 = (b-c1*a-c2*a**2)/(a**3)22 print(c1, c2, c3, bestt)23 if (bestj == 1) and (bestk == 1) :24 d = d*.5Derivation of the steady state for damped, driven oscillationsUsing the trig identities for the sine of a sum and cosine of a sum, we can change equation [2]on page 177 into the form[(−mω 2 + k) cos δ − bω sin δ − F m /A ] sin ωt+ [ (−mω 2 + k) sin δ + bω cos δ ] cos ωt = 0 .Both the quantities in square brackets must equal zero, which gives us two equations we canuse to determine the unknowns A and δ. The results areandδ = tan −1 bωmω 2 − k= tan −1 ωω oQ(ωo 2 − ω 2 )A ==F m√(mω 2 − k) 2 + b 2 ω 2F m.m√(ω 2 − ωo) 2 2 + ωoω 2 2 Q −2Proofs relating to angular momentumUniqueness of the cross productThe vector cross product as we have defined it has the following properties:(1) It does not violate rotational invariance.(2) It has the property A × (B + C) = A × B + A × C.(3) It has the property A × (kB) = k(A × B), where k is a scalar.Theorem: The definition we have given is the only possible method of multiplying two vectorsto make a third vector which has these properties, with the exception of trivial redefinitionswhich just involve multiplying all the results by the same constant or swapping the namesof the axes. (Specifically, using a left-hand rule rather than a right-hand rule corresponds tomultiplying all the results by −1.)Proof: We prove only the uniqueness of the definition, without explicitly proving that it hasproperties (1) through (3).Using properties (2) and (3), we can break down any vector multiplication (A xˆx + A y ŷ +A z ẑ) × (B xˆx + B y ŷ + B z ẑ) into terms involving cross products of unit vectors.A “self-term” like ˆx × ˆx must either be zero or lie along the x axis, since any other directionwould violate property (1). If was not zero, then (−ˆx) × (−ˆx) would have to lie in the opposite912 Chapter Appendix 2: Miscellany