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A Physical Constants1 A Physical Constants1 Constants in Mechanics −11 3 −1 −2 Gravitational constant G =6.672 59(85) × 10 m kg s (1996) −11 3 −1 −2 =6.673(10) × 10 m kg s (2002) −2 Gravitational acceleration g =9.806 65 m s 30 Solar mass M =1.988 92(25) × 10 kg 8 Solar equatorial radius R =6.96 × 10 m 24 Earth mass M⊕ =5.973 70(76) × 10 kg 6 Earth equatorial radius R⊕ =6.378 140 × 10 m 22 Moon mass M =7.36 () × 10 kg 6 Moon radius R =1.738 × 10 m 9 Distance of Earth from Sun Rmax =0.152 1 × 10 m 9 Rmin =0.147 1 × 10 m 9 Rmean =0.149 6 × 10 m 8 Distance of Moon from Earth Rmean =0.380 × 10 m 7 Period of Earth w.r.t. Sun T⊕ = 365.25 d = 3.16 × 10 s 6 Period of Moon w.r.t. Earth T =27.3d=2.36 × 10 s 1 From Particle Data Group, American Institute of Physics 2002 http://pdg.lbl.gov/2002/contents http://physics.nist.gov/constants 326 A Physical Constants Constants in Electromagnetism def Velocity of light c =. 299, 792, 458 ms−1 def. −7 −2 Vacuum permeability μ0 =4π × 10 NA =12.566 370 614 ...× 10−7 NA−2 1 × −12 −1 Vacuum dielectric constant ε0 = 2 =8.854 187 817 ... 10 Fm μ0c Elementary charge e =1.602 177 33 (49) × 10−19 C e2 =1.439 × 10−9 eV m = 2.305 × 10−28 Jm 4πε0 Constants in Thermodynamics −23 −1 Boltzmann constant kB =1.380 650 3 (24) × 10 JK =8.617 342 (15) × 105 eV K−1 23 −1 Avogadro number NA =6.022 136 7(36) × 10 mole Derived quantities : Gas constant R = NLkB Particle number N,molenumbernN= NLn Constants in Quantum Mechanics Planck’s constant h =6.626 068 76 (52) × 10−34 Js h = =1.054 571 596 (82) × 10−34 Js 2π =6.582 118 89 (26) × 10−16 eV s 2 4πε0 −10 Bohr radius a∞ = =0.529 177 208 3 (19) × 10 m me2 # $2 m e2 2 1 e2 Rydberg energy E0 = = 2 = 2 4πε0 2ma0 2 4πε0a0 1 2 2 = 2 mc α =13.605 691 72 (53) eV A Physical Constants 327 −31 Electron rest mass me =9.109 381 88 (72) × 10 kg =0.510 998 902 (21) MeVc−2 −28 Myon rest mass mμ =1.883 566() × 10 kg −27 Proton rest mass mp =1.672 621 58 (13) × 10 kg = 938.271 998(38) MeVc−2 −27 Neutron rest mass mn =1.674 954 3 () × 10 kg 1 g −27 Atomic mass unit (amu) mu = =1.660 538 73 (13) × 10 kg NA mole = 931.494 013(37) MeVc−2 e[c] × −5 −1 Bohr magneton μB = 2mc =5.788 381 749 (43) 10 eV T 2 Fine-structure constant α = e =1/137.035 999 76 (50) 4πε0c Conversion Factors 1eV= 1.602 176 462 (63) × 10−19 J B Scalars, Vectors, Tensors In this appendix some relations are collected, which could be useful in non- relativistic mechanics. B.1 Definitions and Simple Rules B.1.1 Definitions (Mathematical) definition: (i) Tensors (of a given rank) are elements of a vector space (i.e., they obey certain operation rules.1 (ii) Under transformation, a tensor of nth rank (with n indices) has, depending on the type of the index, the behavior of the position vectors. (Sloppy, physical) definition: A vector is a quantity with modulus and direc- tion. Comments: • A coordinate transformation (from unprimed to primed coordinates) of a vector a with the (cartesian) components ai is given by j ai = Ui aj. (B.1) j Correspondingly, the behavior of a tensor nthr rank under a transforma- tion is given by j1 ··· jn Ti1...in = Ui1 Uin Tj1...jn . (B.2) j1...jn 1 See Sect. B.2.1 for vectors. 330 B Scalars, Vectors, Tensors • Scalars and vectors are tensors of zeroth and first rank, respectively. • A scalar is a number. But not each number is a scalar: For example the x component of a vector is a number, but no scalar, because it changes with a coordinate transformation. • Tensors of second rank can be represented by matrices. But not each ma- trix is a tensor: For example the transformation matrix U of (B.1) with the j elements Ui is not a tensor, since it does not refer to a given coordinate system, but gives the relation between two different systems. B.1.2BehaviorUnderInversion Under inversion a tensor of even-numbered rank (in particular thus a scalar) transforms into itself; a tensor of odd-numbered rank (thus in particular a vector) transforms into its negative. Pseudo tensors are likewise elements of vector spaces, have but the wrong behavior under transformation: Under inversion a pseudo tensor of even- numbered rank (in particular a pseudo scalar) transforms into its negative; a pseudo tensor of odd-numbered rank (in particular a pseudo vector) trans- forms into itself. Notation: In order to discriminate, a vector is denoted alternatively as a polar vector and a pseudo vector also as an axial vector. Comments: • The cross product of two polar vectors (e.g., in the case of the angular momentum) is an axial vector (pseudo vector).2 • The tensor product (dyadic product) of two polar vectors is a tensor.3 • The scalar product of a vector with a pseudo vector (i.e. the triple scalar product of three vectors) is a pseudo scalar.4 B.2 Vectors B.2.1 Rules for Vectors One has the following axioms for vectors as elements of a vector space: A sum of vectors a ∈ Rd and b ∈ Rd is defined, which results in a vector c ∈ Rd, a + b = c ⇔ ai + bi = ci. A multiplication is defined of a vector a ∈ Rd withascalarα ∈ K;thespace is linear: d = αa ⇔ di = αai. 2 See Appendix B.2.3. 3 See Appendix B.2.2. 4 See Appendix B.2.4. B.2 Vectors 331 There is an inner product (scalar product), of two vectors resulting in a scalar s, a · b = s. B.2.2 Dyadic Product (Tensor Product) of Vectors The dyadic product (tensor product) ab≡ a ⊗ b of two vectors a, b is a tensor T = ab= a ⊗ b with c · T = c · a ⊗ b = c · a b T · d = a ⊗ b · d = a b · d . Comments: • The behavior of the tensor T under transformation is apparent. • In contrast to the scalar product a · b = b · a the tensor product is not commutative, ab= ba. • The symbol ⊗ is rarely used and is used here only for distinction from the dot product (Fig. B.1). • It is thus of importance, for the distinction from the tensor product to notate the product symbol (dot) in the case of the scalar product!! • The multiplication rule “row times column” for matrices holds likewise for the scalar product of vectors. For the scalar product this is a sum of products (of two terms), while for the tensor product each tensor element consists of just one product. ) *)* . ... · . ⊗ ... Fig. B.1. The scalar product (left) and the tensor product (right) of two vectors with the rule “row times column” B.2.3 Vector Product (Cross Product) of Vectors The vector product (cross product) is restricted to vectors in R3;by c = a × b 332 B Scalars, Vectors, Tensors Fig. B.2. The outer product of two vectors a third vector c is assigned to two vectors a, b with a · c =0=b · c & & & &2& &2 & &2 &a × b& = &a · a& &b · b& − &a · b& a, b, and c = a × b forming a right-handed system (Fig. B.2). The resulting vector c is perpendicular to the vectors a and b; the modulus is equal to the area spanned by the vectors a and b, |a × b| = ab sin α, where α is the angle (in the mathematically positive sense from a to b)be- tween the vectors a and b. The cross product has the following properties: a × b = −b × a a × a =0 a × b =0 ⇒ a =0 or b =0 or a = αb. B.2.4 Triple Scalar Product of Vectors Like the external product thus the triple scalar product is restricted to vectors 3 in R ;with a · b × c = a × b · c according to sections B.2.1 and B.2.2 a (pseudo) scalar is assigned to three (polar) vectors. The triple product is the volume of the parallelepiped spanned by the three vectors and positive if the three vectors form a right-handed system. B.2.5 Multiple Products of Vectors The following relations for vectors in R3 are of use occasionally: a × b × c = a · c b − a · b c (B.3) a × b · c × d = a · c b · d − a · d b · c . (B.4) Comment: a × b × c = a × b × c. C Rectangular Coordinate Systems C.1 Definitions 3 Let a point in R be represented by the (generally curvilinear) coordinates ξi (i =1, 2, 3). Definition: The coordinate sheet fi is the sheet fi(ξi)=0(i =1, 2, 3). Definition: The coordinate line si is the intersecting line of the sheets fj(ξj )=0andfk(ξk)=0(withi, j, k cyclic). Definition: The unit vector ei is the vector tangential to the coordinate line si in the direction of increasing value of ξi with ei · ei =1. C.2 Cartesian Coordinates The position vector is r = x ex + y ey + z ez. The line element is (Fig.
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