Appendix A Abbreviations and Notations For a better overview, we collect here some abbreviations and specific notations. Abbreviations ala Algebraic approach ana Analytical approach ClM Classical mechanics CONS Complete orthonormal system CSCO Complete system of commuting observables DEq Differential equation EPR Einstein–Podolsky–Rosen paradox fapp ‘Fine for all practical purposes’ MZI Mach–Zehnder interferometer ONS Orthonormal system PBS Polarizing beam splitter QC Quantum computer QM Quantum mechanics QZE Quantum Zeno effect SEq Schrödinger equation Operators There are several different notations for an operator which is associated with a phys- ical quantity A; among others: (1) A, that is the symbol itself, (2) Aˆ, notation with hat (3) A, calligraphic typefont, (4) Aop, notation with index. It must be clear from the context what is meant in each case. For special quantities such as the position x, one also finds the uppercase notation X for the corresponding operator. © Springer Nature Switzerland AG 2018 203 J. Pade, Quantum Mechanics for Pedestrians 1, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-030-00464-4 204 Appendix A: Abbreviations and Notations The Hamiltonian and the Hadamard Transformation We denote the Hamiltonian by H. With reference to questions of quantum informa- tion, H stands for the Hadamard transformation. Vector Spaces We denote a vector space by V, a Hilbert space by H. Appendix B Units and Constants B.1 Systems of Units Units are not genuinely natural (although some are called that), but rather are man-made and therefore in some sense arbitrary. Depending on the application or scale, there are various options that, of course, are fixed precisely in each case by convention. Those unit systems in which some fundamental constants are set equal to 1 and are dimensionless are generally referred to as natural unit systems. As we mentioned, the word ‘natural’ here is understood as part of the name and not as a descriptive adjective. We consider the following natural units: Planck units, the unit system of high-energy physics (theoretical units, also called natural units), and the unit system of atomic physics (atomic units). B.1.1 Planck Units Here, the speed of light c, the Planck constant , the gravitational constant G as well as the Boltzmann constant kB and the electric field constant (multiplied by 4π), 4πε0, are all set equal to 1. Their relations to the SI values are shown in the following table: Quantity Expression√ Value (SI) −18 Charge qP = 4πε0c 1.876 × 10 C Length l = G 1.616 × 10−35 m P c3 = c . × −8 Mass m P G 2 177 10 kg 2 Temperature T = m P c 1.417 × 1032 K P kB = lP . × −44 Time tP c 5 391 10 s © Springer Nature Switzerland AG 2018 205 J. Pade, Quantum Mechanics for Pedestrians 1, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-030-00464-4 206 Appendix B: Units and Constants The Planck scale probably marks a limit to the applicability of the known laws of physics. Distances substantially smaller than the Planck length cannot be considered as meaningful. The same holds true for processes that are shorter than the Planck time. Because of lP = ctP , such a process would take place on a distance scale that would be smaller than the Planck length. By comparison, the LHC accelerator has a spatial resolution of about 10−19 m; its accessible energy is of the order of ELHC = 10 TeV. B.1.2 Theoretical Units (Units of High-Energy Physics) Here, c and are set equal to 1; the other constants remain unchanged. The unit of energy is not determined by the choice of c and ; it is usually expressed in eV (or MeV, GeV etc.). Energy and mass have the same unit; this applies also to space and time. Quantity Unit Expression Value (SI) Energy eV 1.602 × 10−19 J Length 1/eV c/eV 1.973 × 10−7 m Mass eV eV/c2 1.783 × 10−36 kg 4 Temperature eV eV/kB 1.160 × 10 K Time 1/eV /eV 6.582 × 10−16 s In SI units, c = 3.1616 · 10−26 Jm = 0.1973 GeVfm. Since c = 1 in theoretical units, we have the rule of thumb 1fm(SI)=ˆ 5/GeV (TE) (B.1) B.1.3 Atomic Units In atomic units, e = me = = 1. These units, which are related to properties of the electron and the hydrogen atom, are mainly used in atomic and molecular physics. All quantities are formally dimensionless which are multiples of the basic units. If they are not dimensionless in SI units, they are generally marked by the formal ‘unit character’ a.u. (the dots are part of the unit symbol). The Hartree unit of energy is twice the ionization potential of the hydrogen atom. Appendix B: Units and Constants 207 Quantity Atomic unit Value (SI) Angular momentum Planck constant 1.055 × 10−34 Js Charge Elementary charge e 1.602 × 10−19 C −18 Energy Hartree energy Eh,H 4.360 × 10 J = . × −11 Length Bohr radius a0 mcα 5 292 10 m −31 Mass Electron mass me 9.109 × 10 kg Time unit Atomic time unit, 1 a.t.u. = 2.419 × 10−17 s Eh B.1.4 Units of Energy Energy is a central concept of physics, and this manifests itself among other things in the multitude of units used. The most common are summarized in the following table. Thereby, one uses E = ω = hν and c = λν. Unit Conversion factor Comment eV 1 Joule 1.602 × 10−19 J Kilowatt hour 4.451 × 10−26 kWh Calorie 3.827 × 10−20 cal Wavelength in nanometer 1239.85nm From E = hc/λ Frequency in Hertz 2.41797 × 1014 Hz From E = h/T (T = time) Wave number 8065.48cm−1 From E = hc/λ Temperature 11, 604.5K From E = kB T (T = temperature) Rydberg 0.07350 Ry Ionization potential of the H atom Hartree 0.03675H Energy equivalent mass, E/c 1.783 × 10−36 kg B.2 Some Constants Derived units in the SI: 1 N (Newton) = 1 kgms−2; 1 W (Watt) = 1 Js−1; 1 C (Coulomb) = 1 As 1 F (Farad) 0 1 AsV−1; 1T(Tesla)=1Vsm−2; 1 WB (Weber) = 1 Vs Important constants in eV: h = 4, 1357 · 10−16 eVs; = 6, 5821 · 10−16 eVs 2 2 2 2 4 −3 mec = 0, 511 MeV; mec α = 2 · 13, 6eV; mec α = 1, 45 · 10 eV 208 Appendix B: Units and Constants Quantity Symbol Value Unit Speed of light in vacuum c 299, 792, 458 (exact) ms−1 −7 −1 Magnetic field constant μo 4π × 10 (exact) TmA −12 −1 Electric field constant εo 8.85419 × 10 Fm Planck constant h 6.62618 × 10−34 Js (Reduced) Planck constant 1.05459 × 10−34 Js Elementary charge e 1.60219 × 10−19 C Newtonian gravitational constant G 6.672 × 10−11 m3 kg−1s−2 −23 −1 Boltzmann constant kB 1.381 × 10 JK −31 Rest mass of the electron me 9.10953 × 10 kg −27 Rest mass of the proton m p 1.67265 × 10 kg Fine structure constant α 1/137.036 Rydberg constant R 2.17991 × 10−18 J −11 Bohr radius ao 5.29177 × 10 m −15 Magnetic flux quantum o 2.068 × 10 Wb Stefan–Boltzmann constant σ 5.671 × 10−8 Wm−2 K−4 −24 −1 Magnetic moment of the electron μe 9.28483 × 10 JT −26 −1 Magnetic moment of the proton μp 1.41062 × 10 JT Large and small: Size comparisons Size of the universe 1028 m; diameter of an atomic nucleus 10−15 m; Planck length 10−35 m Ratio universe/Planck length 1063 B.3 Dimensional Analysis One advantage of physics over mathematics lies in the existence of physical units. Thus, one can test√ results, conjectures etc. by checking their units. Can e.g. the expression T = 2π l · g be correct? No, the unit of the left-hand side is s, while at the right hand side it is m/s. This principle can be applied constructively (so-called dimensional analysis, Buckingham π-theorem). As an example, consider once more the pendulum. The system data are the mass of the pendulum bob, the length of the string and the acceleration of gravity.√ A time (oscillation time or√ period) can be represented only by the combination l/g.Sowemust have T ∼ l/g. In addition, the physical units show that an expression such as eir (so long as r has the unit m) can not be right; alone for dimensional reasons, it must read eikr, where k has the unit m−1. Appendix B: Units and Constants 209 B.4 Powers of 10 and Abbreviations deci, d −1 deka, da 1 centi, c −2 hecto, h 2 milli, m −3kilo,k 3 micro, μ −6mega,M6 nano, n −9giga,G9 pico, p −12 tera, T 12 femto, f −15 peta, P 15 atto, a −18 exa, E 18 zepto, z −21 zetta, Z 21 yocto, y −24 yotta, Y 24 B.5 The Greek Alphabet Name Lower case Upper case Name Lower case Upper case alpha α A nu ν N beta β B xi ξ gamma γ omicron oO delta δ pi π epsilon ε, E rho ρ P zeta ζ Z sigma σ, ς (coda) eta η H tau τ T theta ϑ, θ upsilon υ Y iota ι I phi ϕ, φ kappa κ K chi χ X lambda λ psi ψ mu μ M omega ω Appendix C Complex Numbers C.1 Calculating with Complex Numbers In the algebraic representation, a complex number z is given by z = a + ib (C.1) where a ∈ R is the real part and b ∈ R the imaginary part of z (and not ib); a = Re (z) and b = Im (z).