PHYSICS Fall Semester ODU

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1 Greek Alphabet Capital Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Lowercase α β γ δ ε ζ η θ, ϑ ι κ λ µ Name alpha beta gamma delta epsilon zeta eta theta iota kappa lambda mu Capital Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Lowercase ν ξ ο π ρ σ τ υ φ, ϕ χ ψ ω Name nu xi omicron pi rho sigma tau upsilon phi chi psi omega Fundamental constants: Speed of light: c = m/s (roughly a foot per nanosecond) Planck constant: h = J s;! = h / 2π Fundamental charge unit: e = C Electron mass: m e = kg Coulomb s Law constant: k = / 4πε 0 = Nm 2 /C 2 Gravitational constant: G = Nm 2 /kg 2 Avogadro constant: N A = particles per mol Boltzmann constant: k = J/K = ev/k; R = N A. k = 8.34 J/K/mol Useful conversions: fm (= Fermi ) = 0-5 m, nm = 0-9 m = 0 Å; PHz = 0 5 Hz ( Petahertz ) ev = e. V = J (Energy of elementary charge after V potential difference) kev = 000 ev, MeV= 0 6 ev, GeV = 0 9 ev, TeV = 0 2 ev ( Tera-electronvolt ) New unit of mass m: ev/c 2 = mass equivalent of ev (Relativity!) = kg Momentum p: ev/c = kg m/s; p in ev/c = mass in ev/c 2 times velocity in units of c Planck contant:! = h/2π = ev/c. nm = ev. s = ev/phz Fine-structure constant: α = e 2 / 4πε 0!c = / Electron mass: m e = 50,999 ev/c MeV/c 2 Muon mass: m µ = MeV/c m e Proton mass: m p = MeV/c m e Neutron mass: m n = MeV/c m e Atomic mass unit (/2 of the mass of a 2 C atom): u = MeV/c m e Rydberg constant: Ry = m e c 2 α 2 / 2 = ev Bohr Radius: a 0 =!c / (m e c 2 α) = nm (roughly ½ Å).

2 Special Relativity: For an inertial system S moving along the x-axis of S with constant velocity v < c, and with all axes aligned and the same origin (x = y = z = ct = 0 ó x = y = z = ct = 0): γ = v 2 / c ; x' = γ " $ x v 2 # c ct % & '; ct ' = γ " $ ct v # c x % '; y = y' ; z = z' & Clocks in S appear to S as if they were going slow by factor /γ, and vice versa. Length of object at rest in S appears contracted by factor /γ in S. u ' Velocity addition: u x x c = c + v u ' y c ; u y + u' x v c = γ c. + u' x v c c c c Four-vectors: x µ = (ct, x, y, z); x µ = (ct, x, y, z) (µ=0,,2,3 for ct,x,y,z). Invariant interval between two events (=points in 4-dim. space-time): Δx µ = ( Δct, Δx, Δy, Δz) Δs 2 = Δx µ Δx µ = Δct 2 Δx 2 Δy 2 Δz 2 (same in all inertial systems. ) Positive Δs 2 : time-like separation, Δs 2 = square of elapsed eigentime cτ in a system that travels from the start point (event) to the end point (event) of the interval. Negative Δs 2 : space-like separation, -Δs 2 = square of distance between the two events in a system (which always exists!) where they occur simultaneously. Δs 2 = 0: light-like separation ; a light ray could travel from one event to the other.. Sum of all Four-momentum: P µ = ( E / c, P x, P y, P z ) = (Γmc, Γm u);! Γ = " u 2 / c 2 momenta is conserved in collisions, separately for each component. 0 th component times c is total energy, including kinetic and rest mass energy (E rest = mc 2 ). Transformation of P µ to coordinate system S is analog to x µ (see above). Invariant: P µ P µ = E 2 c 2! P 2 = m 2 c 2 E = m 2 c 4 +! P 2 c 2 ; Quantum Mechanics:!! u c = Pc E. Formal/abstract: All possible knowledge about a system is encoded in its state vector ψ - often only probabilities can be predicted. State vectors are members of a (complex) Hilbert space: they can be added, multiplied by a complex number (scalar), and we can define a scalar product ψ ψ 2 (= some complex number c, with ψ 2 ψ =c * ). All state vectors must be normalizable and by convention are normalized to : ψ ψ =.

3 Example: Motion in Motion in D => state vector represented by wave function ψ(x). Addition: [ψ +ψ 2 ](x) = ψ (x)+ψ 2 (x). Multiplication with scalar: [cψ ](x) = cψ (x). Scalar product: ψ ψ 2 = ψ * (x)ψ 2 (x)dx. Normalizable: ψ ψ =! ψ * (x)ψ(x)dx <. Probability to find particle in interval x x+dx: d Pr(x...x + dx) = ψ(x) 2 dx = ψ(x) * ψ(x)dx (assuming normalized state vector, ψ ψ =). Formal/abstract: Operators are linear functions turning vectors into other vectors: O ψ = ϕ ; O!" c ψ # $ = c ϕ ; O!" ψ + ψ 2 # $ = O ψ + O ψ 2. A vector ϕ ω is called an eigenvector of an operator O with eigenvalue ω (=complex number) IF O ϕ ω = ω ϕ ω. Observables are represented by (Hermitian) operators Ω with only real eigenvalues ω i. Any measurement of the observable must give one of these eigenvalues as result. After we measure ω i, the system will be in the state described by vector ϕ ωi ( collapse of the wave function ). The probability to measure this particular eigenvalue for a state described by ψ is given by Pr(ω i ) = ϕ ωi ψ 2. The average (expectation value) for the observable over many independent trials with the same initial state ψ is Ω ψ = ψ Ω ψ with standard deviationσ Ω = Ω 2 Ω 2. Example: Motion in Motion in D => Important observables: Position Xψ(x) = x ψ(x) eigenvectors ψ x0 (x) = δ(x x 0 ) w/ eigenvalue x 0 ; Momentum Pψ(x) =! eigenvectors ψ p0 (x) = e ip0x/! w/ eigenvalue p 0 ; Hamiltonian (= total i x ψ(x)! energy, kinetic plus potential): Hψ(x) = # P2. " 2m +V(X) $ &ψ(x) =!2 2 % 2m x ψ(x)+v(x)ψ(x) 2 Heisenberg s uncertainty principle: Position x and momentum p cannot be predicted with arbitrary precision simultaneously; σ x σ p!/2. Formal/abstract: Time evolution (Schrödinger Equation): State vector becomes function of time: ψ (t); t ψ (t) = H ψ (t) where H is the Hamiltonian operator. i! Eigenstates of H: H ϕ E = E ϕ E => stationary solutions of Schrödinger Equation: ψ E (t) = ϕ E e iet/! (no time dependence for any observable). Example: Motion in D => Eigenvalue equation:!2 2 ψ(x)+v(x)ψ(x) = Eψ(x). 2 2m x Solution: Stationary States. Eigenstates of the free Hamiltonian (V(x) = 0): i ψ p (x,t) = Ae! px e i p 2! 2m t (simultaneously eigenstates of momentum operator)

4 Gaussian Wave Package: = Linear combination of free Hamiltonian eigenstates (but not an eigenstate itself), with Gaussian weighting over a range of momenta. At time t = 0: ψ GWP (x,t = 0) = 2πσ p ( p p 0 ) 2 i 2 4σ e p! e px dp = e 2πσ x i! p 0x x 2 2 e 4σ x ; σ x =! 2σ p Average momentum p 0, with standard deviation σ p. Average position x = 0; standard deviation for position is σ x =! 2σ p which is the smallest possible given Heisenberg s Uncertainty Relation. However, σ x will increase with time while σ p is constant. Eigenstates of a -dim. square well potential (V(x) = 0 for 0 x L, infinite elsewhere): ϕ n (x) = 2! nπ x $ sin# &; E n = n2 π 2! 2, n =, 2, L " L % 2mL 2 Eigenstates of Harmonic Oscillator: H = P2 2m + mω 2 2 X2! mω ϕ n (x) = AH n # "! x $ &e mω 2! x2 ; E n = (n + 2 % )!ω, n = 0,, H 0 (y) =, H (y) = 2y, H 2 (y) = 4y 2 2; " A 0 = mω % $ ' # π! & /4, A = " mω % $ ' 2 # π! & /4, A 2 = " mω % $ ' 8 # π! & /4. Quantum Mechanics in 3D: Cartesian coordinates: (x,y,z) # ψ(x, y, z); H =!2 2 2m x y + & % 2 (+V(x, y, z); Δ Pr( r,! Δτ ) = ψ(x, y, z) 2 Δτ $ 2 z 2 ' (Small volume Δτ = ΔxΔyΔz located at position (x,y,z)). Separation of variables: Look for solutions for the eigenvalue equation of the type ψ(x, y, z) = ψ (x)ψ 2 (y)ψ 3 (z) Example: Infinite square well in 3D: 8 nπ x jπ y kπ z ϕ njk (x, y, z) = sin sin sin 3 L L L L ; E = njk (n2 + j 2 + k 2 )!2 π 2 2mL 2 Spherical coordinates: r, θ, φ Small volume for probability: Δτ = r 2 Δr sinθδθ Δφ

5 Hamiltonian in spherical coordinates: # H =!2 2m r 2 r r2 r + r 2 sinϑ ϑ sinϑ ϑ + 2 & % (+V(r) $ r 2 sin 2 ϑ ϕ 2 ' =!2 2m r 2 r r2 r +! L 2 +V(r) 2mr 2 Here, L! 2 is the squared orbital angular momentum operator with eigenfunctions Y lm (ϑ,ϕ); L " 2 Y lm = # 2 l(l +)Y lm ; l = 0,, 2 ; L z Y lm = #my lm ; m = l, l +,,l (L z is the z-component of the angular momentum operator). Examples: Y 00 (ϑ,ϕ) = 4π ; Separation of variables: Look for eigenstates of the Hamiltonian of form ψ Elm (r,ϑ,ϕ) = R E,l (r)y lm (ϑ,ϕ) with!2 2m r 2 r r2 r R E,l(r)+!2 l(l +) R 2mr 2 E,l (r)+v(r)r E,l (r) = E R E,l (r) Probability to find particle in volume Δτ at position (r, θ, φ): ψ Elm (r,ϑ,ϕ) 2 Δτ Radial probability distribution: ΔPr(r r+δr) = R E,l (r) 2 r 2 Δr Hydrogen-like atoms: (Nucleus of mass m 2 and charge Ze, bound particle of mass m and charge e) V(r) = Ze2 4πε 0 r = Zα!c α = e 2 / 4πε 0!c r Mass must be replaced by reduced mass of 2-body system with masses m and m 2 : µ r = m m 2 m + m 2 Energy Eigenvalues: E nl = µ r m e Z 2 n 2 Ry (n =, 2, ; Ry = m e c 2 α 2 / 2 = 3.6 ev). Degenerate in l and m; l =0,,, n-, m l = -l +l; also degenerate in electron spin m s = ±/2 => total degeneracy 2n 2. Characteristic radius: a = m e µ r Z a 0 a 0 =!c / (m e c 2 α) = 0.53 Å = nm. Eigenstates: ψ n,l,m (r,ϑ,ϕ) = R n,l (r)y lm (ϑ,ϕ). R n,l (r) (examples): R,0 (r) = 2 a 3/2 e r/a ; R 2,0 (r) = 2 r / a 8a 3/2 e r/2a ; R 2, (r) = r / a e r/2a 3/2 24a

6 Energy of a photon: E γ = hf = hc/λ Momentum of a photon: p γ = h/λ Light emitted or absorbed in transition with energy difference ΔE = E init E final : f = ΔE/h, λ = hc/δe = 2π!c/ΔE Pauli principle: No two identical Fermions (spin-/2, 3/2, particles) can be in the same exact quantum state. (-> See Fermi-Dirac statistics) Nuclear Physics Mass-energy of an atom: (Z protons, N neutrons, A = Z+N): M A c 2 = Z M p c 2 + N M n c 2 + Z m e c 2 BE (Binding energy) typical binding energies BE = 7-8 MeV. A with a maximum for nuclei around iron (A=56). Light nuclei have significantly lower BE per nucleon; beyond iron, the BE per nucleon decreases slowly with A (due to Coulomb repulsion). Energy liberated during a nuclear fusion reaction + 2 -> 3: ΔE = M c 2 + M 2 c 2 M 3 c 2 Energy liberated during a nuclear decay -> 2 + 3: ΔE = M c 2 - M 2 c 2 M 3 c 2 Density: roughly constant ρ = 0.6 Nucleons / fm 3 = kg/m 3 Radioactive nuclei: alpha-decay: (Z,A) (Z-2,A-2) + 4 He + energy beta-plus decay: (Z,A) (Z-, A) + e + + ν e beta-minus decay: (Z,A) (Z+, A) + e - + ν e Decay probability in time Δt: ΔPr(Δt) = Δt/τ (τ = lifetime = T /2 / ln 2) Number of undecayed nuclei at time t (starting with N 0 ): N(t) = N 0 e -t/τ Particle Physics Fundamental Fermions (spin-/2 particles obeying Pauli Exclusion Principle): quarks (up, down, charm, strange, top, bottom) and leptons (electron, muon, tau, electronneutrino, muon-neutrino, tau-neutrino) and their antiparticles:

7 Force-mediating Gauge Bosons (spin- particles obeying Bose-Einstein statistics): Photon γ (electromagnetic interaction), W +, W -, Z 0 (weak interaction), gluons (strong interaction) [graviton (gravity) only conjectured]. All except weak interaction bosons are massless; the latter gain mass (80-9 GeV/c 2 ) through interaction with the Higgs field. All interactions proceed via gauge bosons coupling to various charges: - electromagnetic interaction: electric charge (+ or -) (all Fermions except neutrinos, plus W bosons) - weak interaction: weak charges ( weak isospin and weak hypercharge ) all particles except photons, gluons - strong interaction: color charges ( red, green, blue ) all quarks and gluons. Molecules and Condensed Matter Ionic Bond: One atom gives up (or more) electron(s), the other picks it (them) up; binding through electrostatic attraction. Covalent Bond: Electron(s) shared between two atoms. Example: Let ψ (! r e ) = wave function for hydrogen ground state with proton at position, and ψ 2 (! r e ) for proton at position 2. Symmetric superposition ψ S (! r e ) = 2 ψ (! r e )+ 2 ψ 2 (! r e ) is attractive (net charge between protons), antisymmetric superposition ψ A ( r! e ) = 2 ψ (! r e ) 2 ψ (! 2 r e ) is nonbinding (zero net charge between protons). Metallic Bond: Many electrons (one or more per atom) shared between a large number N of atoms -> positively charged rest atoms in Fermi gas of electrons. Electron energy eigenstates are clustered in bands ; highest (partially or totally unoccupied) band = conduction band, next lower (filled) band = valence band. Each band contains of order N eigenstates. Interaction between electron gas and oscillation modes (=phonons) of the rest atoms gives rise to conductive heating, V = RI, and superconductivity (Bose-Einstein condensation of Cooper pairs of electrons). Conductors: partially filled conduction band and/or overlapping conduction and valence bands. Isolators: Completely empty conduction band, completely filled valence band, large band gap. Semi-conductors: Similar to isolators, but with smaller band gap. Can conduct at finite temperatures (see Fermi-Dirac distribution below). Conductivity increased through electron donor (n-doped) or electron acceptor (p-doped) impurities. pn-junction = diode.

8 Thermal/Statistical Physics Boltzmann Distribution: number n(e) of atoms (molecules, ) out of an ensemble with a total of N atoms ( ) with given energy E in a system with absolute temperature T (in K). Discrete energy levels E i (e.g., quantum systems) with degeneracy g i (= number of eigenstates of the Hamiltonian with energy eigenvalue E i ): n(e i ) = Cg i e E i /kt = e ; C = (E i µ )/kt eµ/kt = N / g i g i e E i /kt (C is a normalization constant; µ is the chemical potential ) Continuous energy levels E (classical system, e.g. monatomic gas) with state density g(e)de (= volume in phase space between energy E and energy E + de): dn(e...e + de) = C g(e)de e E/kT ; C = N / g(e)de e E/kT State density for simple monatomic gas: g(e)de = 4π p 2 dp = 4πm 2mEdE Consequences: Ideal gas law PV = nrt = n N A kt, (n = number of mols; N = n N A ); average energy per degree of freedom (dimension of motion) = ½ kt => total kinetic energy of a monatomic gas = 3/2 kt per atom or E tot = 3 / 2 n N A kt = 3 / 2 nrt Fermi-Dirac Distribution (for a system of indistinguishable Fermions): g n(e i ) = N i ; µ here is right above the Fermi energy = the highest filled e (E i µ )/kt + energy level necessary to accommodate all N fermions, where all lower energy levels are filled with as many Fermions as the Pauli principle allows (= the state of a (degenerate) Fermi gas at (close to) zero temperature). Bose-Einstein Distribution (for a system of indistinguishable bosons): g n(e i ) = N i ; µ here is right below the ground state energy (the lowest e (E i µ )/kt available energy level). If T goes to zero, all levels but that lowest energy level are empty = Bose-Einstein condensation. Photon density for black-body radiation: dn γ (λ λ + dλ) = 8π dv λ 4 dλ e hc/λkt = 8π f 2 c 3 df e hf /kt Energy density (= energy contained in electromagnetic radiation of frequency f or wave length λ, per unit volume V) for black-body radiation (i.e., Bose-Einstein Distribution for a photon gas - Planck s Law): de V = 8πh f 3 df c 3 e hf /kt = 8πhc dλ λ 5 e hc/λkt ; Energy flux/surface area de dadt = 2πhc2 λ 5 dλ e hc/λkt