Physics 371 Spring 2017 Prof. Anlage Review

Transcription

1 Physics 71 Spring 2017 Prof. Anlage Review Special Relativity Inertial vs. non-inertial reference frames Galilean relativity: Galilean transformation for relative motion along the xx xx direction: xx = VVVV + xx and the inverse transformation: xx = xx VVVV yy = yy yy = yy zz = zz zz = zz tt = tt tt = tt vv = vv VV. Michelson-Morley experiment no evidence for an ether that acted as the medium for the propagation of light. Einstein s postulates: 1) If S is an inertial reference frame and if a second frame S moves with constant velocity relative to S, then S is also an inertial reference frame. 2) The speed of light (in vacuum) has the same value c in every direction in all inertial reference frames. Time dilation: tt = γγ tt, where γγ = 1/ 1 ββ 2, and ββ = VV/cc. Length contraction: l = l 0 /γγ, where l 0 is the proper length of an object. Lorentz transformation for relative motion along the xx xx direction: xx = γγ(xx VVVV), yy = yy, zz = zz, tt = γγ(tt xxxx/cc 2 ), and the inverse transformation: xx = γγ(xx + VVVV ), yy = yy, zz = zz, tt = γγ(tt + xx VV/cc 2 ). Spacetime Four-vectors xx 1 xx 2 xx (4) = xx, xx (4) = (xx, cccc), xx 4 The Lorentz transformation as a rotation in 4-dimensional spacetime: γγ 0 0 ββββ xx (4) = Λ xx (4) , Λ = ββββ 0 0 γγ Invariant length of a four-vector: ss xx xx xx 2 xx 2 4, xx (4) yy (4) xx 1 yy 1 + xx 2 yy 2 + xx yy xx 4 yy 4 Relativistic velocity transformation: vv xx = vv xx VV 1 VVvv xx /cc 2, vv yy vv yy =, vv γγ(1 VVvv xx /cc 2 ) zz = vv yy vv zz γγ 1+VVvv xx /cc 2 vv yy =, vv γγ 1+VVvv xx /cc 2 zz = The Relativistic Doppler effect ff AAAAAAAAAAAAAAh = ff 0 1+ββ 1 ββ, ff RRRRRRRRRRRR = ff 0 1 ββ 1+ββ. The light cone; absolute future and absolute past. vv zz γγ(1 VVvv xx /cc 2 ), and the inverse: vv xx = vv xx +VV 1+VVvv xx /cc 2, 1

2 Velocity four-vector: uu (4) = ddxx(4) = γγ(uu)(uu, cc), ddtt 0 Relativistic momentum: pp (4) = mmmm(uu)(uu, cc) = (mmmm(uu)uu, EE/cc), Relativistic energy: EE = γγ(uu)mmcc 2, Three-momentum: pp = mmmm(uu)uu, Kinetic energy: TT EE mmcc 2 = (γγ(uu) 1)mmcc 2, ββ = uu = pp cc/ee, uu = pp = cc γγ(uu)mm pp cc2 /EE. The useful relation: pp (4) pp (4) = (mmmm) 2, EE 2 = (pp cc) 2 + (mmcc 2 ) 2, pp 1 Relativistic four-momentum: pp (4) pp = 2 pp, EE/cc Lorentz transformation of four-momentum: pp (4) = Λ pp (4) Photon: EE = pppp = ħωω, ωω = kkkk, pp = ħωω = ħkk, pp (4) cc γγ = ħ kk, ωω = ħωω (kk, 1), pp (4) (4) cc cc γγ pp γγ = 0, Compton scattering: λλ λλ 0 = h (1 cos θθ), λλ mmmm CC = hcc = 2.4 pppp (for electron mmcc2 scattering) Relativistic four-force: KK (4) = ddpp(4) = γγ ddpp, 1 dddd = γγ FF, 1 uu FF dddd cc dddd cc ddtt 0 Thermodynamics Thermodynamic variables. Extensive vs. Intensive quantities Thermodynamic functions of state (P, V, T, U, S, thermodynamic potentials) have exact differentials An equation of state relates thermodynamic functions of state Ideal gas: PPPP = NNkk BB TT, where kk BB = JJ/KK is Boltzmann s constant. UU = NNkk 2 BBTT. Heat is thermal energy in transit. Heat is not a thermodynamic state function Heat capacity CC = dddd/dddd, specific heat cc = CC/MM. CC VV = dddd/dddd VV, CC PP = dddd/dddd PP, CC PP = γγ the adiabatic index. CC VV Heat and work are not thermodynamic state functions. Inexact differentials ðww = PPPPPP, ðqq = TTTTTT. Work done on a gas ðww = PPPPPP. Zeroth Law of Thermodynamics Thermal equilibrium First law of thermodynamics UU = WW + QQ, or dddd = ðww + ðqq. Reversible vs. irreversible processes Isothermal, adiabatic, isochoric, isobaric processes Heat engine: A machine that produces work from a temperature difference between two reservoirs Carnot cycle (isothermal and adiabatic processes) Carnot efficiency: ηη CCCCCCCCCCCC = WW = 1 TT CCCCCCCC. QQ HHHHHH TT HHHHHH Carnot cycle: QQ HHHHHH = QQ CCCCCCCC, and ðqq rrrrrr = 0, leading to dddd = ðqq rrrrrr. TT HHHHHH TT CCCCCCCC TT TT 2nd Law of Thermodynamics: SS UUUUUUUUUUUUUUUU 0. 2

3 Kelvin statement of the 2 nd law: No process is possible whose sole result is the complete conversion of heat into work. Clausius statement of the 2 nd law: No process is possible whose sole result is the transfer of heat from a colder to a hotter system. Otto cycle internal combustion engine Refrigerator Re-statement of the 1 st -law: dddd = TTTTTT PPPPPP. Thermodynamic potentials: Enthalpy: HH = UU + PPPP; dddd = TTTTTT + VVVVVV. Helmholtz Function: FF = UU TTTT; dddd = pppppp SSSSSS. Gibbs Function: GG = UU + PPPP TTTT; dddd = SSSSSS + VVVVVV. Constraints Fixed V, T Thermally isolated, Fixed V Fixed T, P Thermally isolated, fixed P Minimized Potential Helmholtz Function FF = UU TTTT Entropy SS Gibbs Function GG = UU + PPPP TTTT Enthalpy HH = UU + PPPP Phase transitions: Latent Heat: LL = TT bbbbbbbb SS ssssssssss SS llllllllllll. The order of a phase transition is the order of the lowest differential of GG that shows a discontinuity at TT cc van der Waals equation of state: [PP + (NN 2 /VV 2 )aa][vv NNNN] = NNkk BB TT. Accounts for excluded volume and attractive forces between molecules. PP cc = aa 27bb 2; VV cc = NNNN; TT cc = 8aa, so PP + VV cc 27bb PP cc VV 2 VV 1 = 8 TT, the law of corresponding states VV cc 2 TT cc Isothermal compressibility κκ TT 1 VV TT. TT = 1. Maxwell construction to TT vvvvvv define the vapor/liquid co-existence region: GG vvvvvvvvvv GG llllllllllll = 0 = dddd = LLLLLL vvvvvv VVVVVV. LLLLLL Quantum Physics JJ Thomson charge-to-mass measurement in E, B fields: qq/mm = vv. Millikan oil droplet RRRR experiment: revealed the quantization of electric charge. Blackbody radiation, Stefan-Boltzmann law: RR TTTTTTTTTT = σσtt 4, σσ = WW mm 2 KK 4. Wien displacement law says that λλ mmmmmm TT = mm KK. Radiation power per unit area related to the energy density of a blackbody: RR(λλ) = cc ρρ(λλ). 4 Rayleigh-Jeans (classical equipartition argument) law ρρ(λλ) = 8ππkk BB TT/λλ 4 leads to the ultraviolet catastrophe. Planck blackbody radiation (treat the atoms as having discrete energy states, and the light as having energy EE = hff): ρρ(λλ) = 8ππhcc/λλ5, h = ee hcc/λλkk BB TT JJ ss. Photoelectric effect and the concept of light as a particle (photon with EE = hff): hff = eeee 0 + φφ. Photon collides with one electron and transfers all of its energy, VV 0 is the stopping potential.

4 X-ray production by Bremsstrahlung with cutoff λλ mmmmmm = VV nnnn (Duane-Hunt Rule), explained by Einstein as inverse photoemission with λλ mmmmmm = hcc. Sharp emission lines eevv arise from quantized energy levels in the core shells of atoms. Bragg reflection of x-rays from layers of atoms in crystals: nnnn = 2dd sin θθ, where nn = 1, 2,,, dd is the spacing between the parallel layers. Rutherford scattering (Phys 410) suggested that positive charge is concentrated in a very small volume the nuclear model of the atom. 1 Empirical rule for light emission from hydrogen = RR 1 1 λλ mmmm mm 2 nn2, Rydberg constant RR = RR HH = for Hydrogen. mm Bohr model of the hydrogen atom (assumes stationary states, light comes from transitions between stationary states, electron angular momentum in circular orbits is quantized): LL = rr mmvv = mmmmmm = nnħ, with nn = 1, 2,,, Radius of circular orbits: rr nn = nn2 aa 0 ZZ with aa 0 = 4ππεε 0ħ 2 ZZ = A, Total energy of Hydrogen atom: EE mmee 2 nn = EE 2 0 nn EE 0 = mmcc2 ee 2 /4ππεε 0 2 = mmcc2 αα 2 = 1.6 eeee, αα = ee2 /4ππεε (ħcc) 2 2 ħcc 17 2, with is called the fine structure constant. Explains the Hydrogen atom emission spectrum but not multi-electron atoms. Davisson-Germer experiment shows that matter (electrons) diffract from periodic structures (Ni atoms on a surface) like waves. It is clear that matter has a strong wavelike character when measured under appropriate conditions. debroglie proposed the wavelength of matter waves as λλ dddd = h/pp, where pp is the linear momentum. Classical physics should be recovered in the short-λλ dddd limit the Correspondence Principle 2 Ψ(xx,tt) The time-dependent Schrodinger equation: ħ2 + VV(xx, tt)ψ(xx, tt) = iiħ Ψ(xx,tt) ; 2mm xx 2 Separation of variables leads to Ψ(xx, tt) = ψψ(xx)ee iiiiii/ħ (a property of stationary states); Time-independent Schrodinger equation: ħ2 dd 2 ψψ(xx) + VV(xx) ψψ(xx) = EEEE(xx); 2mm ddxx 2 The wavefunction Ψ(xx, tt) is complex in general and cannot be measured. Born interpretation of the wave function in terms of a probability density PP(xx, tt) = Ψ (xx, tt)ψ(xx, tt); + Normalization condition: ψψ(xx) 2 dddd = 1. Particle of mass mm in an infinite square well between xx = 0 and xx = LL: EE nn = ħ2 2 kk nn = 2mm nn 2 ππ2 ħ 2 with nn = 1, 2,,, and ψψ 2mmLL nn(xx) = 2/LL sin kk 2 nn xx. Finite square well of height VV 0, energy eigenvalues are solutions of the transcendental equation: tan 2mmmm aa = VV 0 EE (even parity solutions). Always at least one solution! ħ EE Harmonic oscillator: ħ2 dd 2 ψψ(xx) 2mm ddxx mmωω2 xx 2 ψψ(xx) = EEEE(xx), EE nn = nn + 1 ħωω, where 2 nn = 0, 1, 2,,, Eigenfunctions: ψψ nn (xx) = CC nn ee mmωω2 xx 2 /2ħ HH nn (xx), involve the Hermite polynomials multiplying a Gaussian in xx. Classical turning points are inflection points in ψψ(xx). General wave uncertainty properties: ( xx) ( kk) 1/2, ( tt) ( ωω) 1/2.

5 Quantum uncertainty properties: ( xx) ( pp) ħ/2, ( tt) ( EE) ħ/2. Expectation values: xx = Ψ (xx, tt) xx Ψ(xx, tt) dddd, and for any function of position: ff(xx) = Ψ (xx, tt) ff(xx) Ψ(xx, tt) dddd Linear momentum operator: pp oooo = iiħ, Hamiltonian operator: H oooo = pp oooo 2 + VV(xx), 2mm the time independent Schrodinger equation written as an operator equation: H oooo ψ(xx) = EE ψ(xx). 0 for xx < 0 Step potential VV(xx) = VV 0 for xx > 0 has reflection rate RR = kk 1 kk 2 2, and transmission kk 1 +kk 2 rate TT = 4 kk 1 kk 2 (kk 1 +kk 2 ) 2, where kk 1 = 2mmmm/ħ and kk 2 = 2mm(EE VV 0 )/ħ. 1 Tunneling rate through a barrier TT = 1 + sinh2 (αααα) 4 EE VV0 1 EE 16 VV0 aa is the barrier width, and αα = 2mm(VV 0 EE)/ħ. EE VV 0 1 EE VV 0 ee 2αααα, where 5