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The mathematics of classical mechanics

The mathematics of classical mechanics. Newton’s Laws of motion: 1) inertia 2) F = ma 3) action:reaction The motion of particle is represented as a differential equation F = ma = m( dv / dt ) = dp / dt. Lagrange: Defines the motion of particle in terms of kinetic and

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The mathematics of classical mechanics

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  1. The mathematics of classical mechanics Newton’s Laws of motion: 1) inertia 2) F = ma 3) action:reaction The motion of particle is represented as a differential equation F = ma = m(dv/dt) = dp/dt Lagrange: Defines the motion of particle in terms of kinetic and potential energy rather than forces. Introduces partial differentials representing each degree of freedom of motion (3D motion) Hamilton: E = K + V — In QM the energy of a particle is determined mathematically by applying the Hamiltonian operator to the wave function representing the particle.

  2. Get Comfortable with units (SI = kg, m, s) Force = Pressure = Energy =

  3. Get Comfortable with units (SI = kg, m, s) Force = m (kg) • a (m s-2) = kg m s-2 Pressure = F/area = kg m s-2 m2 = kg m-1s-2 Energy = F • distance = kg m2s-2 or PV = kg m-1s-2• m3 = kg m2 s-2 Power = E/t = W (watts) = J s-1 = kg m2s-3

  4. classical mechanics vs. quantum mechanics Particle – wave distinction Particle – wave duality Energy is continuous Energy is quantized Deterministic uncertainty principle

  5. l 10-3 10-1 100 103 105 107 109 1011 1013 • n10201 018 10161014 1012 1010 108 106 104 grays xrays UV IR mwaves radio Light (emr) – James C. Maxwell (1831 – 1879) CM: 1) emr is a wave form of energy 2) travels through ‘ether’ 3) E is continuous 4) wave amplitude = intensity QM: 1) Photons of light (Relativity ‘removed’ ether) 2) Energy is quantized: E = hn 3) Intensity = # photons

  6. CLASSICAL MECHANICS failures Blackbody Radiation: Predicts a heated body will emit infinite energy The Photoelectric Effect Predicts that ↑intensity of emr should be sufficient to expel e- The EquipartitionTheory CP of a substance = ½kT for each degree of freedom CP is independent of temperature Atomic structure and spectra: Predicts collapse of electrons into nucleus of atoms No explanation for spectrum of atom

  7. Blackbody Radiation A heated object will emit radiation

  8. r = E/V (energy density J m-3) CM predicts infinite energy would be emitted by a blackbody! slope = dr dl r CM dr= 8pkT/l4 dl l QM  dr= {8phc/l5 • 1/{exp(hc/lkT)-1} dl

  9. QM  dr= {8phc/l5 • 1/{exp(hc/lkT)-1} dl QM predicts a peak of energy density occurring at higher frequencies (lower l) as T↑. This is what is observed. 8120 Sun 5780 K Fe 1811 K NaCl 1074 K Lava~ 1200 K Human 310 K CMBR 2.73K W bulb 3000K dr 6960 5800 4640 l Total power flux (W m-2 or J s-1 m-2) = sT4s = 5.60705 x 10-8 W m-2 K-4 lmax (nm) = 2.90 x 106 (nm • K) ÷ T (K)

  10. Visible range Sun

  11. The Photoelectric Effect Light striking a solid metal surface may result in e- expulsion. This is not the same as the 1st ionization potential which is for gas phase CM – High intensity, low n light should be able to cause e-to be emitted! QM –intensity  # of e-expelled (but 1 photon required for each e-) Threshold nobelow which no e- emitted regardless of intensity. E > hno imparts extra speed to emitted e-. metal work function (eV) Na 2.36 K 2.9 Cs 2.14 Mg 3.66 Ca 2.87 Mn 4.1 Nd 3.2 Ag 4.6 Sn 4.42 Pb 4.25 E = hn= hno + ½mv2 minimum E to expel e- Kinetic energy of e-

  12. Heat Capacity - Oscillations of a solid CM: CV = 3R (constant for all T) QM: CV = 3R at high T - as T  0; CV  0 Einstein-Bose condensation Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy kT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition.

  13. Heat Capacity - Oscillations of a solid CM: CV = 3R (constant for all T) QM: CV = 3R at high T - as T  0; CV  0 where DE < kT where DE > kT Einstein-Bose condensation Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy kT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition.

  14. Atomic Theory: Dalton (1808) – indivisible sphere Thomson (1890s) raisin pudding Rutherford (1908) nuclear model CM predicts that the electron should radiate energy as it orbits the Nucleus and thus eventually collapse into the nucleus. Observation obviously counters that outcome. ● emremitted ●

  15. de Broglie - 1923 E = hn & E = mc2 hn= hc/l = mc2 replace particle velocity for c (derive expression for l) mv2 = hv/l l = h/mv in 1927 Ni crystal observed to diffract e- beam 1st observation of wave properties of ‘particle’

  16. The Bohr Atom - Postulates Energy is Quantized Atoms in stationary state will not emit radiation An atom absorbs or emits radiation as it changes state ● The Bohr Atom 0.529Å Orbit has radius such that angular momentum, L = mvr= nħ The energy of the orbits are quantized (eq. 9.31)

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