Methods of Math. Physics Thus. 2 Dec Brief review of forces, energy, oscillations Bohr atom – quantization of angular momentum E.J. Zita

Forces do work and change energy Work done = force. displacement in the same direction, F x = -dU/dx Ex: Gravity: F = mg, W = Ex: Spring: F = -kx, Conservation of mechanical energy: E tot = K + U = constant Conservative force: Work done doesn’t depend on path taken (curl x F = 0)

Gravitational potential energy and force Near earthfar from Earth Force F Potential energy U

Ch.8-8,9: Energy diagrams and Power Power = rate of change of Energy P = dE/dt Minimum energy = stable state (F=0) Ch.8 (Power, 203) #57, 59, 62, 65, 67, (Diag) 68-71, 94-97

Ch.14: Oscillations Systems oscillate about energy minimum Ex: Spring oscillates about equilibrium x 0 Displacement x(t) = A cos (  t +  ) Ch.14 (p._) # _

Energy in Oscillations Displacement x(t) = A cos (  t) Speed v = dx/dt = Potential energy U(t) = ½ kx 2 = Kinetic energy K(t) = ½ mv 2 =

Frequency of oscillation of spring Angular frequency = angular speed =  = 2  f where frequency f = 1/T and T = period. Differentiate: Simplify: Solve for  2 :

Phys.B: Early atomic models

Observed spectra of Hydrogen and other elements Calculate energies of H lines from their colors: E = hc/ Planck constant h = 6.63 x J.s Energy units: 1 eV = x J

Electrons as waves (1923) DeBroglie postulated: if light can behave like a particle (E = hc/ = pc) then maybe matter could behave like waves! What would be an electron’s wavelength? h/  = p = mv Integer # wavelengths = circumference n = 2  r mv = L = mvr = Quantization of angular momentum! (1927) Davisson and Germer discovered that electrons can diffract as waves! thanks to an accident with their nickel crystal.

Bohr model for the Hydrogen atom Solve for (1) v 2 = Quantize angular momentum: mvr = nh/2  using deBroglie Solve for (2) v 2 = Equate v 2 =v 2 : Solve for r

Energy levels of Bohr atom Calculate H energy levels from theory Compare to energies of observed spectral lines They match!

Quantum synthesis: Bohr + deBroglie Bohr used Rutherford’s model of the orbiting electron and Planck’s quantum applied to angular momentum, later justified by deBroglie’s hypothesis of electron wavelengths: angular momentum is quantized in electron orbits orbit radii and energy levels are derived for H-like atoms. Despite unanswered questions (such as how could such orbits be stable?), Bohr’s model fit observations: * Balmer spectrum* Rydberg constant