Methods of Math. PhysicsThus. 2 Dec. 2010
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, Fx= -dU/dx
Ex: Gravity: F = mg, W =
Ex: Spring: F = -kx,
Conservation of mechanical energy: Etot = K + U = constant
Conservative force: Work done doesn’t depend on path taken
(curl x F = 0)
xx x x
dv dxW m dx m dv mv dv
dt dt F dx
( )W kx dx F dx
Gravitational potential energy and force
Near earth far 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 x0
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 kx2 =
Kinetic energy K(t) = ½ mv2 =
Frequency of oscillation of spring
Angular frequency = angular speed = = 2f
where frequency f = 1/T and T = period.
Differentiate:
Simplify:
Solve for 2:
2
2
2
2( cos ) ( cos )
F ma
d xkx m
dt
dk A t m A t
dt
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 10-34 J.s
Energy units: 1 eV = 1.602 x 10-19 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 = circumferencen = 2r
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
2 2 20 0
2 2
20
( )
4 4
4
F ma
kqQ qQ e ZeF
r r r
Ze vm
r r
Solve for (1) v2=
Quantize angular momentum: mvr = nh/2 using deBroglie Solve for (2) v2=
Equate v2=v2:
Solve for r
Energy levels of Bohr atom
2
20
2
1 12
: 2
,4
UVirial theorem E
ZeF calculateU F dr
r
E
insert r and solve for E
ZE E where E
n
• 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
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