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