1. Rutherford’s Experiment (1911)
gold foil
Thomson atom
a source
collimator
microscope
ZnS screen
dn(q)/dW  (Ze2/Ma va2) /sin4(q/2)
Rutherford atom

2. Bohr’s Model (1913) Bohr’s postulates
1. Electrons in atoms move in circular orbits under the influence of the Coulomb attraction between electron and nucleus, obeying the laws of classical mechanics.
2. Only certain orbits with energies En (n = 1,2,3…) are allowed. An electron changing from one orbit to another emits (or absorbs) a photon with the frequency n = DE/h, where DE is the difference of the energy of the two orbits. The energy spectrum is described by the Balmer formula (DE = hc/l = hcR(1/n2 – 1/m2)  En = – hcR/n2)
3. For large n the behavior of the electrons can be described in terms of classical physics (Correspondence Principle)

3. Matter Waves: de Broglie’s hypothesis
- Also classical particles (e.g. electrons) have a dual character of particle and wave, as observed for light/photons.
- Same relation between energy / momentum and frequency / wavelength apply as for photons:
E = hf ; p = h/l
- Allows a new interpretation of Bohr orbits:
L = nh/2p corresponds to standing electron waves
n = 4

4. Electron Diffraction d a Ds Ds = nl: constructive interference
Source: Merli, Missiroli, Pozzi: Am. J. Phys. Vol. 44, No. 3, March 1976

5. Wave Packages
Heisenberg’s Uncertainty Principle for Position and Momentum: D p D x  ½ ħ
f2(x) = ½(sin((k0+Dk)x)+sin((k0–Dk)x)) = cos(Dkx)sin(k0x)
f(x) =  exp(-(k-k0)2/(2Dk2)) sin(kx) dk
f0(x) = sin(k0x)
f0(x) = sin(k0x)
(x,t) =  exp(-(k-k0)2/(2Dk2)) sin(kx – wt) dk
Uncertainty Principle for Energy and Time: D E D t  ½ ħ

6. Wave Packet – Group and Phase Velocity

7. Wave Packet – Group and Phase Velocity
Group Velocity
Phase Velocity

8. Solution of Schrödinger’s Equations for Potential Wells
Particle in a potential well
with infinitely high walls
Particle in a potential
with finite walls
Wave function: standing wave
with nodes at the walls
Wave function extends into the walls
as decreasing exponential function ∞
n = 1
n = 2
n = 3
Energy
Position a

9. Wave Packet in Potential Well
Particle in a potential well
with infinitely high walls
Wave function: standing wave
with nodes at the walls
Superposition
n = 20
n =100
n =900
± 2
a: 1 µm
Period: 6.0 ps
Period: 55 ps
Period: 260 ps

10. Tunneling Narrow potential barrier Alpha decay Energy Position inside
nucleus
outside
nucleus