LECTURE 14 WAVEPACKETS THE UNCERTAINTY PRINCIPLE PHYS 420-SPRING 2006 Dennis Papadopoulos

an integer number of wavelengths fits into the circular orbit where l is the de Broglie wavelength

ParticleValue of l Electrons of kinetic energy Protons of kinetic energy Thermal neutrons (300K) Neutrons of kinetic energy (14 MeV) He atoms at 300K 1 eV 100 eV eV 12.2 A 1.2 A 0.12A 1 keV 1 MeV 1 GeV A 28.6 F 0.73 F 1.5 A 9.0 F 0.75 A me, walking to the student union for lunch at 2 miles per hour

Fig. 5-6, p. 157

Fig. 5-4, p. 156 d=2.15 A

Fig. 5-5, p. 156  sin(50)=1.65 A

Fig. 5-7, p keV electrons In 400 A alloy film

Fig. 5-8, p. 158 Neutrons in NaCl crystal

Electrons have a wavelength that is much shorter than visible light. The smallest detail that can be resolved is equal to one wavelength.

Human hair Red Blood Cells Table Salt

bulk (thick) foil (thin) Specimen interactions involved in forming an image Backscattered electrons can be used to identify elements in the material. Auger electrons also can give compositional information. Secondary electrons are low in energy and thus can’t escape from the interior of the material. They mostly give information about the surface topography. X-rays are produced by de-excited atoms. Unscattered electrons are those which are transmitted through the material. Since the probability of transmission is proportional to thickness, it can give a reading of thickness variations. Elastically scattered Bragg electrons give info about atomic spacing, crystal orientations, etc. Loss of energy by electrons is characteristic of bulk composition.

Fig. 5-18, p. 165

Beats

Fig. 5-19, p. 166

How does this wave behave at a boundary? at a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave at a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180o phase change)

When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary In this animation, the density of the thick string is four times that of the thin string …

Fig. 5-20, p. 168

The word “particle” in the phrase “wave-particle duality” suggests that this wave is somewhat localized. How do we describe this mathematically? …or this

Fig. 5-17, p. 164

FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite number of harmonic waves spatially localized wave group amplitude of wave with wavenumber k=2 p / l adding varying amounts of an infinite number of waves sinusoidal expression for harmonics Adding several waves of different wavelengths together will produce an interference pattern which begins to localize the wave. To form a pulse that is zero everywhere outside of a finite spatial range Dx requires adding together an infinite number of waves with continuously varying wavelengths and amplitudes.

Fig. 5-21, p. 171

Fig. 5-22, p. 171

Fig. 5-23, p. 172

Fig. 5-24, p. 173

Uncertainty on optics:

Remember our sine wave that went on “forever”? We knew its momentum very precisely, because the momentum is a function of the frequency, and the frequency was very well defined. But what is the frequency of our localized wave packet? We had to add a bunch of waves of different frequencies to produce it. Consequence: The more localized the wave packet, the less precisely defined the momentum.