Two waves of slightly different wavelength and frequency produce beats.

1. NOTE: The spatial distribution depends on the particular frequencies involved x 2  1 k k = x  Two waves of slightly different wavelength and frequency produce beats.

3. Fourier Transforms Generalization of ordinary “Fourier expansion” or “Fourier series” Note how this pairs canonically conjugate variables  and t.

4. xk ~ 2 xp ~ h t ~ 2 tE ~ h

5. 1896 1899 a, b g 1912

6. Henri Becquerel(1852-1908) received the 1903 Nobel Prize in Physics for the discovery of natural radioactivity. Wrapped photographic plate showed clear silhouettes, when developed, of the uranium salt samples stored atop it. • 1896 While studying the photographic images of various fluorescent & phosphorescent • materials, Becquerel finds potassium-uranyl sulfate spontaneously emits radiation • capable of penetrating thick opaque black paper • aluminum plates • copper plates • Exhibited by all known compounds of uranium (phosphorescent or not) • and metallic uranium itself.

8. 1898Marie Curie discovers thorium (90Th) Together Pierre and Marie Curie discover polonium (84Po) and radium (88Ra) 1899Ernest Rutherfordidentifies 2 distinct kinds of rays emitted by uranium  - highly ionizing, but completely absorbed by 0.006 cmaluminum foil or a few cm of air  - less ionizing, but penetrate many meters of air or up to a cm of aluminum. 1900P. Villard finds in addition to  rays, radium emits  - the least ionizing, but capable of penetrating many cm of lead, several feet of concrete

9. a g B-field points into page b 1900-01 Studying the deflection of these rays in magnetic fields, Becquerel and the Curies establish  rays to be charged particles