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Transcript
Wave packets: The solution to the particle-wave dilemma Particle: One point in space Wave packet: Can be squished to a point and stretched to a wave. Wave: Everywhere in space Ch. 13.1 Interference between two waves with different wavelength x Waves add up Waves cancel Forming a wave packet Combine waves with different wavelengths. 2 waves 3 waves 5 waves Waves synchronized here Waves getting out of synch Spatial spread x of a wave packet 8 4 0 -4 x -8 -15 -10 -5 0 J 5 10 15 x, p in the particle and wave limit x x x Particle Wave x 0 x ∞ ∞ p 0 p , p=h/ not defined , p=h/ well-defined The general case: A wave packet There is a tradeoff between particle and wave character. They have to add up to 100% . An electron can be 100% particle: x=0, p=∞ or it can turn into 100% wave: x=∞, p=0 In between it is a wave packet: x0, p0 The uncertainty relation quantifies the tradeoff. The uncertainty relation x · p h/4 x = Position uncertainty p = Momentum uncertainty (Werner Heisenberg 1927) Comparison with diffraction (Lect. 9, Slides 11,12) Real space (x) versus reciprocal space (p): Large objects in real space (large x) Small objects in reciprocal space (small p) Generalization of the uncertainty relation to space-time Real space-time Reciprocal space-time Space x x · p h/4 Momentum p Time t t · E h/4 Energy E The variables (x,t) are incompatible with (p,E). They are also fundamental variables in relativity. Removing h from the uncertainty relation (optional) t · E h/4 Divide by h t · f 1/4 Dividing the uncertainty relation by h and using E = h f gives an uncertainty relation between time and frequency. Planck’s constant of quantum physics h has disappeared. The uncertainty relation is due to the use of wave packets! Connection to mathematics (optional) The mathematical operation that transforms space and time to momentum and energy is a Fourier transform. A Fourier transform describes a wave packet as a sum of plane waves. The Fourier transforms is used widely in signal processing and data compression . The JPEG (MPEG) compression of images (movies) is based on a Fourier transform.The image is compressed by omitting short wavelengths.