This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Gaussian Function • When a set of measurements is made of some quantity x in which the experimental errors are random, the result is often a Gaussian distribution. • The standard deviation σ of the measurements is a measure of the spread of x values about the mean value of x ( x ). σ = 1 N x i x ( 29 2 i = 1 N ∑ y ( x ) = 1 σ 2 π e x x o ( 29 2 2 σ 2 P x 1 x 2 = y ( x ) dx x 1 x 2 ∫ P x ± σ = y ( x ) dx x σ x + σ ∫ = 0.683 Fourier Series The sum of many waves that form a wave packet is called a Fourier series : Summing an infinite number of waves yields the Fourier integral: Narrow Wide σ x σ k = 1 2 Wave Packet Envelope The superposition of two waves yields a wave number and angular frequency of the wave packet envelope. The range of wave numbers and angular frequencies that produce the wave packet have the following relations: A Gaussian wave packet has similar relations: The localization of the wave packet over a small region to describe a particle requires a large range of wave numbers. Conversely, a small range of wave numbers cannot produce a wave packet localized within a small distance. Waves of what? We use a wave packet to describe a particle. In regions where the amplitude Ψ is large then it is more likely to find the particle. The probability for finding it at some point in space is given by Ψ 2 . This description is inherently STATISTICAL. The best you can do is to establish the probable motion of a particle. Atoms, photons, electrons really exist as PARTICLES, but their properties (location in space, p, E) exist only on a contingency basis. Deck of cards: POKER It is only the probability of events that is causally determined by quantum theory, not the outcome of specific events. Particle may be located anywhere within the group at a given time....
View
Full
Document
This note was uploaded on 04/15/2008 for the course PHY 361 taught by Professor Alarcon during the Spring '08 term at ASU.
 Spring '08
 Alarcon
 Physics

Click to edit the document details