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Unformatted text preview: WaveParticle Duality • If electromagnetic radiation has dual character and can be regarded as being particlelike, is it possible that matter could have wavelike properties? In 1924, Louis de Broglie made a revolutionary proposition: Small particles of matter may at times display wavelike properties . de Broglie recognized that standing waves are examples of quantization and suggested that the electron in a Bohr orbit may be associated with a circular standing wave. • • The Heisenberg Uncertainty Principle • Heisenberg investigated this issue in detail and concluded that the waveparticle duality places a fundamental limitation on how precisely we can know the location and momentum of any object at the same instant in time: the Heisenberg Uncertainty Principle . Δ p Δ x ≥ h 4 π Δ p = uncertainty in momentum Δ x = uncertainty in position This limitation only becomes important when the masses are as small as an electron, in which case the particles become "fuzzy" and cannot be localized. • The Schrödinger Equation • • De Broglie's work showed that electrons have wavelike character and the uncertainty principle showed that detailed trajectories cannot be defined if the energy of the atom is welldefined. Schrödinger rationalized that if an electron is a wave, its position and movement in space must be described by a wave equation . For this reason, we are only going to be discussing the probability of locating an electron at a certain point in an atom. • The Schrödinger Equation • H Ψ = E Ψ H is the Hamiltonian operator E is the energy Ψ is the wave function . The Schrodinger equation is a differential equation and the solutions to the equation are the wavefunction and energy . For this course, we only need to know understand the nature of the solutions and not the means to obtain the solutions. Ψ itself has no physical significance (and may be positive or negative), but it contains the information to describe ALL properties of the wave. For example, Ψ 2 is the probability density for finding an electron at a given point. • • • Solutions of the Schrödinger Equation for a Particle in a 1Dimensional Box ZPE • Note that the energy of the particle cannot be zero  this is the socalled "zero point energy" ( ZPE ) and is required by the uncertainty principle . If the energy were to be zero, the momentum would be zero and the uncertainty principle would be violated (because the uncertainty in position could not be infinite – since it is somewhere in the box!). Particles in a potential well always have energy and are never at rest !...
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This note was uploaded on 11/03/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.
 Spring '08
 greenwood
 Calculus

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