Lecture 5

# Lecture 5 - Wave-Particle Duality • If electromagnetic...

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Unformatted text preview: Wave-Particle Duality • If electromagnetic radiation has dual character and can be regarded as being particle-like, 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 wave-like 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 wave-particle 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 wave-like character and the uncertainty principle showed that detailed trajectories cannot be defined if the energy of the atom is well-defined. 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 1-Dimensional Box ZPE • Note that the energy of the particle cannot be zero - this is the so-called "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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Lecture 5 - Wave-Particle Duality • If electromagnetic...

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