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Waves and Particles: Understanding Quantum Mechanics

Explore the dual nature of waves and particles, and how they can be described by wave mechanics and classical mechanics. Learn about wave functions, wave packets, dispersion relations, and the concept of group velocity. Discover the implications of wave dispersion on matter waves.

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Waves and Particles: Understanding Quantum Mechanics

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  1. Waves • Electrons (and other particles) exhibit both particle and wave properties • Particles are described by classical or relativistic mechanics • Waves will be described by wave mechanics = quantum mechanics

  2. Waves • Something disturbing we’ll have to address • The wave and particle seem to be moving at different velocities

  3. Waves • We’ll use Ψ(x,t) to represent the wave function of a particle (electron) • In lecture 1, we wrote down the classical wave equation that describes a propagating disturbance • A solution to the classical wave equation is

  4. Waves

  5. Waves

  6. Waves • We normally write

  7. Waves • More generally we write • Thus for φ=π/2 we have • The phase velocity is defined as the velocity of a point on wave with a given phase (e.g the velocity of the peak of the wave)

  8. Waves • The general harmonic wave can also be written as

  9. Waves

  10. Waves • Notation

  11. Waves • How can we represent a (localized) particle by a (non-localized) wave? • Moire pattern

  12. Waves • We can form a wave packet by summing waves of different wavelengths or frequencies • Summing over an infinitely large number of waves between a range of k values will eliminate the auxiliary wave packets

  13. Waves • Let’s start by adding two waves together

  14. Waves • The first term is the wave and the second term is the envelope

  15. Waves • The combined wave has phase velocity • The group (envelope) has group velocity • More generally the group velocity is

  16. Waves • Recall our problem with the wave velocity being slower than the particle velocity • Using the group velocity we find • Problem solved

  17. Waves • The relation between ω and k is called the dispersion relation • Examples • Photon in vacuum

  18. Waves • Phase and group velocity

  19. Waves • Examples • Photon in a medium

  20. Waves • Examples • de Broglie waves

  21. Waves • Because the group velocity = dω/dk depends on k, the wave packet will disperse with time • This is because each of the waves is moving at a slightly different velocity • Just as white light will be dispersed as it travels through a prism • That’s why ω(k) is called a dispersion relation

  22. Waves • Spreading of a wave packet with time • Does this mean matter waves disperse with time?

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