Wave Packet Expansion: Physical Reason & Conservation Laws

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SUMMARY

Wave packets expand over time due to dispersion, which occurs because waves of different frequencies travel at different speeds. This phenomenon is rooted in the dependence of wave speed on frequency, as seen in classical light waves traveling through various media like glass and water. The discussion also explores the behavior of electron wave packets interacting with step potentials, highlighting the potential for reflection and transmission. Understanding these concepts is crucial for analyzing the evolution of probability density in quantum mechanics.

PREREQUISITES
  • Wave packet theory
  • Dispersion in wave mechanics
  • Quantum mechanics fundamentals
  • Step potential in quantum physics
NEXT STEPS
  • Study the mathematical formulation of wave packets in quantum mechanics
  • Learn about dispersion relations in different media
  • Explore the implications of the Schrödinger equation on wave packet evolution
  • Investigate the behavior of wave packets at potential barriers and wells
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Students and professionals in physics, particularly those focused on quantum mechanics, wave theory, and the behavior of particles in potential fields.

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Why do wave packets expand when they evolve in time?
Is there any physical reason behind this?
Is it a conservation law of some quantities that makes it expanded?
 
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A wave packet is a sum of waves with different frequencies (and wavelengths). Waves with different frequencies travel at different speeds, and this leads to dispersion. If the speed didn't depend on frequency, there would be no dispersion.

The same thing happens classically with light waves in a medium. In general, the speed of light in a medium depends on frequency, so a "white light" pulse disperses as it travels through glass, water, optical fiber, etc.
 
Thanks for that answer :) but let me ask another question.

If we consider an electron as a wave packet moving through space hitting at the step potential, theoretically, electron with low energy could sink into the potential and then jump back to the free space, or, also, be reflected back at first place.

Can we write the wave function as these wave packets(reflect + transmit)?
I mean if I can study the evolution of probability density by time of this electron and a step?
 

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