The frequency of a matter wave

TL;DR

De Broglie's matter waves have a frequency determined by the energy of the particle, completing the wave-particle duality.

Transcript

PROFESSOR: We've talked a lot about de Broglie saying that the wavelength is given by h over p. But we have not said much yet about the frequency of the waves. So what is the frequency of those matter waves? So what is the frequency-- frequency-- of the matter waves. So de Broglie did answer that same question. And the answer was obtained by analog... Read More

Key Insights

• 👋 De Broglie's matter waves have a frequency determined by the energy of the particle, completing the wave-particle duality.
• 👋 Wave packets accurately represent particles' velocities, as plane waves do not carry real information.
• 😄 The group velocity, determined by d omega/dk, is a more meaningful representation of the particle's motion than the phase velocity.
• 😫 De Broglie's theory aligns with special relativity, as energy and momentum form a 4-vector and can be set equal to each other.

Questions & Answers

Q: How did de Broglie determine the frequency of matter waves?

De Broglie determined the frequency of matter waves by analogy to the wavelength, stating that omega (frequency) is equal to e (energy) divided by h bar (Planck's constant).

Q: Why does the phase velocity of matter waves appear to be half the velocity of the particle?

The phase velocity is half the velocity of the particle because plane waves, which have a constant phase, do not accurately represent the motion of particles. Wave packets are needed to convey the information of the particle's velocity.

Q: What is the group velocity, and why is it more meaningful?

The group velocity is determined by d omega/dk, which represents the rate of change of frequency with respect to momentum. It is the velocity at which the wave packet and its maxima move, and it is a more meaningful representation of the particle's motion.

Q: How does de Broglie's theory align with special relativity?

De Broglie's theory aligns with special relativity because, just like position and time form a 4-vector, energy divided by c and momentum form a 4-vector. This makes it plausible to set the energy and momentum of matter waves equal to each other.

Summary & Key Takeaways

• De Broglie's matter waves have a frequency analogously determined by the energy of the particle, completing the wave-particle duality.

• The phase velocity of matter waves, determined by omega/k, appears to be half the velocity of the particle, indicating a need for wave packets rather than plane waves to accurately represent particles.

• The group velocity, determined by d omega/dk, is the more meaningful and accurate velocity to represent the motion of the wave packet.