Freqeuncy of Matter Waves Approaches Infinity as Velocity Approaches c

In summary, the frequency of matter waves increases without bound as the velocity of a particle approaches the speed of light (c). This phenomenon is rooted in quantum mechanics, where the energy of a particle is directly related to its frequency. As the particle accelerates and nears the speed of light, its energy and, consequently, its wave frequency rise significantly, illustrating the fundamental relationship between velocity, energy, and wave behavior in the context of relativistic physics.
  • #1
hisgutsaysmaybe
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0
TL;DR Summary
If phase velocity = fλ, and as group velocity -> c, λ -> 0 but phase velocity -> c, then does this mean that f -> infinity as group velocity -> c?
I have heard that the phase velocity of matter waves can be represented as c^2/v. But if the wavelength of these matter waves goes to zero as momentum approaches infinity and v approaches c, then does this mean that the frequency of the matter waves approaches infinity, to give the matter wave a finite and nonzero speed?
In a way this might make sense, because E = ymc^2, so hf = ymc^2. However, the time of an event dilates for an observer which is moving with respect to the event, also by a factor of gamma. I had believed that these two phenomena might combine to appear to give the matter wave a constant frequency, regardless of the velocity of the observer.
 
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  • #2
hisgutsaysmaybe said:
I have heard
Where? Please give a reference.
 
  • #3
From De Broglie's 1924 paper, page 10 (445 of the journal):
1714455680817.png

Taken from this PDF version: https://www.pwein.at/physics/Lectures/Famous-Papers/Phil-Mag-47-446-1924.pdf

I learned in school that the phase velocity was c^2/v, and I have taken it as fact without trying to understand why yet.
 

FAQ: Freqeuncy of Matter Waves Approaches Infinity as Velocity Approaches c

What does it mean for the frequency of matter waves to approach infinity?

When we say that the frequency of matter waves approaches infinity as velocity approaches the speed of light (c), we are referring to the relationship between the energy of a particle and its frequency, as described by the de Broglie hypothesis. As a particle's velocity increases and approaches the speed of light, its energy increases significantly, which in turn leads to an increase in its associated wave frequency. In the limit, as velocity approaches c, the frequency theoretically tends toward infinity.

Why does the frequency of matter waves increase with velocity?

The frequency of matter waves is related to the energy of the particle through the equation E = hν, where E is energy, h is Planck's constant, and ν is frequency. As the velocity of a particle increases, its relativistic energy increases due to the effects of special relativity. Thus, as the particle moves faster and approaches the speed of light, its energy rises, resulting in a corresponding increase in frequency.

What implications does this have for particles moving at relativistic speeds?

As particles move at relativistic speeds (close to the speed of light), their behavior becomes increasingly non-classical. The increase in frequency implies that the wave-like properties of particles become more pronounced, leading to phenomena such as time dilation and length contraction. Additionally, at these speeds, the particles exhibit significant increases in mass and energy, affecting their interactions and the fundamental forces acting on them.

Are there any physical limitations to reaching the speed of light?

Yes, according to Einstein's theory of relativity, no object with mass can reach the speed of light. As an object approaches c, its relativistic mass increases, requiring an infinite amount of energy to accelerate it to the speed of light. Therefore, while we can describe the behavior of matter waves approaching infinite frequency, physical laws prevent any massive particle from actually reaching or exceeding the speed of light.

How does this concept relate to quantum mechanics?

This concept is deeply rooted in quantum mechanics, particularly in the wave-particle duality of matter. The idea that particles can exhibit both wave-like and particle-like properties is central to quantum theory. The increase in frequency as velocity approaches c highlights the interconnectedness of energy, momentum, and wave behavior in quantum systems, reinforcing the principles of uncertainty and the probabilistic nature of quantum mechanics.

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