Understanding De Broglie's Wave-Frequency Relationship

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In summary, there are various relationships that describe the properties of waves and particles, such as the wave-frequency relationship f = v / \lambda and the de Broglie relations \lambda = h / p and f = E / h. However, it is important to specify which quantities are being referred to and to consider the wave nature of particles in order to fully understand the behavior and limitations of these relationships.
  • #1
redtree
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The wave-frequency relationship is as follows:

f = v / [tex]\lambda[/tex]

Therefore:
v = [tex]\lambda[/tex] * f

The de Broglie relations are as follows:

[tex]\lambda[/tex] = h / p

f = E / h

Using some basic algebra:

v = (h / p) * (E / h)
v = E / p
v = [tex]\gamma[/tex]mc[tex]^{}2[/tex] / [tex]\gamma[/tex]mv
v = c[tex]^{}2[/tex] / v

Now, assuming a natural unit system with c=1

v = 1 / v

That doesn't seem to make sense. Where is my error?
 
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  • #2
are you sure it isn't v=v^2/v
 
  • #3
redtree said:
The de Broglie relations are as follows:

[tex]\lambda[/tex] = h / p

f = E / h
The first gives the de Broglie wavelength. The second relates the frequency of a photon to its energy.
 
  • #4
Total energy is mc[tex]^{}2[/tex] not mv[tex]^{}2[/tex]
 
  • #5
Doc Al said:
The first gives the de Broglie wavelength. The second relates the frequency of a photon to its energy.

The relationship is not just for a photon but for any particle. Frequency and wavelength are related by v = f * [tex]\lambda[/tex]
 
  • #6
redtree said:
The relationship is not just for a photon but for any particle.
What relationship? Not E = hf.
Frequency and wavelength are related by v = f * [tex]\lambda[/tex]
This is yet a third relationship, true for any wave.
 
  • #7
redtree said:
Now, assuming a natural unit system with c=1

v = 1 / v

That doesn't seem to make sense. Where is my error?

It "makes sense" because you have chosen to have velocity be unitless. The solutions to your equation are

v = +1 or v = -1

I.e., the object moves at the speed of light.

Somewhere, you must have used a relation that holds only for objects of zero rest mass. I should be more up on this stuff than I am, and will defer to Doc Al for just where the "error" is.
 
  • #8
That's right. The error is in the equation E = hf, which is the famous Einstein relation for the energy of a photon.

More generally, the error is in writing down a string of equations without defining what any of the quantities refer to.
 
  • #9
People, let's not get crazy and instead actually answer the fellow's question. What our friend has discovered is that the phase velocity of a deBroglie wave is indeed faster than the speed of light
[tex]v_p \frac{\omega}{k} = \frac{c^2}{v}[/tex]
Instead, he should consider particles as being represented by wave packets with some spread in momentum and energy (in order to be localizable). In this case, the relevant quantity is the group velocity
[tex]v_g = \frac{\partial\omega}{\partial k} = \frac{\partial E}{\partial p} = \frac{\partial (\sqrt{p^2 c^2 + m^2 c^4} )}{ p }= \frac{p}{\gamma m} = v[/tex]
I hope this clears things up.

Edit: [tex]E = h f[/tex] is true for particles of mass as well. It's a result of harmonic oscillators in general, of which quantum fields are one type (in the limit where "particle" makes sense).
 

What is De Broglie's Wave-Frequency Relationship?

De Broglie's Wave-Frequency Relationship, also known as the De Broglie equation, is a fundamental equation in quantum mechanics that relates the wavelength of a particle to its frequency. It was first proposed by physicist Louis de Broglie in 1924 and is a key concept in understanding the wave-particle duality of matter.

Why is De Broglie's Wave-Frequency Relationship important?

De Broglie's Wave-Frequency Relationship is important because it helps us understand the behavior of particles at the quantum level. It allows us to describe the wave-like properties of matter, such as electrons and protons, and how they interact with their surroundings. This relationship also plays a crucial role in the development of quantum mechanics and has been confirmed through numerous experiments.

How is De Broglie's Wave-Frequency Relationship derived?

De Broglie's Wave-Frequency Relationship is derived from the principles of wave-particle duality, which states that particles can exhibit both wave-like and particle-like properties. It is based on the equation E=hf, where E is the energy of a particle, h is Planck's constant, and f is the frequency of the particle's wave. By rearranging this equation, we can derive the De Broglie equation, which relates the particle's wavelength (λ) to its momentum (p) and Planck's constant (h): λ = h/p.

What is the significance of De Broglie's Wave-Frequency Relationship in modern physics?

De Broglie's Wave-Frequency Relationship is significant in modern physics because it helped lay the foundation for the development of quantum mechanics. It is a fundamental concept that has been applied in various areas of physics, such as atomic and molecular physics, nuclear physics, and solid-state physics. This relationship has also been confirmed through experiments, further solidifying its importance in our understanding of the behavior of matter at the quantum level.

How does De Broglie's Wave-Frequency Relationship relate to other concepts in physics?

De Broglie's Wave-Frequency Relationship is closely related to other fundamental concepts in physics, such as the uncertainty principle and the wave-particle duality. It also has connections to other equations, such as the Schrödinger equation, which describes the behavior of quantum systems. This relationship is a key building block in our understanding of the behavior of particles and the nature of the universe.

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