Resolving Tiny Structures: De-Broglie-Relation & Relativity

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In summary: This is a covariant relationship, meaning it holds in all reference frames. In summary, the de Broglie relation for particle waves includes relativistic effects and is a covariant relationship that holds in all reference frames.
  • #1
Anton Alice
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Hello,

I was wondering, if the de-Broglie-relation for particle waves already includes relativistic effects?

Suppose I want to resolve an atomic structure of about, say, 0.1 nanometers. For an attempt using particle waves I would need a certain momentum p=h/0.1nm, at least.
Now comes the question: The particle wave sees the contracted version of the atomic structure. Doesnt this mean that I have to include a gamma into the nominator? p would then be gamma*m*v, and then the gammas would cancel, leading to: mv=h/0.1nm, whereby mv stands for the classical momentum. Something is wrong here...

EDIT:
I think my problem is, that I mixed up different reference frames. I have to stay in one reference frame, which I chose to be the the one of the observer. The observer observes an atomic structure of 0.1nm, and observes a particle with speed v (and corresponding p). These two observations have to fit together, i.e. the de-Broglie-relation needs to be fulfilled. And if it is fulfilled in one reference frame, then also in any other. Like saying, if barack obama has been elected in one reference frame, then also he is or will be in any other.
Can you elaborate on this?
 
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  • #2
Anton Alice said:
Like saying, if barack obama has been elected in one reference frame, then also he is or will be in any other.
Can you elaborate on this?

That's pretty much it.
 
  • #3
Anton Alice said:
I was wondering, if the de-Broglie-relation for particle waves already includes relativistic effects?
Are you familiar with four-vectors? The relativistic de Broglie relationship is simply ##p=\hbar k## where p is the four momentum and k is the four-wavevector.
 

1. What is the De-Broglie relation?

The De-Broglie relation, also known as the de Broglie wavelength, is a fundamental concept in quantum mechanics that describes the wave-like behavior of matter. It relates the momentum of a particle to its wavelength and is given by the equation λ = h/mv, where λ is the wavelength, h is Planck's constant, m is the mass of the particle, and v is its velocity.

2. How does the De-Broglie relation help us to resolve tiny structures?

The De-Broglie relation is essential for understanding and resolving tiny structures because it allows us to describe and predict the behavior of particles at the microscopic level. By taking into account the wave-like properties of matter, we can better understand the behavior of particles and their interactions with other particles and structures.

3. What is the significance of relativity in resolving tiny structures?

Relativity, specifically Einstein's theory of special relativity, is crucial in understanding and resolving tiny structures. This theory explains how the laws of physics are the same for all observers, regardless of their relative motion. This is important when studying tiny structures because particles at this scale are often moving at very high speeds, and their behavior is affected by the principles of relativity.

4. How do the De-Broglie relation and relativity work together to resolve tiny structures?

The De-Broglie relation and relativity are closely related and work together to provide a comprehensive understanding of tiny structures. The De-Broglie relation allows us to describe the wave-like behavior of particles, while relativity helps us understand how this behavior is affected by the speed of the particles and their interactions with other particles and structures.

5. What are some real-world applications of the De-Broglie relation and relativity in resolving tiny structures?

The De-Broglie relation and relativity have numerous applications in various fields, such as particle physics, nanotechnology, and medical imaging. In particle physics, these concepts are used to understand the behavior of subatomic particles. In nanotechnology, they are essential for designing and manipulating tiny structures. In medical imaging, they are used to create detailed images of tiny structures in the human body, such as DNA and individual cells.

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