Deriving the De Broglie Wavelength

In summary, E=mc^2 and E=hf are both equations that relate energy to mass and frequency respectively. In Special Relativity, the equation y=h/p can be derived from E=hf by imposing certain constraints on the periodicities of a string in compact space-time. This leads to the energy quantization formula, which is in line with traditional quantum field theory.
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Strafespar
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E=mc^2 and E=hf. In Special Relativity, how can y=h/p be derived from E=hf?
 
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Strafespar said:
E=mc^2 and E=hf. In Special Relativity, how can y=h/p be derived from E=hf?

E2(p) = m2 c4 + p2 c2 [energy in the reference frame with momentum p]

E(p)=h/T(p) [T(p) is the time periodicity in the reference frame p ]

m c2 = E(0) [the mass is the energy in the rest frame p=0]

m c2 = h/T(0) [T(0) is the time periodicity in the reference frame p=0]

by putting all things together you find:

1/T2(p) = 1/T2(0) + c2/y2(p) [from the relativistic dispersion relation]

where y(p)= h / p [is the induced spatial periodicity in the reference frame with momentum p].

See http://arxiv.org/abs/0903.3680" "Compact time and determinism: foundations"

Then if you impose the above periodicities as constraints to a string (field in compact space-time, similarly to the harmonic frequency spectrum of a vibrating string with fixed ends) you obtain the following energy quantization

E2n(p) = n2 E2(p) = n2( M2 c4 + p2 c2)

which is actually the energy quantization coming from the usual field theory with second quantization, after normal ordering. In arXiv:0903.3680 it is shown that this procedure provides an exact matching with ordinary quantum field theory, including Path integral and the commutation relations.
 
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FAQ: Deriving the De Broglie Wavelength

What is the De Broglie wavelength?

The De Broglie wavelength is a concept in quantum mechanics that describes the wavelength associated with a particle. It is named after the physicist Louis de Broglie who proposed the idea that all particles, including matter, have both wave-like and particle-like properties.

How is the De Broglie wavelength derived?

The De Broglie wavelength can be derived using the famous equation E=mc², which relates a particle's energy (E) to its mass (m) and the speed of light (c). By rearranging this equation, we can solve for the wavelength (λ) using the equation λ=h/p where h is Planck's constant and p is the momentum of the particle.

What is the significance of the De Broglie wavelength?

The De Broglie wavelength helps us understand the wave-particle duality of matter. It also has important implications in explaining phenomena such as diffraction and interference, which are typically associated with waves but can also be observed in the behavior of particles on the quantum level.

What is the formula for calculating the De Broglie wavelength?

The formula for calculating the De Broglie wavelength is λ=h/p, where λ is the wavelength, h is Planck's constant (6.626 x 10⁻³⁴ J∙s), and p is the momentum of the particle in kg∙m/s.

What types of particles have a De Broglie wavelength?

All particles, including electrons, protons, and even large molecules, have a De Broglie wavelength. However, this wavelength is typically only noticeable for particles with very small masses, such as in the realm of quantum mechanics.

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