Deriving the De Broglie Wavelength

[Strafespar]

E=mc^2 and E=hf. In Special Relativity, how can y=h/p be derived from E=hf?

 

[naturale]

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

E²_n(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.