Compton's Effective Velocity and de Broglie == numerology?

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Main Question or Discussion Point

In Compton's 1923 paper, he notes that the scattered wavelength's angular dependence is identical to the Doppler Shift due to an electron moving with the incident wave at an effective velocity:
$$\beta = \frac{ h \nu_0 } { h \nu_0 + m_0 c^2 }$$
What's really interesting is that if you calculate the de Broglie wavelength for an electron moving at that speed, it's the same as the Doppler shifted wavelength of the incident electromagnetic wave:
$$\lambda = \sqrt{ \frac{2 h \nu_0 + m_0 c^2 }{ m_0 c^2 } } \lambda_0$$
It's seems odd that these two values are identical. Is this just numerology or can we gather any physical information from it?

Of course, another numerology result is that the de Broglie wavelength at a given speed is identical to the beat frequency "wavelength" of two Doppler-shifted incident electromagnetic waves of the Compton wavelength (from the front and behind).
 

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Baluncore
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It's seems odd that these two values are identical. Is this just numerology or can we gather any physical information from it?
There is a requirement that any field of science be internally consistent. Reaching the same conclusion via two routes is not “numerology” so much as internally consistent “equational humour”.

Quantum mechanics, matter waves, and moving clocks, 2013, Holger Mueller
https://arxiv.org/pdf/1312.6449.pdf
 

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