= de Broglie phase harmony
{c}
In the 1923 note \i[Waves and Quanta], <Louis de Broglie> proposed a periodic process associated with <rest energy>, remaining in phase with a wave along the moving particle's path. He distinguished the internal <frequency> slowed by <time dilation> from the wave <frequency> based on total <energy>.<Waves and Quanta by Louis de Broglie>[Nature 112, p. 540, 13 October 1923, opening paragraph]{ref} The proposal developed into his 1924 thesis.<Nobel Lectures, Physics 1922–1941/Louis de Broglie - Biographical>[Nobel Lectures, Physics 1922–1941; Louis de Broglie, Biographical, printed p. 257: 1924 thesis and doctoral degree]{ref} In the 1929 lecture \i[The wave nature of the electron], he reconstructed the argument using a <Lorentz transformation>.<Nobel Lectures, Physics 1922–1941/The Wave Nature of the Electron by Louis de Broglie>[Nobel Lectures, Physics 1922–1941; 12 December 1929, printed pp. 247–249: rest-frame periodicity, Lorentz transformation and momentum]{ref}
Here is that <quantum phase> argument in modern notation. Set $\omega_0=mc^2/\hbar$ in the <rest frame>. A <Lorentz transformation> to a frame where the particle has <velocity> $v$ gives
$$
t'=\gamma\left(t-\frac{vx}{c^2}\right),\qquad
\gamma=\frac{1}{\sqrt{1-v^2/c^2}},\qquad
\phi=\omega_0t'=\frac{Et-px}{\hbar},\qquad
E=\gamma mc^2,\quad p=\gamma mv.
$$
Along $x=vt$, <proper time> is $\tau=t/\gamma$, so
$$
\phi(vt,t)=\frac{(E-pv)t}{\hbar}
=\frac{mc^2t}{\gamma\hbar}=\omega_0\tau.
$$
This is the harmony: evaluating the spatial wave at the moving particle gives the slower clock phase. At fixed $x$, its <angular frequency> is instead $E/\hbar=\gamma\omega_0$. Since $k=p/\hbar$, the <de Broglie wavelength> follows as $\lambda=2\pi/k=h/p$. The proposed clock was a theoretical construction, not an observed mechanical vibration.
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