User:Johnjbarton/sandbox/Special relativity and de Broglie's phase harmony

Special relativity and de Broglie's phase harmony

While De Broglie's hypothesis initiated modern quantum mechanics, his theory was an application of relativity. In this he faced "a difficulty that has intrigued me for a long time": relativity makes oscillation frequencies appear lower, while making the observed wave frequency higher. His solution applied Einstein's special relativity three different ways, relating them through an idea he called "phase harmony". The animation below illustrations phase harmony for a moving mass (orange line).

Illustration of de Broglie's phase harmony concept in his 1923 paper on quanta. The orange line is the center of mass moving forward. The green line is de Broglie's period phenomenon, oscillating as viewed in our stationary frame. The blue line is the matter wave oscillating in our frame. The green line remains in contact with the blue line as it moves forward: these oscillations are in "phase harmony."

First he used mass–energy equivalence:

One may imagine that, by cause of a meta law of Nature, to each portion of energy with a proper mass $m_{0}$, one may associate a periodic phenomenon of frequency $\nu _{0}$, such that one finds: $h\nu _{0}=m_{0}c^{2}$ The frequency $\nu _{0}$ is to be measured, of course, in the rest frame of the energy packet.

Second, he applied time dilation to the phase oscillations of the particle (green bar in the animation):

By cause of the LORENTZ transformation of time, a periodic phenomenon in a moving object appears to a fixed observer to be slowed down by a factor of ${\textstyle {\sqrt {1-\beta ^{2}}}}$; this is the famous clock retardation.

Third de Broglie applied the Lorentz factor to the total energy of the moving mass:

On the other hand, since energy of a moving object equals ${\textstyle m_{0}c^{2}/{\sqrt {1-\beta ^{2}}}}$, this frequency according to the quantum relation, ..., is given by:

$$\nu ={\frac {1}{h}}{\frac {m_{0}c^{2}}{\sqrt {1-\beta ^{2}}}}$$

Thus the wave frequency appears higher due to relativity: ${\textstyle \nu =\nu _{0}/{\sqrt {1-{v^{2}}/{c^{2}}}}\,.}$ (Blue wave in the animation).

He then shows the two oscillations are in phase (as illustrated in the animation) with his "phase harmony" theorem:

A periodic phenomenon is seen by a stationary observer ... appears constantly in phase with a wave having frequency ${\textstyle \nu =\nu _{0}/{\sqrt {1-{v^{2}}/{c^{2}}}}\,}$ propagating in the same direction with velocity $V=c/\beta$.

In modern terms this velocity $V$ is the matter wave phase velocity