Mechanical analog of quantum bradyons and tachyons

We present a mechanical analog of a quantum wave-particle duality: a vibrating string threaded through a freely moving bead or “masslet.” For small string amplitudes, the particle movement is governed by a set of nonlinear dynamical equations that couple the wave field to the masslet dynamics. Under specific conditions, the particle achieves a regime of transparency in which the field and the particle's dynamics appear decoupled. In that special case, the particle conserves its momentum and a guiding wave obeying a Klein-Gordon equation, with real or imaginary mass, emerges. Similar to the double-solution theory of de Broglie, this guiding wave is locked in phase with a modulating group wave comoving with the particle. Interestingly, both subsonic and supersonic particles can fall into a quantum regime as is the case with the slower-than-light bradyons and hypothetical, faster-than-light tachyons of particle physics.

Figure 1

Sketch of the string-mass mechanical analog. A bead of coordinates ($x_{\mathrm{p}},z_{\mathrm{p}}$) slides without friction on a string with local transverse amplitude $u(t,x)$. A vertical spring acts on the bead as as a vertical restoring force and features an internal clock in the de Broglie double solution of quantum mechanics.

Figure 2

The transparent regime for a bradyonic (i.e., subsonic) particle plotted from the analytical solutions described in the paper. An equivalent animation obtained from the numerical code of the full set of equations is available in the Supplemental Material [26]. For this example, we have $v_{\mathrm{p}}/c=0.1, x_{\mathrm{p},\mathrm{i}}=0.1$ for a string length of $L=1$ and $c=1$ (in natural units). We have chosen a $\psi$-wave temporal period $T_{\text{phase}}=\frac{2\pi }{\omega }=10$ and $\eta =\xi =0$. The continuous thick gray line is the total field $u(x,t)$ which can be decomposed into the phase wave $\psi$ (continuous blue line) and the group wave (dashed gray line). The whole field is shown at two different instants (a) $t=0$ and (b) $t=0.08$ clearly demonstrating the subsonic (or bradyonic) regime of the particle. The particle's initial condition is set so that it is at a maximum of the group phase ${\mathrm{\Phi }}_{\text{brad}}$ [see Eq. (26)] and is clearly locked to it along time. The red arrows in (a) and (b) compare the position of the particle at time $t=0$ and 0.08, whereas the blue arrow shows the position and displacement of a phase maximum at the two instants $t=0$ and 0.08.

Figure 3

The transparent regime for a tachyonic (i.e., supersonic) particle of velocity $w_{\mathrm{p}}/c=10$, and initial coordinate $X_{\mathrm{p},\mathrm{i}}=0$ (in natural units). The two panels correspond to observation times (a) $t=0$ and (b) $t=0.08$. The other parameters of the wave and string are unchanged with respect to Fig. 2. In particular, we have $T_{\text{phase}}=\frac{2\pi }{\mathrm{\Omega }}=\frac{2\pi }{\omega }\frac{c}{v_{\mathrm{p}}}=1$ (see text). Like the subsonic case, the particle is locked to the group wave $cos{\mathrm{\Phi }}_{\text{tach}}$ (dashed gray line) and is clearly faster than the velocity of the phase wave $cos{S}_{\text{tach}}$ on the string (blue continuous line). The red and blue arrows compare the displacement of the particle and phase wave between $t=0$ and 0.08.

Figure 4

Principle of a typical experiment for testing a mechanical analog of wave-particle duality. A rigid fork transmits mechanical vibrations from an excitation source (vibrating pot P) to an oscillating string attached to the fork at both ends. A sliding (red) bead constrained to move along the string is connected to a spring attached to a moving mass (ring on a rail). The bead is acting like a quantum particle excited by the wave propagating along the string (see text).