**T**he animation above illustrates schematically the relationship between
the motion of a particle and the associated de Broglie phase wave.

The particle, which is moving to the right, is represented by the
**red circle**. The "internal periodic phenomenon" assumed by de
Broglie is represented schematically by the up-and-down motion of the particle. Note
that this is for the purpose of illustrating the instantaneous phase of the internal
periodic phenomenon only; it is not intended to convey that the particle is literally
moving up-and-down. Rather, the motion of the particle is smoothly to the right as
illustrated by the steady rightward motion of the vertical line attached between the
"particle" and the horizontal axis.

The de Broglie phase wave is represented by the **green sine
wave**, which also propagates to the right. Note that the speed of the
phase wave (phase velocity) exceeds that of the particle (group velocity). In the actual
phase waves of de Broglie, the product of the phase velocity and the group velocity is
*c*². For massive particles, the group velocity is always less than *c*, and
the phase velocity is always greater than *c*. For massless particles (photons, for
example) the group and phase velocities both equal *c*.

Note that, in accordance with the de Broglie theory, the phase of the phase wave at the
position of the particle always exactly matches the instantaneous phase of the internal
periodic phenomenon associated with the particle, though (except for massless particles)
the particle and the phase wave move, or propagate, with different velocities. The
constant match between the phase of the phase wave and the phase of the internal periodic
phenomenon *at the position of the particle* is illustrated above by the invariable
coincidence of the **red circle** and the
**green sine wave**. For example, when the phase
wave reaches its peak value at the position of the particle, the internal phenomenon also
reaches its peak value. Likewise for all other phase values.

Finally, the tracking of the **red circle**
** on** the

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