What exactly is the phase of a wave – of any sinusoidal wave?

The phase is the passing of crests at regular intervals. If it is is uniform sine (or cosine wave), the phase remains constant.

Phase of a wave for a moving observer (constant velocity)

It is easy to see that the number of crests that pass a moving observer is the same as that for a stationary observer – as they are regularly spaced. One can say that the phase of the wave is relativistically invariant.

Physical Reality of the PHASE of a wave 

This makes the phase of a wave something all observers can agree on – hence, it must represent a real physical quantity. That quantity is simply the NUMBER of wave crests that pass a given physical point (regardless of the frame that the point is viewed from).

What did deBroglie hypothesize about a particle having an associated wave?

With each moving particle, is associated a wave (a REAL wave as per deBroglie). The wave has a certain phase speed (speed at which the wavefronts proceed). This phase speed HAS to somehow tie back to the velocity of the particle. And therein, deBroglie proposed his famous wave particle equation – p = hk (particle momentum = h times wave momentum)

So – what does this have to do with the electric field?

For a single particle, the electric field associated with it is no different from the deBroglie wave. Let us see why this is so.

Starting with the de-broglie wave equation

(1)   \begin{equation*}A(x,t) = A_0 cos(kx - wt) \end{equation*}

The MINUS Sign
The minus sign denotes the fact that if we hold t constant and increase x we are moving “to the right” along the function, whereas if we focus on a fixed spatial location and allow time to increase, we are effectively moving “to the left” along the function (or rather, it is moving to the right and we are stationary).

Reversing the sign gives

(2)   \begin{equation*}A(x,t) = A_0 cos(kx + wt) \end{equation*}

which is the equation of a wave propagating in the negative x direction.

Speed of The SHAPE of the WAVE (phase velocity)
Since \omega is the number of radians of the wave that pass a given location per unit time, and 1/k is the spatial length of the wave per radian, it follows that \omega/k = v is the speed at which the shape of the wave is moving

If we imagine the wave profile as a solid rigid entity sliding to the right, then obviously the phase velocity is the ordinary speed with which the actual physical parts are moving.

For Phase Velocity = c

Since \omega = kc and c^2 = 1 \ ( \epsilon\mu ), where \epsilon and \mu are the permittivity and permeability of free space, respectively, then k^2 = \omega^2\mu\epsilon.

Setting \k^2 in the wave equation (2) , we get

(3)   \begin{equation*} \frac{d^2 \psi}{dx^2} = - (\omega^2 \epsilon \mu) \psi \end{equation*}

This is the Helmholtz equation – which implies that for a MATTER WAVE travelling with speed = c, the wave function (\psi) is no different from the electric field intensity E.

Anuj holds professional certifications in Google Cloud, AWS as well as certifications in Docker and App Performance Tools such as New Relic. He specializes in Cloud Security, Data Encryption and Container Technologies.

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