When the Phase Velocity = c, the wave function becomes the ELECTRIC field

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.

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