Re: [Phys-L] matter waves

[John Denker <jsd@av8n.com>]

On 05/16/2017 05:39 PM, Carl Mungan wrote:

> For what it's worth, I have summarized the standard relations here:
> https://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/MatterWaves.pdf

That all seems correct.

Continuing down that road, let me add that you can obtain the
same results with less work and perhaps obtain some additional
insight by looking at things from the spacetime point of view.

In this case, I suggest drawing the dispersion relation ... i.e.
energy versus momentum ... aka frequency versus wavelength:
  https://www.av8n.com/physics/spacetime-welcome.htm#fig-e-px-hyperbola

In general, it is a hyperbola.  For velocities small compared to c,
it is well approximated by a parabola.

In hyperbolic trigonometry, the angle (i.e. rapidity) is proportional
to the arc length along the hyperbola.  This is profoundly analogous
to prosaic circular trigonometry, where the angle is proportional to
the arc length along the circumference of the circle.

For any state of motion, the thing we normally think of as "the"
velocity is represented by the slope of the curve, i.e. the slope
of the /tangent/ to the curve.  In any particular frame we have:

               dω
        v_g = ----
               dk

                d cosh θ
	   = c ----------
                d sinh θ

                sinh θ
	   = c --------
                cosh θ

which is well behaved at all speeds (large, small, and in between).
This is a ridiculously easy calculation.  You can do it in your head
in less time than it takes to tell about it.  No gamma factors anywhere.


We can use the same tools to examine the phase velocity.  It is the slope
of the /radius/ vector that runs from the origin to the representative
point on the hyperbola.

               ω
        v_p = ---
               k

                cosh θ
	   = c --------
                sinh θ

The radius vector is what it is.  Its slope has dimensions of velocity.
However, there is more to physics than dimensional analysis.

The radius vector has "some" physical significance; it just doesn't
correspond to what we think of as "the" velocity of the particle.