What Is the Doppler Effect in Matter Waves?

[teros]

If I have a matter wave (for example, electron waves in a electronic microscope) at a given wavelength λ, and I move with respect (towards) them at speed $v$, I will measure a Doppler shift in the wave given by:

$\frac{1}{{\lambda '}} = \frac{1}{{\lambda '}}\left( {1 + \frac{v}{{v_e }}} \right)$

where $v_e$ is the velocity of the electron wave. But the phase velocity? or the group velocity?

I know that $v_{ph} v_{gr} = c^2$, so the result can be very different.

 

[mfb]

I think you can calculate the relative velocity of the particle (with the group velocity and your own velocity), and work back to the wavelength and phase velocity afterwards.
The formula might look different - it is not a classical wave.

 

[teros]

But for $v<<<v_e$ it cannot be much different...

 

[teros]

Ok, let us suppose that, instead of matter waves, I speak of sound in a medium where phase and group velocities are different. Which one will go in the Doppler formula?

 

[mfb]

For sound, I would measure the wavelength relative to the medium, independent of the observer. To calculate the frequency, I would use that invariant wavelength and velocity addition with the phase velocity.

 

[Enthalpy]

The product equals c*c for electromagnetic waves.

 

[mfb]

The product equals c*c for electromagnetic waves.

In vacuum. In matter, both phase and group velocity can be below c, for example, therefore the product has to be smaller, too.