Is propagation speed depending of wavelength? (QM)

Question

In case of a single free particle, no potential field, the wave function is:
$$ Psi = A e^{i(kx-omega t)}. $$
As it must fulfill the Schrödinger equation
$$ - frac{hbar^2}{2m} nabla^2  psi(mathbf{r}, t) = ihbarfrac{partial}{partial t} psi (mathbf{r}, t) $$
it follows the restriction:
$$ frac{hbar^2 k^2}{2m} =hbar omega. $$
With this restriction the propagation speed $v$ became:
$$ v = frac{w}{k} = frac{h}{2mlambda} $$
that depends in wavelength, being longer wavelengths slower.
Is this a real conclusion or I made some error in the derivation? Until now, always I've see propagation speed as a property of the medium only.

mike stone · Answer

There's no meaning to the speed $omega/k$ for the Schroedinger equation, as you can add a constant potential without changing anything physical. What matters in the group velocity
$$
v_{rm group} =frac{partial omega}{partial k}= frac{hbar k}{m} = frac{p}{m}
$$
This is the speed at which a wavepacket travels, is independendent of any added constant potential, and as the classical momentum is $p=mv$, it is the speed at which the classical particle would move.

Is propagation speed depending of wavelength? (QM)

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