# Why phase velocity has no physical significance for a matter wave?

The following is quoted from my book:

"The wavelength of a matter wave given by $$λ= \frac{h}{p}$$ has physical significance; its phase velocity $$v_{p}$$ has no physical significance. However, the group velocity of the matter wave is physically meaningful and equals the velocity of the particle."

But why the phase velocity of a matter wave has no physical significance. Can someone please explain? I am so confused. Please help.

• This is a fairly common question and already has answers here: physics.stackexchange.com/q/480700/264848. Commented May 27, 2021 at 5:14
• Commented May 27, 2021 at 6:20

The de Broglie relation you are discussing belongs to the first studies that led to the theory of quantum mechanics. Now it is a useful approximation for the complexity of the final theory.

The formula, used for a massive particle, and an experiment with a collection of this particle gives the wave behavior which is predicted by quantum mechanics for such ensembles. But the quantum mechanical calculations are not for one particle: The probability of finding a particle in an experiment with the same boundary conditions, is given by $$Ψ^*Ψ$$ where $$Ψ$$ is the quantum mechanical solution of the problem. This wave function is a main postulate of quantum mechanics.

As any phases in this mathematical formulation have to do with the probability, the phases in $$Ψ$$ are not connected to the particle's phase velocity in space. For a free particle the wave function is a plane wave:

$$\Psi(x,t)=Ae^{ikx-i\omega t}$$

The velocity is there, but the phase in the expression only affects the probability of measurement of the particle. More on the wavefunction and its connection with the de Broglie relation and Heisenberg uncertainty, here.