Is the Phase Velocity of a Massive Particle Representing Anything Physical?

[LarryS]

Given a massive particle traveling freely. Also assume that it is in a momentum eigenstate - a pure unmodulated sine wave in position space. In non-relativistic quantum mechanics, the phase velocity for that particle would be greater than the velocity of light.

Does the phase velocity for such a particle represent anything physical?

Thank you in advance.

 

[Baluncore]

Does the phase velocity for such a particle represent anything physical?

There can be no dispersion with a pure sine wave so the existence of a phase velocity requires that the wave interact with something else in space. If so, the place and time of those interactions must have a phase velocity ≥ the wave velocity.

 

[LarryS]

There can be no dispersion with a pure sine wave so the existence of a phase velocity requires that the wave interact with something else in space. If so, the place and time of those interactions must have a phase velocity ≥ the wave velocity.

The pure sine wave has no dispersion but it already has, before it interacts with anything else, a phase velocity which is, by definition, the frequency times the wavelength. True?

 

[vanhees71]

Given a massive particle traveling freely. Also assume that it is in a momentum eigenstate - a pure unmodulated sine wave in position space. In non-relativistic quantum mechanics, the phase velocity for that particle would be greater than the velocity of light.

Does the phase velocity for such a particle represent anything physical?

Thank you in advance.

A particle cannot be in a momentum eigenstate since the corresponding wave function ($\hbar=1$),
$$u_{\vec{p}}(\vec{x})=\frac{1}{(2 \pi)^{3/2}} \exp(\mathrm{i} \vec{p} \cdot \vec{x}),$$
is not square integrable.

There is nothing in non-relativistic or relativistic physics that prevents phase or group velocities to be greater than the speed of light. This was puzzling the experimental physicists already in the early days of relativity theory concerning anomalous dispersion of classical electromagnetic waves, where both the phase and group velocities can get $>c$. The answer was given by Sommerfeld in 1907 and worked out later by Sommerfeld and Brillouin in 1913: Only wave fronts of fields with compact support must move with speeds $\leq c$ in order not to violate relativistic causality. All other kinds of wave velocities can exceed the speed of light without trouble.

"Schrödinger waves", of course, may not obey the "speed limit" for the front velocity, because it's a Galilei-covariant but not a Lorentz-covariant model. It's an approximation with a limited range of applicability (as Newtonian mechanics has a range of limited applicability compared to relativistic mechanics).

 

[LarryS]

A particle cannot be in a momentum eigenstate since the corresponding wave function ($\hbar=1$),
$$u_{\vec{p}}(\vec{x})=\frac{1}{(2 \pi)^{3/2}} \exp(\mathrm{i} \vec{p} \cdot \vec{x}),$$
is not square integrable.

There is nothing in non-relativistic or relativistic physics that prevents phase or group velocities to be greater than the speed of light. This was puzzling the experimental physicists already in the early days of relativity theory concerning anomalous dispersion of classical electromagnetic waves, where both the phase and group velocities can get $>c$. The answer was given by Sommerfeld in 1907 and worked out later by Sommerfeld and Brillouin in 1913: Only wave fronts of fields with compact support must move with speeds $\leq c$ in order not to violate relativistic causality. All other kinds of wave velocities can exceed the speed of light without trouble.

"Schrödinger waves", of course, may not obey the "speed limit" for the front velocity, because it's a Galilei-covariant but not a Lorentz-covariant model. It's an approximation with a limited range of applicability (as Newtonian mechanics has a range of limited applicability compared to relativistic mechanics).

I think that it's interesting to look at the phase/group velocity subject from an information transfer perspective. Each unmodulated sine wave carries no information and therefore can travel faster than light (phase velocity). Combine an infinite number of these sine waves choosing the right frequencies and magnitudes and you have the wave packet of a massive particle which carries information (and energy) and therefore travels slower than light (group velocity).

 

[Baluncore]

Each unmodulated sine wave carries no information and therefore can travel faster than light (phase velocity).

When it comes to communications, a sine wave travels at the group velocity which is the speed of light. Energy and information travel in the direction of the poynting vector. It is not possible to transmit information sideways along the wavefront at greater than the group velocity.

The phase velocity is the velocity of coincident interactions with other waves or boundaries. For example a water wave hitting a sea wall may arrive from an angle close to the perpendicular and so generate a splash that travels at a speed very much higher than the water wave. The speed of the splash events along the wall is the phase velocity and will be greater than or equal to the group velocity.

 

[LarryS]

A particle cannot be in a momentum eigenstate since the corresponding wave function ($\hbar=1$),
$$u_{\vec{p}}(\vec{x})=\frac{1}{(2 \pi)^{3/2}} \exp(\mathrm{i} \vec{p} \cdot \vec{x}),$$
is not square integrable.

There is nothing in non-relativistic or relativistic physics that prevents phase or group velocities to be greater than the speed of light. This was puzzling the experimental physicists already in the early days of relativity theory concerning anomalous dispersion of classical electromagnetic waves, where both the phase and group velocities can get $>c$. The answer was given by Sommerfeld in 1907 and worked out later by Sommerfeld and Brillouin in 1913: Only wave fronts of fields with compact support must move with speeds $\leq c$ in order not to violate relativistic causality. All other kinds of wave velocities can exceed the speed of light without trouble.

"Schrödinger waves", of course, may not obey the "speed limit" for the front velocity, because it's a Galilei-covariant but not a Lorentz-covariant model. It's an approximation with a limited range of applicability (as Newtonian mechanics has a range of limited applicability compared to relativistic mechanics).

Physically, a wave that is only nonzero on a closed and bounded subset (compact support) is, IMO, not a wave at all, but a particle.

 

[LarryS]

The answer was given by Sommerfeld in 1907 and worked out later by Sommerfeld and Brillouin in 1913: Only wave fronts of fields with compact support must move with speeds $\leq c$ in order not to violate relativistic causality. All other kinds of wave velocities can exceed the speed of light without trouble.

So far, I have been unable to find the paper, etc. in which Summerfeld and Brillouin worked this out. Was it in one of Summerfeld's books?

 

[vanhees71]

In Sommerfeld's book (vol. 4 of his Lectures on Theoretical Physics) you can find a short version of the calculation. The original papers are

L. Brillouin, Über die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys. (Leipzig) 44, 203 (1914).
http://onlinelibrary.wiley.com/doi/10.1002/andp.19143491003/full

A. Sommerfeld, Über die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys. (Leipzig) 44, 177 (1914).
http://onlinelibrary.wiley.com/doi/10.1002/andp.19143491002/full

 

[LarryS]

A. Sommerfeld, Über die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys. (Leipzig) 44, 177 (1914).
http://onlinelibrary.wiley.com/doi/10.1002/andp.19143491002/full

Do you happen to know if an English translation of this paper exists?

 

[vanhees71]

No, not of the paper, but as I said you find the calculation also in Sommerfeld, Lectures on Theoretical Physics, vol. 4 (optics).

 

[LarryS]

No, not of the paper, but as I said you find the calculation also in Sommerfeld, Lectures on Theoretical Physics, vol. 4 (optics).

OK, thanks.