JustinLevy said:I don't understand the question. How could it possibly be invariant?
By counter example, choose a frame where a particle is moving: velocity x phase velocity is non-zero. Now choose a frame where that same particle isn't moving: velocity is zero so the product is as well.
Thanks. For you and for others interested in my thread I recall what Moller does. He starts with the phase of a plane wave propagating with phase velocity w in I and w' in I'. Among others he derives the addtion law of phase velocities and the transformation equation of the angles along which the wave propagates when detected from I and I' respectively. He has derived previously the transformation equation for the angles along which a tardyon moves with velocity u (u') and the addtion law for u and u'. The conclusion is:"A comparison of the transformation equations for the addition law of u and w respectively and of the angles along which the particle moves and the wave propagates become equal to each other respectively when we put u=cc/w and u'=cc/w'. In other words the velocity of a particle u and its direction n are transformed in the same manner as the corresponding quantities for a wave with the phase velocity w=cc/u and direction n. In his wave theory of elementary particles de Broglie made use of the circumstances by attributing to a particle with the direction of propagation n and the phase velocity w=cc/u a procedure which thus is relativistically invariant."Meir Achuz said:I haven't read Moller in years, but he must be using the classical expression
v=p/E for the "velocity of a tardyon". This gives v=k/w k and w are divided by hbar. The phase velocity of a wave is v_p=w/k, and he gets his result.
BUT, in wave mechanics, the particle's position is described by a
wave packet, whose peak moves with a group velocity, v_g=dw/dk.
This only equals k/w for a massless particle.
Moller seems to be using classical mechanics for v, and wave mechanics
for v_p.
Sorry for the question. So, in wave mechanics, it's wrong to write:Meir Achuz said:I haven't read Moller in years, but he must be using the classical expression
v=p/E for the "velocity of a tardyon". This gives v=k/w k and w are divided by hbar. The phase velocity of a wave is v_p=w/k, and he gets his result.
BUT, in wave mechanics, the particle's position is described by a
wave packet, whose peak moves with a group velocity, v_g=dw/dk.
This only equals k/w for a massless particle.
Moller seems to be using classical mechanics for v, and wave mechanics
for v_p.
Whoops, I made a silly mistake. Thank you light--> for questioning the result.Meir Achuz said:I haven't read Moller in years, but he must be using the classical expression
v=p/E for the "velocity of a tardyon". This gives v=k/w k and w are divided by hbar. The phase velocity of a wave is v_p=w/k, and he gets his result.
BUT, in wave mechanics, the particle's position is described by a
wave packet, whose peak moves with a group velocity, v_g=dw/dk.
This only equals k/w for a massless particle.
Moller seems to be using classical mechanics for v, and wave mechanics
for v_p.
Meir Achuz said:Whoops, I made a silly mistake. Thank you light--> for questioning the result.
The sentence starting with BUT (I only capitalize when I am wrong.)
should have read:
A particle's position is described by a
wave packet, whose peak moves with a group velocity, v_g=dw/dk.
This equals k/w for a particle of any mass.
Then scrap the last sentence.
The v in the expressions is v_g.
The algebra is:(d/dk)\sqrt{k^2+m^2}=k/\sqrt{k^2+m^2}.
(All hbare=1=c)
Sometimes I click submit too before I think.
I would instead write:bernhard.rothenstein said:Do you think that the following derivation holds
Using results of quantum mechanics: wavelength L =h/mv, where v is the particle velocity.Using relativity: energy E = mc^2 = hf, where f is the frequency
associated with E.
Multiply: hfL = mc^2 h/mv
fL = c^2/v
u = c^2/v, where u is the phase velocity
so uv=c^2
Note that this uses for frequency the total energy,including the rest
energy, not just the kinetic energy. And the phase velocity gets larger
as the particle velocity gets smaller!
The invariance of uxw refers to the principle that the product of the velocity (u) and the phase velocity (w) of a wave remains constant, regardless of changes in the medium or frequency of the wave.
This principle is important because it allows us to predict how a wave will behave in different media or at different frequencies. It also helps us understand the relationship between the velocity and frequency of a wave.
The invariance of uxw is a fundamental property of all types of waves, including sound waves, electromagnetic waves, and water waves. It applies to both transverse and longitudinal waves.
While the invariance of uxw is a general principle that applies to many waves, there are some cases in which it may not hold true. For example, in certain nonlinear media, the product of u and w may vary with frequency.
The invariance of uxw is closely related to the Doppler effect, which describes how the frequency of a wave changes when the source or observer is in motion. The Doppler effect does not affect the product of u and w, demonstrating the invariance of uxw.