The discussion centers on the relationships governing matter waves, specifically the equations λ = h / p and E = h f, where E = mc². It establishes that the phase velocity vph = c² / v, with v being the group velocity, leads to infinite wavelength and phase velocity when considering a stationary particle as v approaches zero. The conversation also clarifies the distinction between non-relativistic and relativistic quantum mechanics, emphasizing that the inclusion of rest mass energy alters the frequency but not the de Broglie wavelength, thus maintaining consistent physics across models.
PREREQUISITESPhysicists, quantum mechanics students, and researchers interested in the foundational concepts of matter waves and their implications in both non-relativistic and relativistic frameworks.
I think you are confused going between non-relativistic "energy", which is all kinetic, and relativistic "energy", which includes mass-energy. These are not the same quantities. One is not a low-speed approximation of the other.Philip Koeck said:Are the equations on Wikipedia right?
Post 25 in this thread suggests a different expression connecting E and f.
Yes, I know.PeroK said:I think you are confused going between non-relativistic "energy", which is all kinetic, and relativistic "energy", which includes mass-energy. These are not the same quantities. One is not a low-speed approximation of the other.
This must be correct.Philip Koeck said:Yes, I know.
According to the Wikipedia text E = h f, where E is the total relativistic energy, including the rest mass term.
I don't see that in post #24.Philip Koeck said:According to post 24 the relationship is E - m0c2= h f, if I'm not mistaken.
It's the total relativistic energy. It must be.Philip Koeck said:I'm simply wondering which of the two is correct for relativistic velocities.
In post 24 it says vΦ= ω/k, and then ω is replaced by E - mc2, as I read it. (The h cancels or is set to 1, c is set to 1).PeroK said:I don't see that in post #24.
Okay, I'll let @ergospherical explain that!Philip Koeck said:In post 24 it says vΦ= ω/k, and then ω is replaced by E - mc2, as I read it. (The h cancels or is set to 1, c is set to 1).
Yes, I'm getting that impression.ergospherical said:Particles are not waves and you should not confuse their equations - phase velocity isn't so meaningful for particles.
If it's enough of a problem that its effect on your answer matters, then obviously you are dealing with a case where the non-relativistic approximation won't work for you and you need to use the relativistic equations.Philip Koeck said:At least to me that leads to a problem.
In relativity rest mass is part of the total energy. So it has to be included.Philip Koeck said:I'm simply wondering which of the two is correct for relativistic velocities.
Could you share a source for the relationships you use in the first line?ergospherical said:In the relativistic region the phase velocity is ##v_{\phi} = \omega/k = (E-m)/p##, i.e.\begin{align*}
v_{\phi} = \frac{\gamma - 1}{\gamma v}
\end{align*}For small ##v## expand ##\gamma = (1-v^2)^{-1/2} \sim 1 + v^2/2## (and similarly ##1/\gamma \sim 1-v^2/2##) to get\begin{align*}
v_{\phi} &\sim \frac{(v^2/2)}{v}(1-v^2/2) \sim \frac{v}{2}
\end{align*}omitting terms ##O(v^3)##.
Don't do that. It kinda sort of works if you're just doing a semester or so of special relativity with classical objects (but even that is discouraged these days - why start with something that has to be unlearned to move beyond that first semester or so?) but fails dismally as soon as you start thinking quantum mechanics.Philip Koeck said:When I write m I mean the relativistic mass, not the rest mass.