We all know the speed of light has a absolute value in vacuum measured by all observers from various reference frames and itis the fastest speed in the universe. My question here is, if there is a limitation on how fast matter would travel, is there also a limitation on how slow matter would travel? And are there limitations to all physical quantities such as the smallest particles or heaviest particles?
Since the speed of material particles is relative, it is hard to answer your first question. In other words two things moving at the same velocity are moving at 0 speed relative to each other. Smallest particles question is probably answered by electron neutrinoes. The only things smaller are zero mass things - photons, gluons, and gravitons. Heaviest is a different question. Do you mean fundamental particles (top quark is heaviest), or are you asking about anything - like the entire universe?
Well, that's not really true. Quantum mechanically, there is always the zero-point motion... [tex]E_n=\hbar\omega(n+\frac12)[/tex] To me, that would constitute the 'minimal' velocity. So, near 0 K when two particles move at the same speed, this will result in a minimal velocity difference. Wouldn't you agree?
On other hand, we know that the measure in the space of quantum trajectories concentrates in the continous but no differentiable ones. So the maximum velocity given by relativity is c, and the minimum velocity given by quantum mechanics is, err.. , infinity. Actually I believe the same calculation for relativistic quantum mechanics (an approximate, non existent theory) is intended to give c instead of infinity, so no inconsistency here. In any case, what happens is that forward and backward randomness compensate, and you get a decent finite averaged velocity in the direction you expected to go.
relativistic quantum mechanics (an approximate, non existent theory) Approximate, yes. Non existence (or existence) has not been shown.
[tex]E_n=\hbar\omega(n+\frac12)[/tex] This is for the QHO (quantum harmonic oscillator). You should review how the spectrum of the momentum operator for a QHO is obtained. In the classical case of a mass on the end of a spring, when all the energy is potential energy, the speed of the mass is zero. How does this carry over to the QHO?
Yes, I know. I only used this as a simple example that nearly everybody has seen before: even at the lowest energy, the QHO is not without movement. The zero-point motion is always there. To me, this constitutes in some sense the 'minimal' velocity obtainable. However, the arguments put forward by other replies in this thread made me reconsider (see below). Not easily... In the case of classical mechanics, I agree fully that the 'minimal' velocity obtainable is equal to 0. However, quantum mechanically this is not so simple... As stated in earlier replies, the speed associated with this zero-point motion is either [itex]\infty[/itex] or c, depending on weather you are using the classical or the relativistic version of QM. Now that I thought about it, I think that this is most probably not a usefull definition of the 'minimal' velocity obtainable. So, I'd say that the 'minimal' velocity obtainable from a quantum point of view is just ill defined (as is time in general in classical QM). That leaves me only with the definition from classical (relativistic) mechanics: the 'minimal' velocity obtainable is equal to 0.
Excuse my ignorance, but can anyone please explain to me what QHO and what is the meaning of the equation [tex]E_n=\hbar\omega(n+\frac12)[/tex]?