Momentum vs Velocity
A physicist acquaintance recently wrote the following in an email to myself and a few others:
I’m embarrassed to reveal my ignorance of elementary quantum mechanics, but I hope one of you won’t mind filling this small gap in my knowledge: when considering a single particle of mass m, is there any inherent difficulty with defining a velocity operator by simply dividing the momentum operator by m?
Most people sent in their replies within a day or two, but I spent a while chewing over the question because (in all honesty) it was something that had puzzled me as a quantum physics undergraduate, and it occurred to me now that I had simply forgotten my initial feeling of puzzlement and had never found the time in my busy academic life to sit down and work out an answer for myself.
You see, in Newtonian mechanics, velocity is usually introduced before momentum, as it is something we can readily see with our own eyes: we can much more easily judge the velocity of a moving object relative to ourselves than we can its momentum. That's because the velocity depends purely on the path the object takes as it moves through space, whereas the momentum depends also on its mass, and we cannot easily intuit the mass an object just by looking at it. Specifically, the momentum is related to velocity by the deceptively simple equation p = mv.
So, during my introductory courses on quantum mechanics as an undergraduate, I was puzzled when the emphasis suddenly shifted to talking about the momentum almost exclusively. To some extent this could be explained away by the emphasis on Hamiltonian mechanics in which the momentum rather than the velocity is more prominent, however, what really deepened the mystery was that in quantum mechanics of (say) a single particle moving in a potential, it actually seemed unclear how to define the velocity. Unlike momentum, which one could always safely discuss, it seemed velocity had mysteriously become persona non grata at some point during the move from classical to quantum mechanics.
But, I wondered, why should that be the case? For a single non-relativistic particle, the mass is perfectly well-defined and constant, so shouldn't that old chestnut "p = mv" still work? This was the thought that puzzled me then, until I became distracted by supposedly bigger problems; but the email from my colleague some 15 years later dredged up that old feeling of confusion as an undergraduate and brought it back to the forefront of my mind, showing me that I wasn't the only one to have briefly stumbled over this particular crevice in the curriculum.
I thought: if with all my training as a physicist I still can't come up with a good answer, then it is either time to quit physics, or else publish an article about it!
Well, after thinking about it on and off for just over a week, I send my reply to my colleagues, and it was so well received that I thought I ought to put it in my blog so that others who might have hit upon the same puzzle in their physics journey might find some solace here. The following text is my response:
I think what it boils down to is that momentum and velocity are conceptually very different things. The fact that they are connected by the simple and direct relation p = mv in nonrelativistic classical mechanics of particles is just a happy accident of that particular domain.
Conceptually, momentum is the canonical conjugate of "generalized position" (ie of the system's configuration) and as such it is well-defined in any Hamitonian theory. Velocity, on the other hand, refers to how the physical position of some more-or-less localized entity is changing in time. Which is to say: unlike momentum, which is always well-defined, velocity is only well-defined when you have something like a continuous and not too choppy trajectory.
In the case of non-relativistic QM of a single particle, you can define concepts that more-or-less correspond to velocity, under restricted conditions, but generally speaking there is no concept that neatly fits with what we want "velocity" to mean, because in QM there is not a well-defined trajectory of objects through space. The problem is that the wavefunction does not in general remain localized over time, so you have to decide on a rule for mapping it to a "position" at any given moment, in order to define a trajectory.
On one hand, if you rely on actually measuring its position at some sequence of times, you run into the problem pointed out in another reply from a colleague, who supplied the following quotation from Bohr:
Indeed, the position of an individual at two given moments can be measured with any desired degree of accuracy; but if, from such measurements, we would calculate the velocity of the individual in the ordinary way, it must be clearly realized that we are dealing with an abstraction, from which no unambiguous information concerning the previous or future behaviour of the individual can be obtained.
(Incidentally, an alternative proposal would be to measure the particle's momentum at two different times, take the average, and divide by m. Obviously this is also problematic for the same reason as averaging the position measurements over time; moreover the two methods would not necessarily agree!)
On the other hand, if you don't measure the particle, you have a spatially extended field, so it is ambiguous how to assign a "position" to it, especially if it is not sharply peaked at one place. And presuming you settle on a best way of doing that -- perhaps even treating it as akin to a mass distribution in space and defining its corresponding velocity distribution -- you run into the problem that as soon as a measurement is performed, it dramatically changes the wavefunction in a way that no physical field could possibly change, causing your "velocity" to behave in a really quite uncivilized way.
In contrast, momentum suffers none of these problems because, unlike velocity, its definition comes from "on high", bestowed by the Hamiltonian Gods as it were, and it floats freely of any concept of trajectory or spatial localization. (end of reply)
At least for me, that settles it pretty well. I hope you found it helpful.