Today’s lecture topic was position-space and momentum-space representations of state vectors in quantum mechanics, which once again brought up one of the eternal questions in physics:
Why do we use the symbol p to represent momentum?
I did Google this, but none of the answers looked all that authoritative. And, anyway, I’m sure that the readers of this blog can come up with fanciful speculations that would be far more interesting than the real answer. So, have at it.
(I’m too overextended and exhausted to post anything more substantive right now; maybe later this afternoon.)
When I was taking my first statistical mechanics course, the professor told us, “We call the partition function Z, because Boltzmann spoke German.” I always felt that he skipped a step in that derivation.
My first guess on scientific abbreviations that don’t make sense in English is always that it makes sense in German. Or maybe French. But no, I’ve tried German, French, and Latin in Google Translate with nothing that suggests the word could be from one of those.
If I’m remembering my history of science classes correctly, Descartes actually invented the concept, calling it (in English translation) “quantity of motion” or somesuch. [update: Google is now telling me that Newton also used this phrase.] I can’t remember from our reading of Principia if Newton actually used a symbol for momentum, or just wrote out the word. Unfortunately, my print-outs from that class are in a box in my attic, and I’m too lazy to try to find it online.
mph,
The partition function is one of the things that cause me to always think of German; the German word for partition function is Zustandssumme. (summe is summation or sum; I forget what zustand is exactly, but it’s something like everything or all of it.)
Perhaps it comes from the Hamiltonian formalism, where the variables p and q stand for generalised momentum and position respectively?
On these, while I’m not entirely certain whether the line of descent is exact, Hamilton used the variables p and q in his first paper on the theory of systems of rays in 1828. You can find the paper here:
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Optics.html
With the variables p and q introduced on page 26.
The variables seem to picked in essentially lexicographical order, in a set of 5 variables p, q, r, s, t. Why these letters were picked is unknown. There is talk of a “pencil” of rays, but I don’t think that’s related. (Which isn’t to say that Hamilton didn’t pick p and q deliberately, as they turn out later to be important, p,q, being easier to write I suppose)
As I say, I’m not entirely sure about the connection between Hamilton’s original p,q in an optical system, and the p,q of the later Hamiltonian formalism. But, that’s the extent of my guess anyway.
A little poking around indicated that perhaps Euler was the first to express Newton’s laws mathematically (the Principia states them in language).
Looking at a translation of Euler’s work Mechanica (http://www.17centurymaths.com/contents/mechanica1.html), which is written in Latin, he uses ‘potentia’ to refer to force and uses the symbol p throughout. Not sure where we switched from p=force to F=force and p=momentum, but I’m guessing that this could be the origin.
In this case it’s definitely not German. English to German translation of “momentum” gives several possibilities, none of which start with P; the one that seems to correspond most closely to the physics meaning is Impuls (which is obviously cognate to the English “impulse”).
Google offers several options among the first page of hits, with no two sources agreeing on an explanation and several admitting that the explanation is unknown. Some of them claim a Latin origin, but they differ on which Latin word is the right one. One of the apocryphal explanations offered is that p and q are mirror images intended to convey the notion of action and reaction–the problem is that I have never seen q used as a momentum variable, although (and I stress this is apocryphal) it might plausibly have motivated the choice of q as the canonical position variable in Hamiltonian mechanics, p having been established by then as representing momentum.
Colin @3: Zustand means “state”; thus Zustandsumme is literally a state summation, which is a more useful description than the term “partition function” of what it is.
OK, here is an explanation, totally unfettered by any inconvenient facts:
Obvious symbol, “m”, already used by mass (which would make for a pretty silly-looking equation). Hey, let’s just take the next letter! No, “n” is no good, that gets used for everything. Can’t use “o”, that could be too easily mistaken for zero. How about “p”? It doesn’t look like it is being used for anything that is likely to be in the same equation as momentum. OK! “p” it is!
I have worked for engineers for years. tim eisele’s explanation, while probably not THE explanation, is a good, solid, familiar engineer sort of train of thought. I bet he’s closer to being right than anyone. 🙂
Of course p looks similar enough to the Greek lower-case letter rho (Ï) that it might be worth considering this as a possible origin?
I was once informed that the use of H for enthalpy is actually from the Greek upper-case letter Eta (Î), whether this is correct or not it does seem to make some kind of sense.
I thought momentum was actually represented as rho.
#7:
Russian and German use the word ‘Impuls’/’impulse’ for ‘momentum’. So the letter ‘p’ is fairly logical (‘i’ is used for imaginary numbers, ‘m’ is used for mass).
What about the cartoon that shows Einstein standing before a blackboard scratching his head?
He’s written E = m a2 and crossed it out.
He’s written E – m b2 and crossed that out.
He’s got the chalk in his hand and he’s about to write the next equation and make history!
It may come from projectile, because the concept of momentum was developed in that context.
http://en.wikipedia.org/wiki/Theory_of_impetus
BTW, that Jean Buridan was quite a character…
I found this “best answer” on Yahoo answers at http://answers.yahoo.com/question/index?qid=20100204152900AA701u6
It is similar to a comment above, but with an added specific phrase:
Best Answer – Chosen by Asker
No one is absolutely sure.
It probably dates from the time of Newton, when most people
referred to movement in terms of “impetus”, which shares a
Latin root “pellere”, along with impulsus and impeller. All meaning
to push or impell. Hence p for pellere.
As suggested @4, the starting point in any such analysis should be to find when it was actually used to represent mv. I believe Hamilton’s work is the answer to that question.
It certainly is not anywhere in Newton’s work.
I’ll observe that answers.yahoo does not give a citation for where “p” is found used as an abbreviation from “the time of Newton”. Book and page, please? I’ve never run across any use of “p” for momentum that pre-dates Hamilton.
As for speculation, you see r used for position and s for distance and arc length in old textbooks and other work that predates Hamilton. Those were well established as standard variables, as were x, y, and z. That makes Hamilton’s decision to use “q” as a generalized position variable — one that can represent either r or theta — to be plausibly based on its lexical order and the fact that q just doesn’t get used for anything in mechanics. Choosing an adjacent letter for the conjugate variable makes sense just like x,y,z or Maxwell’s use of triplets of letters to represent the three vector components of each field.
The fact that “p” also appears in impulse and impetus probably sealed the deal for him.
Easier challenge:
When was “p” first used to represent momentum in a physics textbook, and when did it first appear in an introductory level physics text book?
This would be a good exercise for your students if they have never searched for something in a college library that has a vast collection of old books made of paper, particularly if your department has a collection of out-of-date textbooks that go back a century or more.
I believe the answer is circa 1960, in the post-sputnik revolution in science education, but I have never looked beyond the fact that it does not appear in a physics book used at a large university circa 1940 or other old textbooks I have collected or read. They use mv.
Wide use of a symbol for momentum appears to be quite new. Momentum doesn’t even get its own unit.
Thanks to the link @4, which led to
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/
I see that Hamilton used eta for the generalized position in his second essay (linked from above) and some strange character for the generalized momentum. He does, however, use p for the initial values of the generalized momentum (see just below equation A on page 5) and the integrals that show up later in his treatment of perturbation theory.
So there is a hint of it in Hamilton’s work, but since I don’t even know if Lagrange used q for the generalized coordinate, maybe the answer IS in a textbook where someone else rewrote all of it in a consistent notation.