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Units, Units, Units *October 23, 2009*

*Posted by gordonwatts in physics, physics life.*

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Undergraduates know that Physics Professors get all wound up about units. We can’t help ourselves.

But in reading a nytimes article this morning I couldn’t help myself:

In addition, Mr. Holder said, the authorities have seized more than $32 million in American currency, 2,700 pounds of methamphetamine, 4,400 pounds of cocaine, 16,000 pounds of marijuana and 29 pounds of heroin. More arrests are expected.

Well… this is what happens when you wait until the evening to write a blog post you spotted in the morning – they change the article. That 2700 pounds? It was 2700 kilograms (which is significantly more). In short – they had mixed kilograms and pounds. I was going to get on my high horse and… well, seems someone at the times is as sensitive about this as us physicists are.

But it also occured to me that the notion of units is rather flexible. For example, when we do particle physics calculations we often set the speed of light to 1. Normally it is 300000000 meters/second (really fast!). Seriously. We just set it to 1. We are so annoyed by having to carry around that number in our calculations that we just up and set it to one. We do that with an other constant as well (called h-bar). Your unit system ends up being very weird when you do that:

Normal Every Day Units Units in h-bar = c = 1 Energy Energy Time 1/Energy Mass Energy Length 1/Energy

I know this seems weird – but you see it all the time. This is just like making the following unit conversion in the list of drugs: instead of telling us the number of pounds or kilograms, tell us how much pot they got in terms of its street value. And to tell the truth, that would have been a very useful number to have in that article.

Heck, in the old days, the unit of measure in the market was the length of the king’s forearm. When the king changed, the whole country would change its unit system…

Un physics professors getting wound up with units is ironic – we don’t really use them that heavily when we get to more advanced calculations. On the other hand, we can only drop them because we have already learned how to use them. At least, that is what we tell ourselves and everyone else! 😉

I once had a professor who told us to set c = h_bar = alpha = pi = 1.

See an issue with that?

Definately with the pi=1 – with hbar and c you can figure out what you are missing by doing unit analysis – that allows you to get back to the proper number. Pi is unit-less, so you can’t tell how many of them you dropped.

So, if you are doing a-b and a and b have different numbers of pi’s you’ll get a bad difference – this is in particular true if you are plotting a(theta)-b(theta) as a function of theta – you may well get the wrong shape there.

Hah, couldn’t get this out of your mind even after class, huh?

When I first took liner algebra back when I was in high school, my professor told me “it’s commonly said that every math problem is a linear algebra problem.” And the more physics I learn the more I realize that that’s true for physics, too!

An interesting thing to note (well, for theorists) is that the reason we can do this is that the units are really more than just labels. You can think of a similar case when we do the same thing you know backwards-and-forwards. When you want to write a wavefuncion in terms of one basis instead of another. You have a vector |psi> and you want to do a unitary transformation U|psi> to preserve probability and change basis.

But this is just what picking units is! You’ve really got a vector space built out of “basis vectors” that are your units. Changing units is just an “isomorphism” on this vector space, and picking natural units is just a “natural” isomorphism.

Say, for example, you are measuring energy. You have a “units of energy vector space”

[E] = [kg] (x) [m^2] (x) [s^-2]

where (x) is the tensor product of the vector spaces [kg], [m^2] = [m](x)[m], and [s^-2] = [s^-1](x)[s^-1].

So if in this system I measure the energy to be E = 1 J, I am really saying

E = 1 * (1,1,1)

and if I measure a mass m=1kg, I really have

m = 1 * (1,0,0)

Picking natural units says to change basis using the fact that we have cannonical isomorphisms given by c and hbar. For example,

c : [m] -> [s]

identifies the meter and second vector spaces with each other!

and

hbar : [J] -> [s^-1]

so we identify all of the vector spaces together, so now,

[E] = [kg] (x) [m^2] (x) [s^-2] = V (x) V (x) V

[d] = [kg] = V

where V is whatever unit vector space we’d like to use (e.g., V=[J] or [s] or whatever). And what’s happened? We’ve made all of our measurements diagonal in this new vector space. So all of our measurements are now just some number times the identity matrix. So E = E’ * I_{3×3}, and d=d’ * I_{1×1}.

We’ve done a “change of basis” that’s taken us from one complicated vector space to another vector space that looks simple. Just like we can take |psi> from the “complicated” position basis to the “simple” momentum one!

And it shouldn’t worry you that we’ve seemingly changed dimensions from a 3-d vector space to a 1-d one. This happens because we’ve used more than just vector space structure. A similar kind of thing happens when we write a wavefunction in the (say) countable (or even finite) energy basis v.s. the uncountable momentum or position one.

Excellent point, Jeremy – that is a very good way to look at it. It also helps one sort out how many changes you can make before you start loosing information (i.e. the pi=1 comment above).