Why does dimensional analysis to find a characteristic length, time, etc work?

Question

For example, let's say you have an equation with planck's constant h, some mass m, and some velocity v. When you say "the characteristic length of the system is defined as h,m, and v to some powers that dimensionally result in a length," why does that length actually mean something and why isn't it just some random, unrelated length?

A particular example: In a lecture (video) I watched on quantum mechanics, the wave function of a quantum harmonic oscillator is being calculated.

At 29:40, he finds a characteristic energy and length of the system. He says the characteristic energy is pretty close to the ground state energy (double the actual value) and at 32:05 he says that the maximum displacement of the oscillator is probably about the characteristic length. Why does this work in general? It seems so arbitrary.

Also, how do you know what a characteristic quantity will represent in the system before doing any other calculations?

If it's relevant, I am good with calculus and differential equations, but I have only taken high school physics which didn't use calculus and every answer was always precisely calculated from formulas, we never did any characteristic quantities of systems.

Answer 1

When you say "the characteristic length of the system is defined as h,m, and v to some powers that dimensionally result in a length," why does that length actually mean something and why isn't it just some random, unrelated length?

In fact it does not really mean anything in the proper sense, but let us look at an example: take a simple pendulum of length $L$ and let us assume that, under the action gravity $g$, we want to calculate the period of its oscillations. The only parameters that can constitute an equation are, thus, $L$ and $g$ and the only combination thereof that results in a unit of time is $$ \sqrt{\frac{L}{g}} $$ as such, the period of a pendulum must be some coefficient times the above expression. Obviously, if you take the above per sé, it does not mean anything but eventually you find out that the proper expression is$^1$ $$ 2\pi \sqrt{\frac{L}{g}}. $$ Likewise, for more complicated matters, one can "cook" together the characteristic parameters appearing in the theory to have an "a priori" sense of what the proper scale may be: eventually, it will just be a matter of some numerical coefficients to multiply it by.

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$^1$ The above was actually an entry question for general physics at the famous "Scuola Normale Superiore di Pisa" in Italy, where the candidate was asked to choose what the "most feasible" expression for the period of oscillation can be (had they not known the correct formula) between $ 2\pi \sqrt{\frac{g}{L}}$ and $\sqrt{\frac{L}{g}}$: the former has the correct coefficient $2\pi$ but swaps $L$ and $g$ (making it, thus, not a unit of time); the latter did not have the right coefficient, but still actually it could by all means be it, if you have to guess. The ones who chose the former were addressed in the answer as "not having the proper mindset for physics" (or something along those lines).

Answer 2

Honestly, it is because the chosen parameters are not randomly chosen; they are chosen based on experience with the dynamic system in consideration and may depend on the regime one is considering.

It takes time to get a feel for the physics behind certain situations but after a while you get familiar with the different phenomena which drive the physical system and allow you to guess how certain aspects of it like the typical length or time scales are related to parameters which characterize the system.

Sometimes we encounter problems which are not analytically solvable; in such cases it is useful to do a dimensional analysis to get a feel for the system. But then how do you chose the correct parameters? You could of course list all possible parameters but some may not be relevant. Instead it could be useful to make a simplifying (but not entirely correct) assumption which simplifies the problem to allow for an analytic solution. Having gotten some characteristic times and length scales from the simplified solution, it may then be extended to do the dimensional analysis for the more complex problem.

Answer 3

When you say "the characteristic length of the system is defined as h,m, and v to some powers that dimensionally result in a length," why does that length actually mean something and why isn't it just some random, unrelated length?

In general, no, the particular quantity you get by manipulating the variables to get a length doesn't mean anything. But what this quantity does for you is give you a sense of scale for what you're dealing with.

To give you a different example, think about driving in a car. On the road, there is a speed limit, giving you the maximum speed you are legally allowed to drive. When driving, you also usually have a destination in mind. The distance to your destination is another meaningful quantity for the problem. If you want to know how long it will take to drive there, you need to take a lot of different things into account: do you drive above or below the speed limit, how many red lights will you encounter, how much traffic is there. But if you just divide the driving distance by the speed limit, you have a really quick idea of the kinds of time scales you're dealing with. If the speed limit is 70 mph and the destination is 200 miles away, there is a characteristic time of about 3 hours. That doesn't mean that your drive will be 3 hours, but it definitely won't be 10 minutes or 1 year.

Why does this work in general?

It works in general because most systems only have one mass scale, one length scale, one velocity scale, etc. With a harmonic oscillator, there is only one object moving, it's maximum distance doesn't vastly change over time, it only has one mass, etc. There are certainly systems where that is not the case. One example is light diffracting through a grating. The grating must be on a similar length scale as the wavelength of the light (measured in 100s of nanometers), but once it passes the diffraction grating, in order to measure the direction of the light, it has to travel over meters scale distances. But even then, the meters scale can be abstracted away by instead talking about the angle of the diffraction instead of the distance.

Also, how do you know what a characteristic quantity will represent in the system before doing any other calculations?

A characteristic quantity only represents a scale typical of the problem. All other meaningful quantities of the same dimensionality will in some way be related to that characteristic scale. This will either be by some numerical factor, or because the meaningful quantities will be calculated in a similar way. In the quantum harmonic oscillator example, the characteristic energy is $\hbar \omega$ while the ground state energy is $\frac{1}{2}\hbar\omega$, a numerical factor of $1/2$. In the quantum harmonic oscillator case, the characteristic energy is physically meaningful on in its own way because it is the energy difference between adjacent energy levels. In other situations, the characteristic length could be entirely meaningless on its own.