In the Coursera course From the Big Bang to Dark Energy on several occasions dimensional analysis was used to estimate the scale of quantities. This almost seems like a contradiction in terms to me, since you cannot strictly infer anything about dimensionless factors. Nevertheless it seems to work if you know some rules of thumb.
The general procedure seemed to be to take the speed of light, Planck's constant and Boltzmann's constant, and then use dimensional analysis to relate these to the known quantities to get a quantity of the desired dimension.
- wavelength of cosmic microwave background from temperature
- kinetic energy of quarks from proton radius
- energy of W and Z bosons from length scale of weak force
- radius of hydrogen atom from electron mass via electron wavelength
In the last case the dimensionless fine structure constant has to be thrown in.
I understand that when a force is involved, you will have to include some proportionality constant that cannot be obtained from dimensional analysis alone (like the fine structure constant above), but maybe that often is all that is needed.
Is there just a handful of rules of thumb, using which more or less reliable estimates can be made?
Apart from that, I would also be very interested in general comments on this theme.
As pointed out by QMechanic, a related (and very interesting) thread in this forum is found here In dimensional analysis, why the dimensionless constant is usually of order 1?. There certainly are similarities between the questions, but I am more specifically asking how to use dimensional analysis to obtain such estimates, and didn't find concrete answers in that thread.
I find it fascinating that it can be done, but wouldn't dare to apply it myself. Rather than asking why the constant is usually of order 1, maybe my question is when the constant is of order 1, or more generally, how to estimate its value.