1

The system of equations in question is $$ \dot{n} = GnN - kn$$ $$\dot{N} = GnN - fN + p$$

Where ${N(t)}$ is the number of excited atoms, ${n(t)}$ is the number of photons, ${G}$ is the gain coefficient, ${f}$ is the rate of decay for spontaneous emission, and ${p}$ is the pump strength. All parameters except ${p}$ are positive, ${p}$ can be either +ve or -ve

The problem has us assume $\dot{N} \approx 0$, with part D asking us for what range of parameters we can use this approximation. Following examples set earlier in the text, I'd like to non-dimensionalize this equation, but I cannot for the life of me seem to figure out what units each of these coefficients should be, especially ${G}$ and ${N}$. If someone could give me hints without giving away under what conditions we can neglect the parameters (giving the dimensions of all the things or less, ideally) that would be amazing.

Qmechanic
  • 201,751
Tuatarian
  • 113
  • Anything that's a "number" of something is unitless. $n$ and $N$, being the number of photons and the number of excited atoms, respectively, are both units. You should be able to get the units of $\dot{n}$, $\dot{N}$, $G$, $k$, and $p$ from there. – march Feb 14 '22 at 04:18

1 Answers1

1

As march already commented, the variables described as "numbers", $n$ and $N$, can safely be taken to be dimensionless (though, of course, we do use pseudo-units "dozen", "mole", etc. — see this question), and thus $\dot{n}=dn/dt$ will have the inverse unit of time, and so on for the other quantities.

stafusa
  • 12,435