Exploring Yukawa Potential with Coupling Constant 0.3

[K.J.Healey]

If I had was given a yukawa potential of the form:

$$\text{YL}[\text{r\_}]=\left(\left.g_L{}^{\wedge}2\right/r\right) \text{Exp}\left[-\mu _Lr (c/\hbar )\right]$$

I put the c/h in the exponent to make it unitless, but what do I do about the outside?

i'm given the coupling constant is 0.3. Just 0.3. So should the potential need the units of eV?

So should i then take what I have and multiply it by "c hbar"? Its an inverse length right now, so i figure "ch" = "m eV s/s"
is this right?

 

[Valerie Park]

No, the outside of the potential should not be multiplied by anything. The unit of the Yukawa potential (YL) is already in eV. The coupling constant gL is dimensionless, so it has no units associated with it. Therefore, you do not need to multiply anything by cℏ to make it unitless.

 

[sir-pinski]

Yes, you are correct. In order to make the potential unitless, you need to include the units of c and hbar in the exponent. Since c and hbar have units of meters per second and electron-volts per second, respectively, their product will have units of meters times electron-volts. Therefore, you should multiply the potential by "c hbar" to make it unitless. This will also give you the units of energy, as you correctly stated, in electron-volts. So the final form of the potential would be:

YL[r_] := ((gL^2)/r) Exp[-(muL r c/hbar)]

where gL is the coupling constant and muL is the mass parameter. This potential now has units of electron-volts and is unitless, as desired.