Re: [Phys-L] weighting in the wings ... damped harmonic oscillator ... bandwidth ... algebra ... bug hunting

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding JD's comment:

>  ...                          In particular, the
> effective bandwidth "B" that shows up in Nyquist's formula
> for the Johnson noise (4 kT B) is *not* the FWHM of the
> resonance.  It's bigger by a factor of π/2.

I might be doing something incorrectly here, but it seems to me that this above factor is only for the special case when the resonance in question has a Q-factor that is quite small compared to 1/2, (i.e. the case of a very overdamped resonance).

It seems the above mismatch factor ought to be actually

π/(2*sqrt(1 + 4*Q^2)).

The total integrated (over a white noise spectrum) weight of the Lorentzian line (in Hz) for the Nyquist formula bandwidth is

B = (π*f_0)/(2*Q)

where f_0 is the resonant frequency.  But the FWHM bandwidth (in Hz) for the resonance is

B = f_0*sqrt(1/Q^2 + 4).

This makes their quotient π/(2*sqrt(1 + 4*Q^2)).

BTW JD's stimulating post reminded me of a kind of cute paper I had come across a while ago that used the measured thermal noise in a resistor connected to an amplifier circuit and the ambient temperature to experimentally determine Boltzmann's constant and absolute zero, and then also uses the measured shot noise (which has Poisson statistics rather than Gaussian statistics) & to experimentally determine the electron elementary charge.  All these determined fundamental constants are based on essentially just noise measurements.  The paper can be found at http://web.mit.edu/dvp/Public/noise-paper.pdf .

David Bowman