# Measurement uncertainty due to thermal noise

• A
• kelly0303
In summary, the particle is in a Penning trap and the equation of motion is a simple harmonic oscillator. The particle is damped with a damping coefficient and the noise power comes from the warm resistance in the tuned circuit.
kelly0303
I have a charged particle in a Penning trap. The particle motion is non-relativistic and the energy is high enough such that we can assume it is not in the quantum regime. For the purpose of the question I am interested only in the axial motion of the particle, so basically this is a classical simple harmonic oscillator. In the ideal case, the equation of motion is simply ##z(t)=z_0cos(\omega_0 t)##, where ##\omega_0## is the resonant axial frequency and ##z_0## is the amplitude given by the initial energy of the particle.

Now assume I connect the trap to a resonant RLC circuit, with resonant frequency ##\omega_0##, at temperature ##T##, such that the motion of the particle is damped with a damping coefficient ##\gamma##. Now the motion can be simply described by a damped harmonic oscillator equation.

However, I am interested in how the thermal noise of the circuit comes into play. I read a bit about noise theory and there is a pretty straight forward formula for the noise spectrum as a function of the frequency. However I am interested in how does the uncertainty in the position at a given time t is affected by the noise.

Of course we can't precisely tell the position anymore, as the thermal noise is random, but I would like to assign an uncertainty to it. Basically, I have ##z(t)## from the damped harmonic oscillator formula and I would like to add to it a ##dz(t)##, which is the uncertainty for a given time and position (as a function of probably ##\gamma## and ##T##). How can I do this? Thank you!

vanhees71
I'd consider to describe this by a Langevin equation, i.e., in addition to the force due to the em. field in the Penning trap you have a friction (dissipative) force and a random force, which you can assume to be described by white noise (diffusion). The friction/drag and the diffusion coefficients are related by the Einstein dissipation-fluctuation relation, ensuring that in the long time limit the particle comes in to thermal equilibrium with the heat bath.

https://en.wikipedia.org/wiki/Langevin_equation

DrClaude
May I suggest a simplistic answer? Assuming a series RLC circuit, the noise power Pn will be 2kTB, where k is Boltzmann's Constant, and the noise current will be sqrt (Pn/R). Bandwidth B = centre frequency/Q. And Q=Xl / R. As this current is entirely carried by the motion of the electron, which is in series with LCR, the RMS velocity of the electron will be fluctuate by vel=I/q. This fluctuation velocity will be added continuosly to any other motion of the electron.

The distribution of a sinusoid in Gaussian noise is the Rice distribution. You can easily look up it up to find the variance or standard deviation. Damped sinusoids are much more complicated, however. Here is a paper that gives the Cramer-Rao lower bound on the variance of the estimated amplitude, frequency and damping constant for a damped sinusoid in Gaussian noise. The CRLB, which is valid at medium to high SNR, is the absolute lowest uncertainty possible. You may achieve this bound or not, depending on your measurement and analysis techniques. (In the radar world, for example, maximum likelihood estimation (MLE) achieves the CRLB for range estimation but does not when used to estimate angle of arrival.)

berkeman and vanhees71
It appears to me that the noise power which is jiggling the electron arises from the warm resistance in the tuned circuit. If the tuned circuit also has damped oscillation that is separate and unrelated. The current and voltage from the thermal noise and the damped oscillation just add but do not interact.

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