This is a question about the proper form for thermal noise from a resistor. This is purely academic for me - I always work in the regime where [itex] \hbar \omega << k T [/itex] so the noise spectrum is simply P \approx kT. When this no longer holds, quantum effects matter of course. Then I have seen two different expressions for the spectral density. There is the Planck result(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

P = \frac{\hbar \omega}{ exp(\hbar \omega / k T) - 1}

[/tex]

that we learn about in basic modern physics. Then there is the result that uses the actual energy levels of the quantum harmonic oscillator, which adds in the zero point energy

[tex]

P = \frac{\hbar \omega}{2} + \frac{\hbar \omega}{exp(\hbar \omega / k T) - 1}.

[/tex]

For the thermal noise problem, the Planck result shows up in Nyquist's original 1928 paper of course, while the last result shows up in Callen and Welton's 1951 paper. (my grad work was in plasma physics, so the fluctuation dissipation theorem is a tool I suposedly learned at one point - but I have only used the classical limit). I have tried google, google scholar, and stat mech books in the library at work, but have not been able to get a definitively clear understanding of which is correct, and more importantly why! Many books (Pathria 3rd edition stat mech book for example) simply sweep it under the rug with less than one sentence of discussion.

One argument I have seen for the Planck result is that the energy available for noise generation is from transitions between states, so the reference/zero-point energy is irrelevant. Seems reasonable to me. An argument the other way goes like this: the Planck result yields zero energy as T->0, violating the energy-time uncertainty principle. Given my lack of true understanding of quantum mechanics I do not know how good/bad this argument is, especially since if this expression is multiplied by the density of modes in a cavity the blackbody radiation formula once again has the ultraviolet catastrophe that Planck was trying to fix!

By the way, this is a question from an engineer who has taken only 1 quantum mechanics class ~20 years ago and has worked mostly in the classical world ever since. My stat-mech background is essentially self-taught out of Statistical and Thermal Physics by Reif - I primarily worked through the classical chapters and did solve a fair number of the problems (in other words, I learned enough to be able to survive my plasma physics courses). Perhaps I just do not understand enough physics to be able to sort this out - but I am hoping someone here can enlighten me!

Thanks!

jason

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# Zero point energy in thermal noise spectrum?

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