Editing Johnson–Nyquist noise (section)

=== Quantum effects at high frequencies or low temperatures ===
With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near [[absolute zero]]), the multiplying factor <math>\eta(f)</math> mentioned earlier is in general given by:<ref>{{Cite journal |last1=Callen |first1=Herbert B. |last2=Welton |first2=Theodore A. |date=1951-07-01 |title=Irreversibility and Generalized Noise |url=https://link.aps.org/doi/10.1103/PhysRev.83.34 |journal=Physical Review |volume=83 |issue=1 |pages=34–40 |doi=10.1103/PhysRev.83.34|url-access=subscription }}</ref>
:<math>\eta(f) = \frac{hf/k_\text{B} T}{e^{hf/k_\text{B} T} - 1}+\frac{1}{2}
\frac{h f}{k_\text{B} T} \, .</math>
At very high frequencies (<math>f \gtrsim \tfrac{k_\text{B} T}{h}</math>), the spectral density <math>S_{v_n v_n}(f)</math> now starts to exponentially decrease to zero. At room temperature this transition occurs in the terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set <math>\eta(f)=1</math> for conventional electronics work.

==== Relation to Planck's law ====
Nyquist's formula is essentially the same as that derived by Planck in 1901 for electromagnetic radiation of a blackbody in one dimension—i.e., it is the one-dimensional version of [[Planck's law|Planck's law of blackbody radiation]].<ref>{{cite book |title=Fundamentals of Microwave Photonics |page=63 |url=https://books.google.com/books?id=mg91BgAAQBAJ&pg=PA63 |first1=V. J.|last1=Urick|first2=Keith J.|last2=Williams|first3=Jason D.|last3=McKinney|isbn=9781119029786 |date=2015-01-30 |publisher=John Wiley & Sons }}</ref> In other words, a hot resistor will create electromagnetic waves on a [[transmission line]] just as a hot object will create electromagnetic waves in free space.

In 1946, [[Robert H. Dicke]] elaborated on the relationship,<ref>{{Cite journal| doi = 10.1063/1.1770483| volume = 17| issue = 7| pages = 268–275| last = Dicke| first = R. H.| title = The Measurement of Thermal Radiation at Microwave Frequencies| journal = Review of Scientific Instruments| date = 1946-07-01| pmid=20991753| bibcode = 1946RScI...17..268D| s2cid = 26658623| doi-access = free}}</ref> and further connected it to properties of antennas, particularly the fact that the average [[antenna aperture]] over all different directions cannot be larger than <math>\tfrac{\lambda^2}{4\pi}</math>, where λ is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law.