Make the interval math steps to explain the next text
At room temperatures, the average number of photons at optical frequencies is very small (on the order of 10^(-40)). At the surface temperature of the sun (6000 K) and at the frequency of yellow light (6 x 10^14 Hz, λ = 500 nm) the average photon number is about 10^(-2). On the other hand, the average photon number rapidly increases with increasing wavelength. Again at room temperature, /bar(n) ≈ 1 for λ in the range λ = 10-100 μm. In the microwave part of the spectrum, /bar(n) >> 1. From Eq. (2.141) it follows that
exp(-ℏ(ω)/(k_B)T) = (tilde(n))/(1 + /bar(n))
and from Eqs. (2.137) and (2.138) it follows that hat(ρ)_Th can be written in terms of
/bar(n) as
hat(ρ)_Th = (1)/(1 + /bar(n)) ∑_(n=0)^(∞) ((tilde(n))/(1 + /bar(n)))^n |n:
The probability of finding n photons in the field is given in terms of /bar(n) as
P_n = (/bar(n)^(**))/((1 + /bar(n))^(n + 1)).
In Fig. 2.3 we plot P_n versus n for two different values of /bar(n). It is clear in both
cases that the most probable photon number is the vacuum, P_n decreasing mono-
tonically with n. There is obviously nothing special about P_n for n near or at /bar(n)
(which need not be an integer).
The fluctuations in the average photon number are given as
(:(Δn)^(2):) = (:hat(n)^(2):) - (:hat(n):)^(2).
It can be shown, in a manner similar to the derivation of, tilde(n) that
((:)/(bar(n)^(2):) = Tr(/bar(n)^(2)hat(ρ)_m)
()/(bar) = /bar(n) + (2)/(bar)bar(n)^(2)
so that
(:(Δn)^(2):)()/(bar) = /bar(n) + (2)/(bar)bar(n)^(2)
from which it is apparent that the fluctuations of hat(n) are larger than the average /bar(n).
The root-mean-square (r.m.s.) deviation is
Δn = ((tilde(n)) + tilde(n)^(2))^((1)/(2))
which for /bar(n) >> 1 is approximately
Δn ≈ (√(/)bar(n) + (1)/(2)).
The relative uncertainty is given by the ratio Δ(n)/(/)bar(n), which is approximately 1
for /bar(n) >> 1 and is approximately (1)/(√(/))bar(n) for /bar(n) << 1. Obviously, Δ(n)/(/)bar(n) -> ∞ as
/bar(n) -> 0. Wigner quasiprobability distribution
Make the interval math steps to explain the next text
At room temperatures, the average number of photons at optical frequencies is very small (on the order of 10^40). At the surface temperature of the sun (6000 K) and at the frequency of yellow light (6 x 10^13 Hz, λ = 500 nm) the average photon number is about 10^2. On the other hand, the average photon number rapidly increases with increasing wavelength. Again at room temperature, fi ≈ 1 for λ in the range λ = 10-100 m. In the microwave part of the spectrum, ni > 1. From Eq. (2.141) it follows that
(2.143)
and from Eqs. (2.137) and (2.138) it follows that r can be written in terms of ni as Pn=1(1)m(1. (2.144) The probability of finding n photons in the field is given in terms of as P, = (2.145) 1+** In Fig. 2.3 we plot P, versus n for two different values of ri. It is clear in both cases that the most probable photon number is the vacuum, P, decreasing mono- tonically with n. There is obviously nothing special about P, for n near or at i (which need not be integer). The fluctuations in the average photon number are given as (n)2)=(2)(A)2. (2.146) It can be shown, in a manner similar to the derivation of, that (2)= Tr(2n) (2.147) = i+ 22
((n)2)= +2
(2.148)
from which it is apparent that the ffucfuations of ii are larger than the average ii. The root-mean-square (r.m.s.) deviation is = ( +2)1/2 (2.149) which for >> 1 is approximately
(2.150)
The relative uncertainty is given by the ratio /ii, which is approximately 1 for > 1 and is approximately 1/ for 1. Obviously, / as n0.
