Planck’s sum (Equation 2.83) for the average energy per mode of blackbody radiation is
It is convenient to introduce the variable , so
Next consider the quantity
Using the chain rule to take the derivative yields
has the form , so
The Stefan–Boltzmann law for the integrated brightness of blackbody radiation at temperature (Equation 2.89) is
is Planck’s law and is the Stefan–Boltzmann constant. Although the Stefan–Boltzmann law and constant were first determined experimentally, both can be derived mathematically from Planck’s law. For simplicity, define
can be expanded in terms of the infinite series
and the integral becomes
Each integral in this series can be integrated by parts three times:
converges quickly and is the value of the Riemann zeta function . Thus
Finally, the integrated brightness of blackbody radiation is
is the value of the Stefan–Boltzmann constant.
Similarly, the integral
is needed to evaluate the number density of blackbody photons:
Following the derivation above,
A complex exponential , where and is any dimensionless real variable, is a complex number in which the real and imaginary parts are sines and cosines given by Euler’s formula
Euler’s formula can be derived from the Taylor series
Complex exponentials (or sines and cosines) are widely used to represent periodic functions in physics for the following reasons:
They comprise a complete and orthogonal set of periodic functions. This set of functions can be used to approximate any piecewise continuous function, and they are the basis of Fourier transforms (Appendix A.1).
They are eigenfunctions of the differential operator—that is, the derivatives of complex exponentials are themselves complex exponentials:
Most physical systems obey linear differential equations, a low-pass filter consisting of a resistor and a capacitor, for example. A sinusoidal input signal will yield a sinusoidal output signal of the same frequency (but not necessarily with the same amplitude and phase), while a square-wave input will not yield a square-wave output. The response to a square-wave input can be calculated by treating the input square wave as a sum of sinusoidal waves, and the filter output is the sum of these filtered sinusoids. This is the reason why periodic waves or oscillations are almost always treated as combinations of complex exponentials (or sines and cosines).
Real periodic signals can be expressed as the real parts of complex exponentials:
Adding and subtracting the equations
gives the identities
The advantage of complex exponentials over the equivalent sums of sines and cosines is that they are easier to manipulate mathematically. For example, you can use complex exponentials to calculate the output spectrum of a square-law detector (Section 3.6.2) without having to remember trigonometric identities. A square-law detector is a nonlinear device whose output voltage is the square of its input voltage. If the input voltage is , the output voltage is
The output spectrum has two frequency components: one at twice the input frequency and the other at zero frequency (DC).
The normalized Gaussian function is usually written as
where is its rms width. To calculate its Fourier transform
it is easier to use the form , for which . Then
To evaluate this one-dimensional integral, break it into the product of two integrals and change one dummy variable from to to suggest Cartesian coordinates in a plane:
Next transform to polar coordinates so and :
Finally, substitute and to get
The Fourier transform of a Gaussian is a Gaussian.
The voltage of random noise has a Gaussian probability distribution
where is the differential probability that the voltage will be within the infinitesimal range to and is the root mean square (rms) voltage. The probability of measuring some voltage must be unity, so
The normalization of in Equation B.17 can be confirmed by evaluating the integral
Equation B.15 immediately yields the definite integral
Substituting gives the desired result:
The rms (root mean square) of a normalized distribution is defined by
For the symmetric Gaussian distribution, , so
The definite integral
can be derived by integrating Equation B.21 by parts. Inserting yields
confirming that in Equation B.17 is the rms of the Gaussian distribution.
A square-law detector multiplies the input voltage by itself to yield an output voltage that is proportional to the input power. The input voltage distribution is a Gaussian with rms (Equation B.17),
The same value of is produced by both positive and negative values of and , so
for all . Because ,
for . The distribution of detector output voltage is sharply peaked near and has a long exponentially decaying tail (Figure 3.33).
The mean detector output voltage follows from Equation B.28: .
The rms of the detector output voltage is
Integrating Equation B.27 by parts yields the definite integral
and substituting gives
The rms of the detector output voltage is times the mean output voltage . The rms uncertainty in each independent sample of the measured noise power is times the mean noise power. If independent samples are averaged, the fractional rms uncertainty of the averaged power is . This result is the heart of the ideal radiometer equation (Equation 3.154). According to the central limit theorem, the distribution of these averages approaches a Gaussian as becomes large.
Let be the speed of a particle (e.g., an electron) of mass in a gas in LTE at temperature . From thermodynamics, recall that the average kinetic energy is per degree of freedom (e.g., per spatial coordinate for a single particle), so
Collisions eventually bring the gas into LTE, leading to identical Gaussian distributions (Appendix B.5) for , , and . Writing out only the -coordinate distribution yields
where is the rms (root mean square) value of . The definition of this rms is
In three dimensions, by isotropy,
All velocities in the spherical shell of radius correspond to the speed , so
This is the nonrelativistic Maxwellian distribution of speeds for particles of mass at temperature . If we normalize the speeds by the rms speed , the Maxwellian speed distribution looks like Figure 4.6.