Question

Derive the constant A for a normalized one-dimensional Maxwellian

distribution

^

f uð Þ¼ Aexp

mu2=2KT

 

such that

ð

1

1

^

f uð Þdu ¼ 1

Answer 1

$$a=\frac{m}{kT}\\F(u)=Ae^{-0.5au^2}\\\int^\infty_{-\infty}Ae^{-0.5au^2}du=1 \\\int^\infty_{-\infty}e^{-0.5au^2}du=\sqrt{\frac{2\pi}{a}}\\A=\sqrt{\frac{m}{2\pi kT}}$$