Some Essential Concepts
\[ \newcommand\v[1]{\boldsymbol{#1}} \newcommand\vdot[2]{\v{#1}\cdot\v{#2}} \]
\[ \newcommand{\dd}[2][]{\mathrm{d}^{#1}#2} \]
\[ \newcommand\f[2]{\dfrac{#1}{#2}} \newcommand\da[2]{\dfrac{d #1}{d #2}} \newcommand\daa[2]{\dfrac{d^2 #1}{d #2^2}} \newcommand\pa[2]{\dfrac{\partial #1}{\partial #2}} \newcommand\pal[3]{\left.\dfrac{\partial #1}{\partial #2}\right|_{#3}} \newcommand\paa[2]{\dfrac{\partial^2 #1}{\partial #2^2}} \newcommand\pab[3]{\dfrac{\partial^2 #1}{\partial #2 \partial #3}} \def\half{\frac{1}{2}} \]
\[ \newcommand\ket[1]{\left|#1\right\rangle} % |x> \newcommand\exp[1]{\left\langle#1\right\rangle} % <x> \newcommand\lr[1]{\left(#1\right)} % (x) \newcommand\lrs[1]{\left[#1\right]} % [x] \newcommand\lra[1]{\left|#1\right|} % |x| \newcommand\lrc[1]{\left\{#1\right\}} % {x} \newcommand\lrl[3]{\left.#1\right|_{#2}^{#3}} % x_limits \]
\[ \def\ie{\text{i.e.}\quad} \]
\[ \newcommand{\me}[3][\rightarrow]{{#2}#1{#3}} % Typical reaction \newcommand{\nr}[3]{{#1}\,({#2})\,{#3}} % Nuclear reactions \newcommand{\ele}[4]{_{#2}^{#3}{\textbf{#1}}_{#4}} \newcommand{\el}[2]{^{#2}\text{#1}} \newcommand{\til}[2][\mathbf{\sim}]{\underset{#1}{{#2}}} \newcommand{\b}[1]{\overline{#1}} \]
\[ \newcommand{\coulomb}[1]{\dfrac{#1}{4\pi\epsilon_0}} \def\degree{^*} \]
\[ \def\amu{\text{ amu}} \def\J{\text{ J}} \def\K{\text{ K}} \def\s{\text{ s}} \def\second{\text{ s}} \def\Psecond{\text{ s}^{-1}} \def\m{\text{ m}} \def\mm{\text{ mm}} \def\km{\text{ km}} \def\fm{\text{ fm}} \def\mum{\text{ $\mu$m}} \def\mus{\text{ $\mu$s}} \def\mPERs{\text{ ms$^{-1}$}} \def\eV{\text{ eV}} \def\keV{\text{ keV}} \def\GeV{\text{ GeV}} \def\MeV{\text{ MeV}} \def\MeVc{\text{ MeV/c$^2$}} \def\KeVc{\text{ KeV/c$^2$}} \newcommand\ttp[1]{\times 10^{#1}} % To The Power \]
\[ \def\en{\nu_e} \def\Aen{\b{\nu_e}} \def\mun{\nu_\mu} \def\Amun{\b{\nu}_\mu} \]
\[ \def\uraniumA{\ele{U}{92}{233}{}} \def\uraniumB{\ele{U}{92}{235}{}} \def\uraniumC{\ele{U}{92}{238}{}} \def\carbonB{\ele{C}{6}{14}{}} \def\nitrogenB{\ele{N}{7}{14}{}} \def\oxygenA{\ele{O}{8}{14}{}} \def\oxygenB{\ele{O}{8}{16}{}} \]
\[ % \def\com{\text{\scriptsize CM}} \def\com{\mathrm{\scriptsize CM}} \]
\[ \def\l{l} \]
\[ \def\g{\gamma} \]
Units of Energy
It is often convenient to express rest mass in units of energy by using the mass-energy equivalence embodied in:
\[ E^2 = p^2c^2 + m^2c^4 \Rightarrow E = mc^2 \]
So the proton with a mass of \(m_p = 1.672\ 626\ 192 \times 10^{-27}\) kg may be given as: \[ \begin{align*} m_p &= (1.672\ 626\ 192 \times 10^{-27} \text{ kg}) \times (2.997\ 92 \times 10^8 \text{ m/s})^2\\ &= 1.503\ 277\ 616 \times 10^{-14} \text{ J}\\ &= 938.272 \text{ MeV} \end{align*} \]
Often you see some who explicitly write MeV/c\(^2\) as the units.
Some Useful Numbers
| \(\amu\) | \(931.494\MeV\) |
| \(\hbar\) | \(6.582\ttp{-22}\MeV\cdot \second\) \(197.3 \MeV \cdot \fm /c\) |
| \(c\) | \(2.998\ttp{23}\fm\cdot \Psecond\) |
| \(c^2\) | \(931.494 \MeV/\text{u}\) |
| \(k_B\) | \(1.381\ttp{-23}\J/\K\) \(8.617\ttp{-11}\MeV/\K\) |
| \(\hbar c\) | \(197.3 \MeV\cdot\fm\) |
| \(\dfrac{e^2}{4\pi \epsilon_0}\) | \(1.44\MeV \cdot \fm\) |
| \(\alpha=\dfrac{e^2}{4\pi \epsilon_0\,\hbar c}\) | \(\dfrac{1}{137.04}\) |
Fourier Transforms
The Fourier transform \(F(k,\omega)\) of a function \(f(x,t)\) that varies in both space and time may be written as \[ F(k,\omega) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,t)\, e^{-i(kx-\omega t)}\, \dfrac{\dd x}{\sqrt{2\pi}}\,\dfrac{\dd t}{\sqrt{2\pi}} . \tag{1}\]
The associated inverse Fourier transform expresses \(f(x,t)\) as a superposition of complex plane waves: \[ f(x,t) = \frac{1}{(2\pi)^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} F(k,\omega)\, e^{i(kx-\omega t)}\, \dfrac{\dd k}{\sqrt{2\pi}}\,\dfrac{\dd \omega}{\sqrt{2\pi}}. \tag{2}\]