Fundamental relations
Light (electromagnetic radiation) is characterized by its wavelength \( \lambda \) (m), frequency \( \nu \) (Hz), and the energy per photon \( E \) (J). The two central relations are
\[ c = \lambda\,\nu \]
\[ E = h\,\nu \]
Here \( c \) is the speed of light in vacuum and \( h \) is Planck’s constant. Using these two relations together lets you convert between any two of \( \lambda \), \( \nu \), and \( E \).
Useful derived forms
Replace \( \nu = c/\lambda \) into \( E=h\nu \) to get energy in terms of wavelength:
\[ E = \frac{h c}{\lambda}. \]
It’s often convenient to work in electronvolts and nanometres. Useful constants (SI and derived):
| Constant | Value | Practical Value |
|---|---|---|
| Speed of light \(c\) | \(2.99792458\times10^8\ \text{m·s}^{-1}\) | \(2.998\times10^8\ \text{m·s}^{-1}\) |
| Planck constant \(h\) | \(6.62607015\times10^{-34}\ \text{J·s}\) | \(6.626\times10^{-34}\ \text{J·s}\) |
| Elementary charge \(e\) | \(1.602176634\times10^{-19}\ \text{C}=\text{J·eV}^{-1}\) | \(1.602\times10^{-19}\ \text{C}=\text{J·eV}^{-1}\) |
| Product \(hc\) | \(1.986445857\times10^{-25}\ \text{J·m}\) | \(1.986\times10^{-25}\ \text{J·m}\) |
| \(hc\) in eV·nm | \(\displaystyle \frac{hc}{e}\times10^{9}\approx 1239.841984\ \text{eV·nm}\) | \(\displaystyle \frac{hc}{e}\times10^{9}\approx 1240\ \text{eV·nm}\) |
Using these values, the photon energy in electronvolts for a wavelength in nm is approximately
\[ E(\text{eV}) \approx \frac{1239.842}{\lambda(\text{nm})}. \]
Quick rearrangements (handy algebra)
\[ \nu = \frac{c}{\lambda}, \qquad \lambda = \frac{c}{\nu}, \qquad E = h\nu = \frac{hc}{\lambda}. \]
You can also express energy in wave-number units (common in spectroscopy). Define the wave number \( \tilde{\nu} = 1/\lambda \) (when \( \lambda \) is in cm, \( \tilde{\nu} \) has units cm\(^{-1}\)):
\[ \tilde{\nu}(\text{cm}^{-1}) = \frac{10^{7}}{\lambda(\text{nm})}. \]
Then \( E \) in J is \( E = h c \tilde{\nu} \) (with \( \tilde{\nu} \) in m\(^{-1}\) or cm\(^{-1}\) using consistent units).