Key Equations

Wien’s displacement law	$λ_{\max} T = 2.898 \times 10^{- 3} m \cdot K$
Stefan’s law	$P (T) = σ A T^{4}$
Planck’s constant	$h = 6.626 \times 10^{- 34} J \cdot s = 4.136 \times 10^{- 15} eV \cdot s$
Energy quantum of radiation	$Δ E = h f$
Planck’s blackbody radiation law	$I (λ, T) = \frac{2 π h c^{2}}{λ^{5}} \frac{1}{e^{h c / λ k_{B} T} - 1}$
Maximum kinetic energy of a photoelectron	$K_{\max} = e Δ V_{s}$
Energy of a photon	$E_{f} = h f$
Energy balance for photoelectron	$K_{\max} = h f - ϕ$
Cut-off frequency	$f_{c} = \frac{ϕ}{h}$
Relativistic invariant energy equation	$E^{2} = p^{2} c^{2} + m_{0}^{2} c^{4}$
Energy-momentum relation for photon	$p_{f} = \frac{E_{f}}{c}$
Energy of a photon	$E_{f} = h f = \frac{h c}{λ}$
Magnitude of photon’s momentum	$p_{f} = \frac{h}{λ}$
Photon’s linear momentum vector	${\vec{p}}_{f} = ℏ \vec{k}$
The Compton wavelength of an electron	$λ_{c} = \frac{h}{m_{0} c} = 0.00243 nm$
The Compton shift	$Δ λ = λ_{c} (1 - \cos θ)$
The Balmer formula	$\frac{1}{λ} = R_{H} (\frac{1}{2^{2}} - \frac{1}{n^{2}})$
The Rydberg formula	$\frac{1}{λ} = R_{H} (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}}), n_{i} = n_{f} + 1, n_{f} + 2, \dots$
Bohr’s first quantization condition	$L_{n} = n ℏ, n = 1, 2, \dots$
Bohr’s second quantization condition	$h f = \| E_{n} - E_{m} \|$
Bohr’s radius of hydrogen	$a_{0} = 4 π ε_{0} \frac{ℏ^{2}}{m_{e} e^{2}} = 0.529 Å$
Bohr’s radius of the nth orbit	$r_{n} = a_{0} n^{2}$
Ground-state energy value, ionization limit	$E_{0} = \frac{1}{8 ε_{0}^{2}} \frac{m_{e} e^{4}}{h^{2}} = 13.6 eV$
Electron’s energy in the nth orbit	$E_{n} = - E_{0} \frac{1}{n^{2}}$
Ground state energy of hydrogen	$E_{1} = - E_{0} = - 13.6 eV$
The nth orbit of hydrogen-like ion	$r_{n} = \frac{a_{0}}{Z} n^{2}$
The nth energy of hydrogen-like ion	$E_{n} = - Z^{2} E_{0} \frac{1}{n^{2}}$
Energy of a matter wave	$E = h f$
The de Broglie wavelength	$λ = \frac{h}{p}$
The frequency-wavelength relation for matter waves	$λ f = \frac{c}{β}$
Heisenberg’s uncertainty principle	$Δ x Δ p \geq \frac{1}{2} ℏ$