Quantum Mechanics Equations Cheat Sheet

Planck's radiation law $$ U(v,T) = \frac{8 \pi v^2}{c^3} \frac{hv}{e^{\frac{hv}{kt}}-1} $$

$U \rightarrow $ Energy density, $ T \rightarrow $ Temperature, $v \rightarrow $ frequency, $k \rightarrow $ Boltzmann's constant, $h \rightarrow $ Planck's constant

Planck's Constant and Reduced Planck constant $$ \hbar = \frac{h}{2\pi} $$ $h \rightarrow $ Planck's constant, $ \hbar \rightarrow $ Reduced Planck's constant

Compton Wavelength Shift $$ \Delta \lambda = \frac{2h}{m_0 c} \sin^2 \frac{\theta}{2} $$

$\Delta \lambda \rightarrow$ wavelength shift, $\theta \rightarrow $ scattering angle, $m_0 \rightarrow $ rest mass of the electron

Rydberg-Ritz formula $$ \frac{1}{\lambda} = R \bigg( \frac{1}{n_1^2} - \frac{1}{n_2^2} \bigg) $$ $ R \rightarrow $ Rydberg constant $ (1.097 \times 10^7 m^{-1} ) $, $ \lambda \rightarrow$ wavelength, $n_1$ and $n_2$ are positive integers with $n_1 < n_2$ and $n_2 = n_1 + 1$

Hydrogen series for different $n_1$ $\\$ $n_1 = 1 \rightarrow$ Lyman series - Ultraviolet region $\\$ $n_1 = 2 \rightarrow$ Balmer series - Visible region $\\$ $n_1 = 3 \rightarrow$ Paschen series - Infrared region $\\$ $n_1 = 4 \rightarrow$ Brackett series - Infrared region $\\$ $n_1 = 5 \rightarrow$ Pfund series - Infrared region

With every material particle, a wave is associated, having a wavelength (de Broglie wavelength) $$ \lambda = \frac{h}{p} $$ $h \rightarrow $ Plancks constant $(6.625 \times 10^{-34} Js)$, $p \rightarrow$ momentum

de Broglie wavelength in non-relativistic kinetic energy $K$ $$ \lambda = \frac{h}{\sqrt{2mK}} $$

de Broglie wavelength in non-relativistic accelerating potential difference $V$ $$ \lambda = \frac{12.3}{\sqrt{V}} \mathring{A}$$ using $K=qV$, $q \rightarrow$ charge

de Broglie wavelength in relativistic kinetic energy $K$ $$ \lambda = \frac{hc}{\sqrt{K(K+2m_0c^2)}} $$ using $E^2 = p^2c^2 + m_0^2c^4$

de Broglie wavelength in relativistic accelerating potential difference $V$ $$ \lambda = \frac{h}{\sqrt{2m_0qV \bigg( 1 + \frac{\alpha}{2} \bigg)}} $$ where $\alpha = \frac{qV}{m_0c^2}$

Heisenberg's Uncertainty relation for position and momentum $$ \Delta x \Delta p \ge \hbar $$

Heisenberg's Uncertainty relation for energy and time $$ \Delta E \Delta t \ge \hbar $$

Hamiltonian Operator $$ H = - \frac{\hbar^2 \nabla^2}{2m} + V(r,t) $$

Normalization Condition $$ \int | \Psi (r,t) |^2 dr =1 $$

Probability Current Density $$ J = Re[\Psi^* \frac{\hbar}{im} \nabla \Psi ]$$

Time-Dependent Schrödinger Equation(TDSE) $$ i \hbar \frac{\partial}{\partial t} \Psi (r,t) = \bigg[ - \frac{\hbar^2 \nabla^2}{2m} + V(r,t) \bigg] \Psi(r,t) $$ $\Psi \rightarrow$ wavefunction. TDSE is used to study dynamic behaviour, such as wave packet spreading, interference, and quantum tunnelling. This Schrödinger Equation is the basic equation of non-relativistic quantum mechanics. This is not derived and can't prove to be true. But we can verify it with experimental results.

Time-Independent Schrödinger Equation(TISE) $$ \bigg[ - \frac{\hbar^2 \nabla^2}{2m} + V(r) \bigg] \Psi(r) = E \Psi(r) $$