Equation sheet

Equation sheet#

From classical to Quantum#

Blackbody radiation#

Description	Equations
Energy quantization	$E = nh\nu$
Average energy of an oscillating dipole	$\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}$
Spectral radiation density of blackbody (Planck)	$\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3}\dfrac{1}{e^{h\nu / k_B T} - 1} \ d\nu$
Spectral radiation density of blackbody (classical)	$\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3} k_B T d\nu$

Wave-particle duality#

Description	Equations
Energy of light	$E = h \nu$
Photoelectric effect Kinetic energy of ejected photoelectron	$E_k = h \nu - \Phi$
de Broglie relation	$p = \dfrac{h}{\lambda}$
Kinetic energy	$E_k = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}$

Atomic spectra of hydrogen and Bohr’s model#

Description	Equations
Hydrogen emission lines $n_2 > n_1$	$\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$
Bohr’s radius	$r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}$
Energy level in Bohr’s model	$E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}$
Emission of hydrogen atom $n_2 > n_1$	$\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$

Waves#

Description	Equations
Classical nondispersive wave equation	$\dfrac{\partial \Psi(x, t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial \Psi(x, t)}{\partial t^2}$
Wave number	$k = \dfrac{2\pi}{\lambda}$
Frequency	$\nu = \dfrac{1}{T}$
Angular frequency	$\omega = \dfrac{2\pi}{T} = 2\pi\nu$
Wave speed	$v = \lambda\nu$
Euler’s formula	$e^{i\theta} = \cos\theta + i\sin\theta$
Solution of wave equation	$\begin{aligned}\Psi(x, t) &= A \sin(kx - \omega t + \phi) \\\ &= \mathrm{Re}(Ae^{i(kx-\omega t + \phi')})\end{aligned}$
Interfering traveling waves give standing wave	$\begin{aligned}\Psi(x, t) &= A[\sin(kx - \omega t) + \sin(kx + \omega t)] \\\ &= 2A \sin(kx)\cos(\omega t) \\\ &= \psi(x)\cos(\omega t) \end{aligned}$
Time-independent Schrodinger equation	$-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$
Time-dependent Schrodinger equation	$-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x, t)}{\partial x^2} + V(x, t)\Psi(x, t) = i\hbar\Psi(x, t)$
Stationary states are standing waves	$\Psi(x, t) = \psi(x) e^{-i(E/\hbar)t}$
Normalization	$\Vert f(x) \Vert = \int_D f^* f \ dx = 1$
Orthogonality	$\int_D f^* g \ dx = 0$
Use quantum mechanics when …	1. $\lambda_{\text{particle}} \sim L_{\text{problem}}$ 2. $\Delta E \gtrsim k_bT$ (discrete energy spectrum)

Quantum-Mechanical Postulates#

The state of a quantum-mechanical particle is completely specified by a wave function $\Psi(x, t)$. The probability that the particle will be found at time $t_0$ in a spatial interval of width $dx$ centered at $x_0$ is given by $\Psi^*(x_0, t_0)\Psi(x_0, t_0) dx$
For every measurable property of a system, there exists a corresponding operator.
In any single measurement of the observable that corresponds to the operator $\hat{A}$, the only values that will ever be measured are the eigenvalues of that operator.
If the system is in a state described by the wave function $\Psi(x, t)$, and the value of the observatle $a$ is measured once on each of many identically prepared systems, the average value (expectation value) of all of the measurements is given by $$ \langle a \rangle = \dfrac{\displaystyle\int_{-\infty}^{\infty} \Psi^* \hat{A} \Psi \ dx}{\displaystyle\int_{-\infty}^{\infty} \Psi^*\Psi \ dx} $$
The evolution in time of a quantum-mechanical system is governed by the time-dependent Schrödinger equation $$ \hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t} $$

Operators#

Description	1D	3D
Position	$\hat{x} = x$	$\mathbf{\hat{x}} = \mathbf{x}$
Linear momentum	$\hat{p}_x = -i\hbar \dfrac{d}{dx}$	$\mathbf{\hat{p}} = -i\hbar\mathbf{\nabla}$
Kinetic energy	$\hat{T} = \dfrac{\hat{p}_x^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}$	$\mathbf{\hat{T}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2$
Potential energy	$\hat{V} = V(x)$	$\mathbf{\hat{V}} = V(\mathbf{x})$
Total energy Hamiltonian	$\hat{H} = \hat{T} + \hat{V} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x)$	$\mathbf{\hat{H}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x})$

Simple Quantum Systems#

Stationary states#

Description	Equations
Time dependent Schrodinger equation	$\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}$
Time independent Schrodinger equation	$\hat{H}\psi_n(x) = E_n \psi_n(x)$
Stationary state wave function	$\Psi(x, t) = \psi(x) T(t)$
Time component of wave function	$T(t) = e^{iEt/\hbar}$
Probability of finding particle in an interval	$\mathrm{Prob}(x, x+dx) = \vert \Psi(x, t) \vert^2 dx = \vert \psi(x) \vert^2 dx$
General solution as linear combination of stationary states	$\psi(x) = \sum\limits_n c_n \phi_n(x)$
Expansion coefficients	$c_n = \langle \phi_n \vert \psi \rangle = \int \phi_n^* \psi \ dx$
Normalization	$\sum\limits_n c_n = 1$

Particle in a 1D box#

Description	Equations
Time independent Schrodinger equation	$\left[ -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x) \right] \psi(x) = E\psi(x)$
Wave function $n = 0, 1, 2, ...$	$\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin\left(\dfrac{n \pi x}{L}\right)$
Energy eigenvalues	$E_n = \dfrac{h^2}{8mL^2} n^2 = \dfrac{\hbar^2 \pi^2}{2mL^2}n^2$

Particle in a 3D box#

Description	Equations
Time independent Schrodinger equation	$\left[ -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})$
Wave function $n_x = 0, 1, 2, ...$ $n_y = 0, 1, 2, ...$ $n_z = 0, 1, 2, ...$	$\begin{aligned}&\psi_{n_x, n_y, n_z}(\mathbf{x}) \\\ =& \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\\ =& \sqrt{\dfrac{2}{L_x}}\sqrt{\dfrac{2}{L_y}}\sqrt{\dfrac{2}{L_z}} \sin\left(\dfrac{n_x \pi x}{L_x}\right)\sin\left(\dfrac{n_y \pi y}{L_y}\right)\sin\left(\dfrac{n_z \pi z}{L_z}\right)\end{aligned}$
Energy eigenvalues	$E_n = \dfrac{h^2}{8m} \left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)$

Finite potential well#

Description	Equations
Potential	$V(x) = \begin{cases}0 & x\in [0, L] \\\ V_0 & \mathrm{elsewhere}\end{cases}$
Reflection probability	$R = \dfrac{(\sqrt{E} - \sqrt{E - V_0})^2}{(\sqrt{E} + \sqrt{E - V_0})^2}$
Transmission probability	$T = \dfrac{4\sqrt{E(E - V_0)}}{(\sqrt{E} - \sqrt{E - V_0})^2}$

Commutators and Uncertainty#

Description	Equations
Commutator	$[A, B] = AB - BA$
Condition of commutation	$[A, B] = 0$
Standard deviation (uncertainty)	$\begin{aligned}\sigma_A &= \sqrt{\langle (A - \langle A \rangle^2 \rangle)} \\\ &= \sqrt{\langle A^2 \rangle - \langle A \rangle^2}\end{aligned}$
Heisenberg uncertainty principle (general)	$\sigma_A \sigma_B \ge \frac{1}{2} \vert\langle[\hat{A}, \hat{B}]\rangle\vert$
Heisenberg uncertainty principle (position-momentum)	$\sigma_x \sigma_p \ge \frac{\hbar}{2}$

Spectroscopy#

Vibration: quantum harmonic oscillator#

Description	Equations
Vibrational Schrodinger equation	$\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + V(r) \right]\psi(\mathbf{x}) = E\psi(\mathbf{x})$
Wave function	$\psi(\mathbf{x}) = R(r)Y(\theta, \phi)$
Harmonic approximation	$V(r) \approx \frac{1}{2}kr^2$
Spring constant	$k = \mu\omega_0^2$
Vibrational Schrodinger equation	$\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{1}{2}kr^2 \right]\psi(r) = E\psi(r)$
Wave function $n = 0, 1, 2, ...$	$\psi(r) = \dfrac{1}{\sqrt{2^2 n!}}\left(\dfrac{\alpha}{\pi}\right)^{1/4} H_n(\sqrt{\alpha} r) e^{-\alpha r^2 / 2}$
Hermite polynomials	$H_n(r) = (-)^n e^{x^2}\left(\dfrac{d^n}{dx^n}\right)e^{-x^2}$
Constant	$\alpha = \dfrac{m\omega_0}{\hbar}$
Energy eigenvalue $n = 0, 1, 2, ...$	$E_n = (n + \frac{1}{2})\hbar \omega_0$
Transition dipole moment	$\vec{\mu}_{fi} = \dfrac{d\vec{\mu}(x_0)}{dx} \langle \psi_f \vert \hat{x} \vert \psi_i \rangle$
Vibrational selection rule	$\Delta n = \pm 1$

Rotation: quantum rigid rotor#

Classical rigid rotor#

Description	Equations
Angular momentum	$\mathbf{L} = \mathbf{x} \times \mathbf{p} = I\vec{\omega}$
Linear velocity	$\vec{v} = R_0 \vec{\omega}$
Moment of inertia	$I = mR_0^2$
Rotational kinetic energy	$E = \dfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}$

Quantum rigid rotor#

Description	Equations
Angular momentum operator	$\hat{\mathbf{L}} = \hat{\mathbf{x}} \times \hat{\mathbf{p}}$
z-component of angular momentum operator	$L_x = \dfrac{\hbar}{i}\dfrac{\partial}{\partial\phi}$
Magnitude of angular momentum operator	$\hat{\mathbf{L}}^2 = L^2 = -\hbar^2 \left[ \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta} \left( \sin\theta\dfrac{\partial}{\partial\theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2}{\partial\phi^2} \right) \right]$
Components of $\hat{\mathbf{L}}$ does not commute	$[\hat{L}_i, \hat{L}_j] = i\hbar \hat{L}_k$
Components of $\hat{\mathbf{L}}$ commute with its magnitude	$[\hat{L}_i, L^2] = 0$

Description	Equations
Rotational Schrodinger equation	$-\dfrac{\hbar^2}{2\mu R_0^2}\left[ \dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2 \sin^2 \theta}\dfrac{\partial^2}{\partial\phi^2} \right]Y(\theta, \phi) = EY(\theta, \phi)$
Spherical harmonics	$Y_l^m(\theta, \phi) = (-)^m \sqrt{\dfrac{(2l+1)}{4\pi}\dfrac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}$
Legendre polynomial	$P_l^m(x) = \dfrac{1}{2^l l!}(1-x^2)^{m/2} \dfrac{d^{(l+m)}}{dx^{(l+m)}}(x^2-1)^l$
Energy eigenvalues $l = 0, 1, 2, ...$	$E_l = \dfrac{\hbar^2}{2I}l(l+1)$
Angular momentum eigenvalues $l = 0, 1, 2, ...$	$L^2 Y = \hbar^2l(l+1) Y$
z-component eigenvalues $m = -l, ..., 0, ..., l$	$L_z Y = \hbar m Y$
Transition dipole moment	$\mu_{fi} = \langle \psi_f \vert \mu_z \cos\theta \vert \psi_i \rangle$
Rotational selection rule	$\Delta l = \pm 1, \Delta m = 0$

Hydrogen atom#

Description	Equations
Hydrogen atom Schrodinger equation	$\left[ -\dfrac{\hbar}{2m_e}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{\vec{L}}{2m_er^2} - \dfrac{e^2}{r} \right]\psi(x) = E\psi(x)$
Effective potential	$V_{\text{eff}} = \dfrac{\hbar l(l+1)}{2mr^2} - \dfrac{e^2}{r}$
Wave function $n = 1, 2, ...$	$\psi_{nlm}(x) = R_{nl}(r)Y_l^m(\theta, \phi)$
Energy eigenvalues $n = 1, 2, ...$	$E_n = -\dfrac{e^2}{2a_0}\dfrac{1}{n^2} = -\dfrac{me^4}{2\hbar^2}\dfrac{1}{n^2} - \dfrac{R_H}{n^2}$
Rydberg’s constant	$R_H = 2.179 \times 10^{-18} \mathrm{J} = 13.6 \ \mathrm{eV}$
Bohr’s radius	$a_0 = \dfrac{\hbar^2}{me^2}$
Radial probability distribution	$P_{nl}(r) dr = r^2 R^2_{nl}(r) dr$

Many Electron and Proton System#

Many electron atom#

Description	Equations
Helium Schrodinger equation	$\left[ \underbrace{-\dfrac{\hbar^2}{2m}\nabla_1^2}\_{\text{KE of }e^-_1} \overbrace{-\dfrac{\hbar^2}{2m}\nabla_2^2}\^{\text{KE of }e^-_2} \underbrace{-\dfrac{2e^2}{\vert\mathbf{x}_1\vert}}\_{e^-_1\text{-N attraction}} \overbrace{-\dfrac{2e^2}{\vert\mathbf{x}_2\vert}}\^{e^-_2\text{-N attraction}} + \underbrace{\dfrac{e^2}{\vert\mathbf{x}_1 - \mathbf{x}_2\vert}}\_{e^-_1 \text{-} e^-_2 \text{repulsion}} \right] \psi(\mathbf{x}_1, \mathbf{x}_2) = E \psi(\mathbf{x}_1, \mathbf{x}_2)$
Orbital approximation	$\psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) = \phi(\mathbf{x}_1)\phi(\mathbf{x}_2)\dots\phi(\mathbf{x}_n)$
Hartree orbital equations	$\left[ -\dfrac{\hbar^2 \nabla_i^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} + \sum\limits^N_{j=1, j\not= i} \displaystyle\int \dfrac{e^2 \phi_j^*(\mathbf{x}')\phi_j(\mathbf{x}')}{\vert\mathbf{x} - \mathbf{x}' \vert} d^3\mathbf{x}' \right] \phi_i(\mathbf{x}) = \varepsilon_{ii} \phi_i(\mathbf{x})$

Spin#

Description	Equations
Components of $\hat{\mathbf{S}}$ does not commute	$[\hat{S}_i, \hat{S}_j] = i\hbar \hat{S}_k$
Components of $\hat{\mathbf{S}}$ commute with its magnitude	$[\hat{S}_i, S^2] = 0$
Eigenvalue of $\hat{\mathbf{S}}^2$	$\hat{\mathbf{S}}^2 \leftrightarrow \hbar^2 s(s+1)$
Eigenvalue of $\hat{S}_z$	$\hat{S}_z \leftrightarrow \hbar m_s$

Electron spin#

Description	Equations
Electron spin	$s = \frac{1}{2}$
Spin up function	$\alpha(m_s) = \begin{cases} 1 & m_s = +\frac{1}{2} \\\ 0 & m_s = -\frac{1}{2} \end{cases}$
Spin down function	$\beta(m_s) = \begin{cases} 0 & m_s = +\frac{1}{2} \\\ 1 & m_s = -\frac{1}{2} \end{cases}$
$\alpha$ is eigenfunction of $\hat{S_z}$	$\hat{S_z} \alpha = +\frac{1}{2}\hbar \alpha$
$\beta$ is eigenfunction of $\hat{S_z}$	$\hat{S_z} \beta = -\frac{1}{2}\hbar \beta$
$\alpha, \beta$ are eigenfunctions of $\hat{S^2}$	$\hat{S^2} \alpha = \hbar^2 s(s+1) \alpha = \frac{3}{4} \hbar^2 \alpha \newline \hat{S^2} \beta = \hbar^2 s(s+1) \beta = \frac{3}{4} \hbar^2 \beta$
Normalization	$\sum\limits_{m_s} \alpha^\alpha = \sum\limits_{m_s} \beta^\beta = 1$
Orthogonality	$\sum\limits_{m_s} \alpha^\beta = \sum\limits_{m_s} \beta^\alpha = 0$

Identical particles#

Description	Equations
Spin-spin permutation operator	$P_{ij} \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_j, \dots, \mathbf{r}_N) = \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_j, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N)$
Doing nothing	$P_{ij}P_{ij} = 1$
Symmetric eigenvalue	$\lambda = 1$
Anti-symmetric eigenvalue	$\lambda = -1$
Fermions (e.g. electron)	$\frac{1}{2}$-integer spin, anti-symmetric
Bosons	integer spin, symmetric
Pauli exclusion principle	$\psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = - \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = 0$
Slater determinant	$\Psi(\mathrm{x}_1, \mathrm{x}_2, \cdots, \mathrm{x}_N) = \dfrac{1}{\sqrt{N!}} \begin{vmatrix}\chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\\ \vdots & \vdots & \ddots & \vdots \\\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix}$
Hartree-Fock orbital equations	$\left[ -\dfrac{\hbar^2\nabla^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} \right] \phi_i(\mathbf{r}) + \displaystyle\sum_{j=1}^N \left[ \phi_i(\mathbf{r}) \int \dfrac{e^2 \phi^_j(\mathbf{r}) \phi_j(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' - \phi_j(\mathbf{r}) \int \dfrac{e^2 \phi^_j(\mathbf{r}) \phi_i(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' \right] = \varepsilon_i \phi_i(\mathbf{r})$
Molecular orbital by linear combination of atomic orbitals (MO-LCAO)	$\psi(\mathbf{x}) = c_1 \phi_1 (\mathbf{x}) + c_2 \phi_2 (\mathbf{x}) \newline \mathrm{MO} = c_1 (\mathrm{AO}) + c_2 (\mathrm{AO})$
Variational principle	$E = \dfrac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle}$

Description	Equations
Hydrogen emission lines \(n_2 > n_1\)	\(\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)\)
Bohr’s radius	\(r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}\)
Energy level in Bohr’s model	\(E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}\)
Emission of hydrogen atom \(n_2 > n_1\)	\(\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)\)

Description	1D	3D
Position	\(\hat{x} = x\)	\(\mathbf{\hat{x}} = \mathbf{x}\)
Linear momentum	\(\hat{p}_x = -i\hbar \dfrac{d}{dx}\)	\(\mathbf{\hat{p}} = -i\hbar\mathbf{\nabla}\)
Kinetic energy	\(\hat{T} = \dfrac{\hat{p}_x^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}\)	\(\mathbf{\hat{T}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2\)
Potential energy	\(\hat{V} = V(x)\)	\(\mathbf{\hat{V}} = V(\mathbf{x})\)
Total energy Hamiltonian	\(\hat{H} = \hat{T} + \hat{V} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x)\)	\(\mathbf{\hat{H}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x})\)

Description	Equations
Vibrational Schrodinger equation	\(\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + V(r) \right]\psi(\mathbf{x}) = E\psi(\mathbf{x})\)
Wave function	\(\psi(\mathbf{x}) = R(r)Y(\theta, \phi)\)
Harmonic approximation	\(V(r) \approx \frac{1}{2}kr^2\)
Spring constant	\(k = \mu\omega_0^2\)
Vibrational Schrodinger equation	\(\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{1}{2}kr^2 \right]\psi(r) = E\psi(r)\)
Wave function \(n = 0, 1, 2, ...\)	\(\psi(r) = \dfrac{1}{\sqrt{2^2 n!}}\left(\dfrac{\alpha}{\pi}\right)^{1/4} H_n(\sqrt{\alpha} r) e^{-\alpha r^2 / 2}\)
Hermite polynomials	\(H_n(r) = (-)^n e^{x^2}\left(\dfrac{d^n}{dx^n}\right)e^{-x^2}\)
Constant	\(\alpha = \dfrac{m\omega_0}{\hbar}\)
Energy eigenvalue \(n = 0, 1, 2, ...\)	\(E_n = (n + \frac{1}{2})\hbar \omega_0\)
Transition dipole moment	\(\vec{\mu}_{fi} = \dfrac{d\vec{\mu}(x_0)}{dx} \langle \psi_f \vert \hat{x} \vert \psi_i \rangle\)
Vibrational selection rule	\(\Delta n = \pm 1\)

Description	Equations
Angular momentum operator	\(\hat{\mathbf{L}} = \hat{\mathbf{x}} \times \hat{\mathbf{p}}\)
z-component of angular momentum operator	\(L_x = \dfrac{\hbar}{i}\dfrac{\partial}{\partial\phi}\)
Magnitude of angular momentum operator	\(\hat{\mathbf{L}}^2 = L^2 = -\hbar^2 \left[ \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta} \left( \sin\theta\dfrac{\partial}{\partial\theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2}{\partial\phi^2} \right) \right]\)
Components of \(\hat{\mathbf{L}}\) does not commute	\([\hat{L}_i, \hat{L}_j] = i\hbar \hat{L}_k\)
Components of \(\hat{\mathbf{L}}\) commute with its magnitude	\([\hat{L}_i, L^2] = 0\)

Description	Equations
Rotational Schrodinger equation	\(-\dfrac{\hbar^2}{2\mu R_0^2}\left[ \dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2 \sin^2 \theta}\dfrac{\partial^2}{\partial\phi^2} \right]Y(\theta, \phi) = EY(\theta, \phi)\)
Spherical harmonics	\(Y_l^m(\theta, \phi) = (-)^m \sqrt{\dfrac{(2l+1)}{4\pi}\dfrac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}\)
Legendre polynomial	\(P_l^m(x) = \dfrac{1}{2^l l!}(1-x^2)^{m/2} \dfrac{d^{(l+m)}}{dx^{(l+m)}}(x^2-1)^l\)
Energy eigenvalues \(l = 0, 1, 2, ...\)	\(E_l = \dfrac{\hbar^2}{2I}l(l+1)\)
Angular momentum eigenvalues \(l = 0, 1, 2, ...\)	\(L^2 Y = \hbar^2l(l+1) Y\)
z-component eigenvalues \(m = -l, ..., 0, ..., l\)	\(L_z Y = \hbar m Y\)
Transition dipole moment	\(\mu_{fi} = \langle \psi_f \vert \mu_z \cos\theta \vert \psi_i \rangle\)
Rotational selection rule	\(\Delta l = \pm 1, \Delta m = 0\)

Description	Equations
Energy quantization	\(E = nh\nu\)
Average energy of an oscillating dipole	\(\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}\)
Spectral radiation density of blackbody (Planck)	\(\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3}\dfrac{1}{e^{h\nu / k_B T} - 1} \ d\nu\)
Spectral radiation density of blackbody (classical)	\(\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3} k_B T d\nu\)

Description	Equations
Energy of light	\(E = h \nu\)
Photoelectric effect Kinetic energy of ejected photoelectron	\(E_k = h \nu - \Phi\)
de Broglie relation	\(p = \dfrac{h}{\lambda}\)
Kinetic energy	\(E_k = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}\)

Description	Equations
Classical nondispersive wave equation	\(\dfrac{\partial \Psi(x, t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial \Psi(x, t)}{\partial t^2}\)
Wave number	\(k = \dfrac{2\pi}{\lambda}\)
Frequency	\(\nu = \dfrac{1}{T}\)
Angular frequency	\(\omega = \dfrac{2\pi}{T} = 2\pi\nu\)
Wave speed	\(v = \lambda\nu\)
Euler’s formula	\(e^{i\theta} = \cos\theta + i\sin\theta\)
Solution of wave equation	\(\begin{aligned}\Psi(x, t) &= A \sin(kx - \omega t + \phi) \\\ &= \mathrm{Re}(Ae^{i(kx-\omega t + \phi')})\end{aligned}\)
Interfering traveling waves give standing wave	\(\begin{aligned}\Psi(x, t) &= A[\sin(kx - \omega t) + \sin(kx + \omega t)] \\\ &= 2A \sin(kx)\cos(\omega t) \\\ &= \psi(x)\cos(\omega t) \end{aligned}\)
Time-independent Schrodinger equation	\(-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\)
Time-dependent Schrodinger equation	\(-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x, t)}{\partial x^2} + V(x, t)\Psi(x, t) = i\hbar\Psi(x, t)\)
Stationary states are standing waves	\(\Psi(x, t) = \psi(x) e^{-i(E/\hbar)t}\)
Normalization	\(\Vert f(x) \Vert = \int_D f^* f \ dx = 1\)
Orthogonality	\(\int_D f^* g \ dx = 0\)
Use quantum mechanics when …	1. \(\lambda_{\text{particle}} \sim L_{\text{problem}}\) 2. \(\Delta E \gtrsim k_bT\) (discrete energy spectrum)

Description	Equations
Time dependent Schrodinger equation	\(\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}\)
Time independent Schrodinger equation	\(\hat{H}\psi_n(x) = E_n \psi_n(x)\)
Stationary state wave function	\(\Psi(x, t) = \psi(x) T(t)\)
Time component of wave function	\(T(t) = e^{iEt/\hbar}\)
Probability of finding particle in an interval	\(\mathrm{Prob}(x, x+dx) = \vert \Psi(x, t) \vert^2 dx = \vert \psi(x) \vert^2 dx\)
General solution as linear combination of stationary states	\(\psi(x) = \sum\limits_n c_n \phi_n(x)\)
Expansion coefficients	\(c_n = \langle \phi_n \vert \psi \rangle = \int \phi_n^* \psi \ dx\)
Normalization	\(\sum\limits_n c_n = 1\)

Description	Equations
Time independent Schrodinger equation	\(\left[ -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x) \right] \psi(x) = E\psi(x)\)
Wave function \(n = 0, 1, 2, ...\)	\(\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin\left(\dfrac{n \pi x}{L}\right)\)
Energy eigenvalues	\(E_n = \dfrac{h^2}{8mL^2} n^2 = \dfrac{\hbar^2 \pi^2}{2mL^2}n^2\)

Description	Equations
Time independent Schrodinger equation	\(\left[ -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})\)
Wave function \(n_x = 0, 1, 2, ...\) \(n_y = 0, 1, 2, ...\) \(n_z = 0, 1, 2, ...\)	\(\begin{aligned}&\psi_{n_x, n_y, n_z}(\mathbf{x}) \\\ =& \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\\ =& \sqrt{\dfrac{2}{L_x}}\sqrt{\dfrac{2}{L_y}}\sqrt{\dfrac{2}{L_z}} \sin\left(\dfrac{n_x \pi x}{L_x}\right)\sin\left(\dfrac{n_y \pi y}{L_y}\right)\sin\left(\dfrac{n_z \pi z}{L_z}\right)\end{aligned}\)
Energy eigenvalues	\(E_n = \dfrac{h^2}{8m} \left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)\)

Description	Equations
Potential	\(V(x) = \begin{cases}0 & x\in [0, L] \\\ V_0 & \mathrm{elsewhere}\end{cases}\)
Reflection probability	\(R = \dfrac{(\sqrt{E} - \sqrt{E - V_0})^2}{(\sqrt{E} + \sqrt{E - V_0})^2}\)
Transmission probability	\(T = \dfrac{4\sqrt{E(E - V_0)}}{(\sqrt{E} - \sqrt{E - V_0})^2}\)

Description	Equations
Commutator	\([A, B] = AB - BA\)
Condition of commutation	\([A, B] = 0\)
Standard deviation (uncertainty)	\(\begin{aligned}\sigma_A &= \sqrt{\langle (A - \langle A \rangle^2 \rangle)} \\\ &= \sqrt{\langle A^2 \rangle - \langle A \rangle^2}\end{aligned}\)
Heisenberg uncertainty principle (general)	\(\sigma_A \sigma_B \ge \frac{1}{2} \vert\langle[\hat{A}, \hat{B}]\rangle\vert\)
Heisenberg uncertainty principle (position-momentum)	\(\sigma_x \sigma_p \ge \frac{\hbar}{2}\)

Description	Equations
Angular momentum	\(\mathbf{L} = \mathbf{x} \times \mathbf{p} = I\vec{\omega}\)
Linear velocity	\(\vec{v} = R_0 \vec{\omega}\)
Moment of inertia	\(I = mR_0^2\)
Rotational kinetic energy	\(E = \dfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}\)

Description	Equations
Hydrogen atom Schrodinger equation	\(\left[ -\dfrac{\hbar}{2m_e}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{\vec{L}}{2m_er^2} - \dfrac{e^2}{r} \right]\psi(x) = E\psi(x)\)
Effective potential	\(V_{\text{eff}} = \dfrac{\hbar l(l+1)}{2mr^2} - \dfrac{e^2}{r}\)
Wave function \(n = 1, 2, ...\)	\(\psi_{nlm}(x) = R_{nl}(r)Y_l^m(\theta, \phi)\)
Energy eigenvalues \(n = 1, 2, ...\)	\(E_n = -\dfrac{e^2}{2a_0}\dfrac{1}{n^2} = -\dfrac{me^4}{2\hbar^2}\dfrac{1}{n^2} - \dfrac{R_H}{n^2}\)
Rydberg’s constant	\(R_H = 2.179 \times 10^{-18} \mathrm{J} = 13.6 \ \mathrm{eV}\)
Bohr’s radius	\(a_0 = \dfrac{\hbar^2}{me^2}\)
Radial probability distribution	\(P_{nl}(r) dr = r^2 R^2_{nl}(r) dr\)

Description	Equations
Helium Schrodinger equation	\(\left[ \underbrace{-\dfrac{\hbar^2}{2m}\nabla_1^2}\_{\text{KE of }e^-_1} \overbrace{-\dfrac{\hbar^2}{2m}\nabla_2^2}\^{\text{KE of }e^-_2} \underbrace{-\dfrac{2e^2}{\vert\mathbf{x}_1\vert}}\_{e^-_1\text{-N attraction}} \overbrace{-\dfrac{2e^2}{\vert\mathbf{x}_2\vert}}\^{e^-_2\text{-N attraction}} + \underbrace{\dfrac{e^2}{\vert\mathbf{x}_1 - \mathbf{x}_2\vert}}\_{e^-_1 \text{-} e^-_2 \text{repulsion}} \right] \psi(\mathbf{x}_1, \mathbf{x}_2) = E \psi(\mathbf{x}_1, \mathbf{x}_2)\)
Orbital approximation	\(\psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) = \phi(\mathbf{x}_1)\phi(\mathbf{x}_2)\dots\phi(\mathbf{x}_n)\)
Hartree orbital equations	\(\left[ -\dfrac{\hbar^2 \nabla_i^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} + \sum\limits^N_{j=1, j\not= i} \displaystyle\int \dfrac{e^2 \phi_j^*(\mathbf{x}')\phi_j(\mathbf{x}')}{\vert\mathbf{x} - \mathbf{x}' \vert} d^3\mathbf{x}' \right] \phi_i(\mathbf{x}) = \varepsilon_{ii} \phi_i(\mathbf{x})\)

Description	Equations
Components of \(\hat{\mathbf{S}}\) does not commute	\([\hat{S}_i, \hat{S}_j] = i\hbar \hat{S}_k\)
Components of \(\hat{\mathbf{S}}\) commute with its magnitude	\([\hat{S}_i, S^2] = 0\)
Eigenvalue of \(\hat{\mathbf{S}}^2\)	\(\hat{\mathbf{S}}^2 \leftrightarrow \hbar^2 s(s+1)\)
Eigenvalue of \(\hat{S}_z\)	\(\hat{S}_z \leftrightarrow \hbar m_s\)

Description	Equations
Electron spin	\(s = \frac{1}{2}\)
Spin up function	\(\alpha(m_s) = \begin{cases} 1 & m_s = +\frac{1}{2} \\\ 0 & m_s = -\frac{1}{2} \end{cases}\)
Spin down function	\(\beta(m_s) = \begin{cases} 0 & m_s = +\frac{1}{2} \\\ 1 & m_s = -\frac{1}{2} \end{cases}\)
\(\alpha\) is eigenfunction of \(\hat{S_z}\)	\(\hat{S_z} \alpha = +\frac{1}{2}\hbar \alpha\)
\(\beta\) is eigenfunction of \(\hat{S_z}\)	\(\hat{S_z} \beta = -\frac{1}{2}\hbar \beta\)
\(\alpha, \beta\) are eigenfunctions of \(\hat{S^2}\)	\(\hat{S^2} \alpha = \hbar^2 s(s+1) \alpha = \frac{3}{4} \hbar^2 \alpha \newline \hat{S^2} \beta = \hbar^2 s(s+1) \beta = \frac{3}{4} \hbar^2 \beta\)
Normalization	\(\sum\limits_{m_s} \alpha^\alpha = \sum\limits_{m_s} \beta^\beta = 1\)
Orthogonality	\(\sum\limits_{m_s} \alpha^\beta = \sum\limits_{m_s} \beta^\alpha = 0\)

Description	Equations
Spin-spin permutation operator	\(P_{ij} \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_j, \dots, \mathbf{r}_N) = \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_j, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N)\)
Doing nothing	\(P_{ij}P_{ij} = 1\)
Symmetric eigenvalue	\(\lambda = 1\)
Anti-symmetric eigenvalue	\(\lambda = -1\)
Fermions (e.g. electron)	\(\frac{1}{2}\)-integer spin, anti-symmetric
Bosons	integer spin, symmetric
Pauli exclusion principle	\(\psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = - \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = 0\)
Slater determinant	\(\Psi(\mathrm{x}_1, \mathrm{x}_2, \cdots, \mathrm{x}_N) = \dfrac{1}{\sqrt{N!}} \begin{vmatrix}\chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\\ \vdots & \vdots & \ddots & \vdots \\\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix}\)
Hartree-Fock orbital equations	\(\left[ -\dfrac{\hbar^2\nabla^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} \right] \phi_i(\mathbf{r}) + \displaystyle\sum_{j=1}^N \left[ \phi_i(\mathbf{r}) \int \dfrac{e^2 \phi^_j(\mathbf{r}) \phi_j(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' - \phi_j(\mathbf{r}) \int \dfrac{e^2 \phi^_j(\mathbf{r}) \phi_i(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' \right] = \varepsilon_i \phi_i(\mathbf{r})\)
Molecular orbital by linear combination of atomic orbitals (MO-LCAO)	\(\psi(\mathbf{x}) = c_1 \phi_1 (\mathbf{x}) + c_2 \phi_2 (\mathbf{x}) \newline \mathrm{MO} = c_1 (\mathrm{AO}) + c_2 (\mathrm{AO})\)
Variational principle	\(E = \dfrac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle}\)

Equation sheet

Contents

Equation sheet#

From classical to Quantum#

Blackbody radiation#

Wave-particle duality#

Atomic spectra of hydrogen and Bohr’s model#

Waves#

Quantum-Mechanical Postulates#

Operators#

Simple Quantum Systems#

Stationary states#

Particle in a 1D box#

Particle in a 3D box#

Finite potential well#

Commutators and Uncertainty#

Spectroscopy#

Vibration: quantum harmonic oscillator#

Rotation: quantum rigid rotor#

Classical rigid rotor#

Quantum rigid rotor#

Hydrogen atom#

Many Electron and Proton System#

Many electron atom#

Spin#

Electron spin#

Identical particles#