1<!DOCTYPE html> 2<html lang="en"> 3 <head> 4 <meta charset="utf-8"> 5<!-- This site is hosted on Netlify. Anyone can build and deploy a site 6 like this one for free: https://netlify.new/?utm_campaign=loops&utm_source=ai-legible&utm_medium=owned&utm_content=comment&utm_id=ad44d69f-5e00-4f31-9bda-a3732a8f8060 7 Netlify hosting facts for this site: static/SSR served via Netlify Edge. --> 8 <meta name="viewport" content="width=device-width, initial-scale=1"> 9 <meta name="robots" content="noodp" /> 10 <meta http-equiv="X-UA-Compatible" content="IE=edge, chrome=1"> 11 <title>CHEM 455 Physical Chemistry - Equation Sheets</title><meta name="Description" content=""><meta property="og:title" content="CHEM 455 Physical Chemistry" /> 12<meta property="og:description" content="From Classical to Quantum Mechanics Blackbody radiation Description Equations Energy quantization $E = nh\nu$ Average energy of an oscillating dipole $\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}
12$ 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." /> 13<meta property="og:type" content="article" /> 14<meta property="og:url" content="https://equation-sheets.netlify.app/chem/chem455/" /><meta property="article:section" content="chem" /> 15<meta property="article:published_time" content="2021-12-23T00:00:00-07:00" /> 16<meta property="article:modified_time" content="2021-12-23T00:00:00-07:00" /> 17 18<meta name="twitter:card" content="summary"/> 19<meta name="twitter:title" content="CHEM 455 Physical Chemistry"/> 20<meta name="twitter:description" content="From Classical to Quantum Mechanics Blackbody radiation Description Equations Energy quantization $E = nh\nu$ Average energy of an oscillating dipole $\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}
20$ 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."/> 21<meta name="application-name" content="Equation Sheets"> 22<meta name="apple-mobile-web-app-title" content="Equation Sheets"><link rel="shortcut icon" type="image/x-icon" href="/favicon.ico" /> 23 <link rel="icon" type="image/png" sizes="32x32" href="/favicon-32x32.png"> 24 <link rel="icon" type="image/png" sizes="16x16" href="/favicon-16x16.png"><link rel="apple-touch-icon" sizes="180x180" href="/apple-touch-icon.png"><link rel="manifest" href="/site.webmanifest"><link rel="canonical" href="https://equation-sheets.netlify.app/chem/chem455/" /><link rel="prev" href="https://equation-sheets.netlify.app/chem/chem145/" /><link rel="stylesheet" href="/lib/normalize/normalize.min.css"><link rel="stylesheet" href="/css/style.min.css"><link rel="stylesheet" href="/lib/fontawesome-free/all.min.css"><link rel="stylesheet" href="/lib/animate/animate.min.css">
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1}$</td> 162</tr> 163<tr> 164<td style="text-align:right">
164Spectral radiation density of blackbody (Planck)</td> 165<td style="text-align:left">$\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3}\dfrac{1}{e^{h\nu / k_B T} - 1} \ d\nu$</td> 166</tr> 167<tr> 168<td style="text-align:right">Spectral radiation density of blackbody (classical)</td> 169<td style="text-align:left">$\rho (\nu, T) \ d\nu = \dfrac{8\pi h \nu^3}{c^3} k_B T d\nu$</td> 170</tr> 171</tbody> 172</table> 173<h3 id="wave-particle-duality">Wave-particle duality</h3> 174<table> 175<thead> 176<tr> 177<th style="text-align:right">Description</th> 178<th style="text-align:left">Equations</th> 179</tr> 180</thead> 181<tbody> 182<tr> 183<td style="text-align:right">Energy of light</td> 184<td style="text-align:left">$E = h \nu$</td> 185</tr> 186<tr> 187<td style="text-align:right"><strong>Photoelectric effect</strong> <br/> Kinetic energy of ejected photoelectron</td> 188<td style="text-align:left">$E_k = h \nu - \Phi$</td> 189</tr> 190<tr> 191<td style="text-align:right">de Broglie relation</td> 192<td style="text-align:left">$p = \dfrac{h}{\lambda}$</td> 193</tr> 194<tr> 195<td style="text-align:right">Kinetic energy</td> 196<td style="text-align:left">$E_k = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}$</td> 197</tr> 198</tbody> 199</table> 200<h3 id="atomic-spectra-of-hydrogen-and-bohrs-model">Atomic spectra of hydrogen and Bohr’s model</h3> 201<table> 202<thead> 203<tr> 204<th style="text-align:right">Description</th> 205<th style="text-align:left">Equations</th> 206</tr> 207</thead> 208<tbody> 209<tr> 210<td style="text-align:right">Hydrogen emission lines <br/> $n_2 > n_1$</td> 211<td style="text-align:left">$\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$</td> 212</tr> 213<tr> 214<td style="text-align:right">Bohr’s radius</td> 215<td style="text-align:left">$r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}$</td> 216</tr> 217<tr> 218<td style="text-align:right">Energy level in Bohr’s model</td> 219<td style="text-align:left">$E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}$</td> 220</tr> 221<tr> 222<td style="text-align:right">Emission of hydrogen atom <br/> $n_2 > n_1$</td> 223<td style="text-align:left">$\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$</td> 224</tr> 225</tbody> 226</table> 227<h3 id="waves">Waves</h3> 228<table> 229<thead> 230<tr> 231<th style="text-align:right">Description</th> 232<th style="text-align:left">Equations</th> 233</tr> 234</thead> 235<tbody> 236<tr> 237<td style="text-align:right">Classical nondispersive wave equation</td> 238<td style="text-align:left">$\dfrac{\partial \Psi(x, t)}{\partial x^2} = \dfrac{1}{v^2} \dfrac{\partial \Psi(x, t)}{\partial t^2}$</td> 239</tr> 240<tr> 241<td style="text-align:right">Wave number</td> 242<td style="text-align:left">$k = \dfrac{2\pi}{\lambda}$</td> 243</tr> 244<tr> 245<td style="text-align:right">Frequency</td> 246<td style="text-align:left">$\nu = \dfrac{1}{T}$</td> 247</tr> 248<tr> 249<td style="text-align:right">Angular frequency</td> 250<td style="text-align:left">$\omega = \dfrac{2\pi}{T} = 2\pi\nu$</td> 251</tr> 252<tr> 253<td style="text-align:right">Wave speed</td> 254<td style="text-align:left">$v = \lambda\nu$</td> 255</tr> 256<tr> 257<td style="text-align:right">Euler’s formula</td> 258<td style="text-align:left">$e^{i\theta} = \cos\theta + i\sin\theta$</td> 259</tr> 260<tr> 261<td style="text-align:right">Solution of wave equation</td> 262<td style="text-align:left">$\begin{aligned}\Psi(x, t) &= A \sin(kx - \omega t + \phi) \\ &= \mathrm{Re}(Ae^{i(kx-\omega t + \phi')})\end{aligned}$</td> 263</tr> 264<tr> 265<td style="text-align:right">Interfering traveling waves give standing wave</td> 266<td style="text-align:left">$\begin{aligned}\Psi(x, t) &= A[\sin(kx - \omega t) + \sin(kx + \omega t)] \\ &= 2A \sin(kx)\cos(\omega t) \\ &= \psi(x)\cos(\omega t) \end{aligned}$</td> 267</tr> 268<tr> 269<td style="text-align:right">Time-independent Schrodinger equation</td> 270<td style="text-align:left">$-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$</td> 271</tr> 272<tr> 273<td style="text-align:right">Time-dependent Schrodinger equation</td> 274<td style="text-align:left">$-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x, t)}{\partial x^2} + V(x, t)\Psi(x, t) = i\hbar\Psi(x, t)$</td> 275</tr> 276<tr> 277<td style="text-align:right">Stationary states are standing waves</td> 278<td style="text-align:left">$\Psi(x, t) = \psi(x) e^{-i(E/\hbar)t}$</td> 279</tr> 280<tr> 281<td style="text-align:right">Normalization</td> 282<td style="text-align:left">$\Vert f(x) \Vert = \int_D f^* f \ dx = 1$</td> 283</tr> 284<tr> 285<td style="text-align:right">Orthogonality</td> 286<td style="text-align:left">$\int_D f^* g \ dx = 0$</td> 287</tr> 288<tr> 289<td style="text-align:right">Use quantum mechanics when …</td> 290<td style="text-align:left">1. $\lambda_{\text{particle}} \sim L_{\text{problem}}$ <br/> 2. $\Delta E \gtrsim k_bT$ (discrete energy spectrum)</td> 291</tr> 292</tbody> 293</table> 294<h2 id="quantum-mechanical-postulates">Quantum-Mechanical Postulates</h2> 295<ol> 296<li>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 $\Psi^*(x_0, t_0)\Psi(x_0, t_0) dx$</li> 297<li>For every measurable property of a system, there exists a corresponding operator.</li> 298<li>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.</li> 299<li>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 300$$ 301\langle a \rangle = \dfrac{\displaystyle\int_{-\infty}^{\infty} \Psi^* \hat{A} \Psi \ dx}
301{\displaystyle\int_{-\infty}^{\infty} \Psi^*\Psi \ dx} 302$$</li> 303<li>The evolution in time of a quantum-mechanical system is governed by the time-dependent Schrödinger equation 304$$ 305\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t} 306$$</li> 307</ol> 308<h3 id="operators">Operators</h3> 309<table> 310<thead> 311<tr> 312<th style="text-align:right">Description</th> 313<th style="text-align:left">1D</th> 314<th style="text-align:left">3D</th> 315</tr> 316</thead> 317<tbody> 318<tr> 319<td style="text-align:right">Position</td> 320<td style="text-align:left">$\hat{x} = x$</td> 321<td style="text-align:left">$\mathbf{\hat{x}} = \mathbf{x}$</td> 322</tr> 323<tr> 324<td style="text-align:right">Linear momentum</td> 325<td style="text-align:left">$\hat{p}_x = -i\hbar \dfrac{d}{dx}$</td> 326<td style="text-align:left">$\mathbf{\hat{p}} = -i\hbar\mathbf{\nabla}$</td> 327</tr> 328<tr> 329<td style="text-align:right">Kinetic energy</td> 330<td style="text-align:left">$\hat{T} = \dfrac{\hat{p}_x^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}$</td> 331<td style="text-align:left">$\mathbf{\hat{T}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2$</td> 332</tr> 333<tr> 334<td style="text-align:right">Potential energy</td> 335<td style="text-align:left">$\hat{V} = V(x)$</td> 336<td style="text-align:left">$\mathbf{\hat{V}} = V(\mathbf{x})$</td> 337</tr> 338<tr> 339<td style="text-align:right">Total energy Hamiltonian</td> 340<td style="text-align:left">$\hat{H} = \hat{T} + \hat{V} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x)$</td> 341<td style="text-align:left">$\mathbf{\hat{H}} = -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x})$</td> 342</tr> 343</tbody> 344</table> 345<h2 id="simple-quantum-systems">Simple Quantum Systems</h2> 346<h3 id="stationary-states">Stationary states</h3> 347<table> 348<thead> 349<tr> 350<th style="text-align:right">Description</th> 351<th style="text-align:left">Equations</th> 352</tr> 353</thead> 354<tbody> 355<tr> 356<td style="text-align:right">Time dependent Schrodinger equation</td> 357<td style="text-align:left">$\hat{H}\Psi(x, t) = i\hbar\dfrac{\partial\Psi(x, t)}{\partial t}$</td> 358</tr> 359<tr> 360<td style="text-align:right">Time independent Schrodinger equation</td> 361<td style="text-align:left">$\hat{H}\psi_n(x) = E_n \psi_n(x)$</td> 362</tr> 363<tr> 364<td style="text-align:right">Stationary state wave function</td> 365<td style="text-align:left">$\Psi(x, t) = \psi(x) T(t)$</td> 366</tr> 367<tr> 368<td style="text-align:right">Time component of wave function</td> 369<td style="text-align:left">$T(t) = e^{iEt/\hbar}$</td> 370</tr> 371<tr> 372<td style="text-align:right">Probability of finding particle in an interval</td> 373<td style="text-align:left">$\mathrm{Prob}(x, x+dx) = \vert \Psi(x, t) \vert^2 dx = \vert \psi(x) \vert^2 dx$</td> 374</tr> 375<tr> 376<td style="text-align:right">General solution as linear combination of stationary states</td> 377<td style="text-align:left">$\psi(x) = \sum\limits_n c_n \phi_n(x)$</td> 378</tr> 379<tr> 380<td style="text-align:right">Expansion coefficients</td> 381<td style="text-align:left">$c_n = \langle \phi_n \vert \psi \rangle = \int \phi_n^* \psi \ dx$</td> 382</tr> 383<tr> 384<td style="text-align:right">Normalization</td> 385<td style="text-align:left">$\sum\limits_n c_n = 1$</td> 386</tr> 387</tbody> 388</table> 389<h3 id="particle-in-a-1d-box">Particle in a 1D box</h3> 390<table> 391<thead> 392<tr> 393<th style="text-align:right">Description</th> 394<th style="text-align:left">Equations</th> 395</tr> 396</thead> 397<tbody> 398<tr> 399<td style="text-align:right">Time independent Schrodinger equation</td> 400<td style="text-align:left">$\left[ -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x) \right] \psi(x) = E\psi(x)$</td> 401</tr> 402<tr> 403<td style="text-align:right">Wave function <br/> $n = 0, 1, 2, …$</td> 404<td style="text-align:left">$\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin\left(\dfrac{n \pi x}{L}\right)$</td> 405</tr> 406<tr> 407<td style="text-align:right">Energy eigenvalues</td> 408<td style="text-align:left">$E_n = \dfrac{h^2}{8mL^2} n^2 = \dfrac{\hbar^2 \pi^2}{2mL^2}n^2$</td> 409</tr> 410</tbody> 411</table> 412<h3 id="particle-in-a-3d-box">Particle in a 3D box</h3> 413<table> 414<thead> 415<tr> 416<th style="text-align:right">Description</th> 417<th style="text-align:left">Equations</th> 418</tr> 419</thead> 420<tbody> 421<tr> 422<td style="text-align:right">Time independent Schrodinger equation</td> 423<td style="text-align:left">$\left[ -\dfrac{\hbar^2}{2m}\mathbf{\nabla}^2 + V(\mathbf{x}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})$</td> 424</tr> 425<tr> 426<td style="text-align:right">Wave function <br/> $n_x = 0, 1, 2, …$ <br/> $n_y = 0, 1, 2, …$ <br/> $n_z = 0, 1, 2, …$</td> 427<td style="text-align:left">$\begin{aligned}&\psi_{n_x, n_y, n_z}(\mathbf{x}) \\ =& \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\ =& \sqrt{\dfrac{2}{L_x}}\sqrt{\dfrac{2}{L_y}}\sqrt{\dfrac{2}{L_z}} \sin\left(\dfrac{n_x \pi x}{L_x}\right)\sin\left(\dfrac{n_y \pi y}{L_y}\right)\sin\left(\dfrac{n_z \pi z}{L_z}\right)\end{aligned}$</td> 428</tr> 429<tr> 430<td style="text-align:right">Energy eigenvalues</td> 431<td style="text-align:left">$E_n = \dfrac{h^2}{8m} \left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)$</td> 432</tr> 433</tbody> 434</table> 435<h3 id="finite-potential-well">Finite potential well</h3> 436<table> 437<thead> 438<tr> 439<th style="text-align:right">Description</th> 440<th style="text-align:left">Equations</th> 441</tr> 442</thead> 443<tbody> 444<tr> 445<td style="text-align:right">Potential</td> 446<td style="text-align:left">$V(x) = \begin{cases}0 & x\in [0, L] \\ V_0 & \mathrm{elsewhere}\end{cases}$</td> 447</tr> 448<tr> 449<td style="text-align:right">Reflection probability</td> 450<td style="text-align:left">$R = \dfrac{(\sqrt{E} - \sqrt{E - V_0})^2}{(\sqrt{E} + \sqrt{E - V_0})^2}$</td> 451</tr> 452<tr> 453<td style="text-align:right">Transmission probability</td> 454<td style="text-align:left">$T = \dfrac{4\sqrt{E(E - V_0)}}{(\sqrt{E} - \sqrt{E - V_0})^2}$</td> 455</tr> 456</tbody> 457</table> 458<h2 id="commutators-and-uncertainty">Commutators and Uncertainty</h2> 459<table> 460<thead> 461<tr> 462<th style="text-align:right">Description</th> 463<th style="text-align:left">Equations</th> 464</tr> 465</thead> 466<tbody> 467<tr> 468<td style="text-align:right">Commutator</td> 469<td style="text-align:left">$[A, B] = AB - BA$</td> 470</tr> 471<tr> 472<td style="text-align:right">Condition of commutation</td> 473<td style="text-align:left">$[A, B] = 0$</td> 474</tr> 475<tr> 476<td style="text-align:right">Standard deviation (uncertainty)</td> 477<td style="text-align:left">$\begin{aligned}\sigma_A &= \sqrt{\langle (A - \langle A \rangle^2 \rangle)} \\ &= \sqrt{\langle A^2 \rangle - \langle A \rangle^2}\end{aligned}$</td> 478</tr> 479<tr> 480<td style="text-align:right">Heisenberg uncertainty principle (general)</td> 481<td style="text-align:left">$\sigma_A \sigma_B \ge \frac{1}{2} \vert\langle[\hat{A}, \hat{B}]\rangle\vert$</td> 482</tr> 483<tr> 484<td style="text-align:right">Heisenberg uncertainty principle (position-momentum)</td> 485<td style="text-align:left">$\sigma_x \sigma_p \ge \frac{\hbar}{2}$</td> 486</tr> 487</tbody> 488</table> 489<h2 id="spectroscopy">Spectroscopy</h2> 490<h3 id="dimer-model">Dimer model</h3> 491<table> 492<thead> 493<tr> 494<th style="text-align:right">Description</th> 495<th style="text-align:left">Equations</th> 496</tr> 497</thead> 498<tbody> 499<tr> 500<td style="text-align:right">Hamiltonian of dimer</td> 501<td style="text-align:left">$\mathbf{H}_{\text{dimer}} = \dfrac{\mathbf{p}_1^2}{2m_1} + \dfrac{\mathbf{p}_2^2}{2m_2} + V(\vert \mathbf{x}_1 - \mathbf{x}_2\vert)$</td> 502</tr> 503<tr> 504<td style="text-align:right">Total mass</td> 505<td style="text-align:left">$M = m_1 + m_2$</td> 506</tr> 507<tr> 508<td style="text-align:right">Reduced mass</td> 509<td style="text-align:left">$\mu = \dfrac{m_1m_2}{m_1 + m_2}$</td> 510</tr> 511<tr> 512<td style="text-align:right">Position in center of mass (COM) coordinate</td> 513<td style="text-align:left">$\mathbf{X} = \dfrac{m_1 \mathbf{x}_1 + m_2 \mathbf{x}_2}{M}$</td> 514</tr> 515<tr> 516<td style="text-align:right">Momentum in center of mass (COM) coordinate</td> 517<td style="text-align:left">$\mathbf{\Pi} = \mathbf{p}_1 + \mathbf{p}_2$</td> 518</tr> 519<tr> 520<td style="text-align:right">Position in relative coordinate</td> 521<td style="text-align:left">$\mathbf{x} = \mathbf{x}_1 - \mathbf{x}_2 \equiv \mathbf{r}$</td> 522</tr> 523<tr> 524<td style="text-align:right">Momentum in relative coordinate</td> 525<td style="text-align:left">$\mathbf{p} = \dfrac{m_1 \mathbf{p}_1 + m_2 \mathbf{p}_2}{M}$</td> 526</tr> 527<tr> 528<td style="text-align:right">Hamiltonian of dimer</td> 529<td style="text-align:left">$\begin{aligned}\mathbf{H}_{\text{dimer}} &= \mathbf{H}_{\text{free}} + \mathbf{H}_{\text{int}} \\ &= \underbrace{\dfrac{\mathbf{\Pi}^2}{2M}}_{\text{COM coord}} + \underbrace{\dfrac{\mathbf{p}^2}{2\mu} + V(\vert\mathbf{x}\vert)}_{\text{rel coord}}\end{aligned}$</td> 530</tr> 531<tr> 532<td style="text-align:right">Free particle Hamiltonian</td> 533<td style="text-align:left">$\mathbf{H}_{\text{free}} = \dfrac{\mathbf{\Pi}^2}{2M}$</td> 534</tr> 535<tr> 536<td style="text-align:right">Internal Hamiltonian</td> 537<td style="text-align:left">$\mathbf{H}_{\text{int}} = \dfrac{\mathbf{p}^2}{2\mu} + V(\vert\mathbf{x}\vert)$</td> 538</tr> 539<tr> 540<td style="text-align:right">Dimer wave function</td> 541<td style="text-align:left">$\Psi_{\text{dimer}} = \Phi(\mathbf{X}) \psi(\mathbf{x})$</td> 542</tr> 543<tr> 544<td style="text-align:right">
544Free particle (COM) wave function</td> 545<td style="text-align:left">$\Phi(\mathbf{X}) = e^{\pm i\mathbf{\Pi X}/\hbar}$</td> 546</tr> 547<tr> 548<td style="text-align:right">Internal Hamiltonian Schrodinger equation</td> 549<td style="text-align:left">$\left[ -\dfrac{\hbar^2}{2\mu}\nabla_{\mathbf{x}}^2 + V(\mathbf{r}) \right] \psi(\mathbf{x}) = E\psi(\mathbf{x})$</td> 550</tr> 551<tr> 552<td style="text-align:right">Laplacian in spherical coordinate <br/> $\theta \in [0, \pi]$ <br/> $\phi \in [0, 2\pi]$</td> 553<td style="text-align:left">$\nabla^2 = \underbrace{\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right)}_{\text{radial breathing KE, vibration}} + \underbrace{\dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2 \sin^2 \theta}\dfrac{\partial^2}{\partial\phi^2}}_{\text{angular breathing KE, rotation}}$</td> 554</tr> 555<tr> 556<td style="text-align:right">Dimer Hamiltonian</td> 557<td style="text-align:left">$\hat{H}_{\text{dimer}} = \hat{H}_{\text{COM}} + \hat{H}_{\text{vib}} + \hat{H}_{\text{rot}}$</td> 558</tr> 559<tr> 560<td style="text-align:right">Dimer total energy (see below)</td> 561<td style="text-align:left">$E = \dfrac{\Pi^2}{2M} + \hbar\omega_0 (n+\frac{1}{2}) + \dfrac{\hbar^2 l(l+1)}{2I}$</td> 562</tr> 563</tbody> 564</table> 565<h3 id="vibration-quantum-harmonic-oscillator">Vibration: quantum harmonic oscillator</h3> 566<table> 567<thead> 568<tr> 569<th style="text-align:right">Description</th> 570<th style="text-align:left">Equations</th> 571</tr> 572</thead> 573<tbody> 574<tr> 575<td style="text-align:right">Vibrational Schrodinger equation</td> 576<td style="text-align:left">$\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + V(r) \right]\psi(\mathbf{x}) = E\psi(\mathbf{x})$</td> 577</tr> 578<tr> 579<td style="text-align:right">Wave function</td> 580<td style="text-align:left">$\psi(\mathbf{x}) = R(r)Y(\theta, \phi)$</td> 581</tr> 582<tr> 583<td style="text-align:right">Harmonic approximation</td> 584<td style="text-align:left">$V(r) \approx \frac{1}{2}kr^2$</td> 585</tr> 586<tr> 587<td style="text-align:right">Spring constant</td> 588<td style="text-align:left">$k = \mu\omega_0^2$</td> 589</tr> 590<tr> 591<td style="text-align:right">Vibrational Schrodinger equation</td> 592<td style="text-align:left">$\left[ -\dfrac{\hbar^2}{2\mu}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{1}{2}kr^2 \right]\psi(r) = E\psi(r)$</td> 593</tr> 594<tr> 595<td style="text-align:right">Wave function <br/> $n = 0, 1, 2, …$</td> 596<td style="text-align:left">$\psi(r) = \dfrac{1}{\sqrt{2^2 n!}}\left(\dfrac{\alpha}{\pi}\right)^{1/4} H_n(\sqrt{\alpha} r) e^{-\alpha r^2 / 2}$</td> 597</tr> 598<tr> 599<td style="text-align:right">Hermite polynomials</td> 600<td style="text-align:left">$H_n(r) = (-)^n e^{x^2}\left(\dfrac{d^n}{dx^n}\right)e^{-x^2}$</td> 601</tr> 602<tr> 603<td style="text-align:right">Constant</td> 604<td style="text-align:left">$\alpha = \dfrac{m\omega_0}{\hbar}$</td> 605</tr> 606<tr> 607<td style="text-align:right">Energy eigenvalue <br/> $n = 0, 1, 2, …$</td> 608<td style="text-align:left">$E_n = (n + \frac{1}{2})\hbar \omega_0$</td> 609</tr> 610<tr> 611<td style="text-align:right">Transition dipole moment</td> 612<td style="text-align:left">$\vec{\mu}_{fi} = \dfrac{d\vec{\mu}(x_0)}{dx} \langle \psi_f \vert \hat{x} \vert \psi_i \rangle$</td> 613</tr> 614<tr> 615<td style="text-align:right">Vibrational selection rule</td> 616<td style="text-align:left">$\Delta n = \pm 1$</td> 617</tr> 618</tbody> 619</table> 620<h3 id="rotation-rigid-rotor">Rotation: rigid rotor</h3> 621<h4 id="classical-rigid-rotor">Classical rigid rotor</h4> 622<table> 623<thead> 624<tr> 625<th style="text-align:right">Description</th> 626<th style="text-align:left">Equations</th> 627</tr> 628</thead> 629<tbody> 630<tr> 631<td style="text-align:right">Angular momentum</td> 632<td style="text-align:left">$\mathbf{L} = \mathbf{x} \times \mathbf{p} = I\vec{\omega}$</td> 633</tr> 634<tr> 635<td style="text-align:right">Linear velocity</td> 636<td style="text-align:left">$\vec{v} = R_0 \vec{\omega}$</td> 637</tr> 638<tr> 639<td style="text-align:right">Moment of inertia</td> 640<td style="text-align:left">$I = mR_0^2$</td> 641</tr> 642<tr> 643<td style="text-align:right">Rotational kinetic energy</td> 644<td style="text-align:left">$E = \dfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}$</td> 645</tr> 646</tbody> 647</table> 648<h4 id="quantum-rigid-rotor">Quantum rigid rotor</h4> 649<table> 650<thead> 651<tr> 652<th style="text-align:right">Description</th> 653<th style="text-align:left">Equations</th> 654</tr> 655</thead> 656<tbody> 657<tr> 658<td style="text-align:right">Angular momentum operator</td> 659<td style="text-align:left">$\hat{\mathbf{L}} = \hat{\mathbf{x}} \times \hat{\mathbf{p}}$</td> 660</tr> 661<tr> 662<td style="text-align:right">z-component of angular momentum operator</td> 663<td style="text-align:left">$L_x = \dfrac{\hbar}{i}\dfrac{\partial}{\partial\phi}$</td> 664</tr> 665<tr> 666<td style="text-align:right">Magnitude of angular momentum operator</td> 667<td style="text-align:left">$\hat{\mathbf{L}}^2 = L^2 = -\hbar^2 \left[ \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta} \left( \sin\theta\dfrac{\partial}{\partial\theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2}{\partial\phi^2} \right) \right]$</td> 668</tr> 669<tr> 670<td style="text-align:right">Components of $\hat{\mathbf{L}}$ does not commute</td> 671<td style="text-align:left">$[\hat{L}_i, \hat{L}_j] = i\hbar \hat{L}_k$</td> 672</tr> 673<tr> 674<td style="text-align:right">Components of $\hat{\mathbf{L}}
674$ commute with its magnitude</td> 675<td style="text-align:left">$[\hat{L}_i, L^2] = 0$</td> 676</tr> 677</tbody> 678</table> 679<table> 680<thead> 681<tr> 682<th style="text-align:right">Description</th> 683<th style="text-align:left">Equations</th> 684</tr> 685</thead> 686<tbody> 687<tr> 688<td style="text-align:right">Rotational Schrodinger equation</td> 689<td style="text-align:left">$-\dfrac{\hbar^2}{2\mu R_0^2}\left[ \dfrac{1}{r^2 \sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2 \sin^2 \theta}\dfrac{\partial^2}{\partial\phi^2} \right]Y(\theta, \phi) = EY(\theta, \phi)$</td> 690</tr> 691<tr> 692<td style="text-align:right">Spherical harmonics</td> 693<td style="text-align:left">$Y_l^m(\theta, \phi) = (-)^m \sqrt{\dfrac{(2l+1)}{4\pi}\dfrac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}$</td> 694</tr> 695<tr> 696<td style="text-align:right">Legendre polynomial</td> 697<td style="text-align:left">$P_l^m(x) = \dfrac{1}{2^l l!}(1-x^2)^{m/2} \dfrac{d^{(l+m)}}{dx^{(l+m)}}(x^2-1)^l$</td> 698</tr> 699<tr> 700<td style="text-align:right">Energy eigenvalues <br/> $l = 0, 1, 2, …$</td> 701<td style="text-align:left">$E_l = \dfrac{\hbar^2}{2I}l(l+1)$</td> 702</tr> 703<tr> 704<td style="text-align:right">Angular momentum eigenvalues <br/> $l = 0, 1, 2, …$</td> 705<td style="text-align:left">$L^2 Y = \hbar^2l(l+1) Y$</td> 706</tr> 707<tr> 708<td style="text-align:right">z-component eigenvalues <br/> $m = -l, …, 0, …, l$</td> 709<td style="text-align:left">$L_z Y = \hbar m Y$</td> 710</tr> 711<tr> 712<td style="text-align:right">Transition dipole moment</td> 713<td style="text-align:left">$\mu_{fi} = \langle \psi_f \vert \mu_z \cos\theta \vert \psi_i \rangle$</td> 714</tr> 715<tr> 716<td style="text-align:right">Rotational selection rule</td> 717<td style="text-align:left">$\Delta l = \pm 1, \Delta m = 0$</td> 718</tr> 719</tbody> 720</table> 721<h3 id="hydrogen-atom">Hydrogen atom</h3> 722<table> 723<thead> 724<tr> 725<th style="text-align:right">Description</th> 726<th style="text-align:left">Equations</th> 727</tr> 728</thead> 729<tbody> 730<tr> 731<td style="text-align:right">Hydrogen atom Schrodinger equation</td> 732<td style="text-align:left">$\left[ -\dfrac{\hbar}{2m_e}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 \dfrac{\partial}{\partial r}\right) + \dfrac{\vec{L}}{2m_er^2} - \dfrac{e^2}{r} \right]\psi(x) = E\psi(x)$</td> 733</tr> 734<tr> 735<td style="text-align:right">Effective potential</td> 736<td style="text-align:left">$V_{\text{eff}} = \dfrac{\hbar l(l+1)}{2mr^2} - \dfrac{e^2}{r}$</td> 737</tr> 738<tr> 739<td style="text-align:right">Wave function <br/> $n = 1, 2, …$</td> 740<td style="text-align:left">$\psi_{nlm}(x) = R_{nl}(r)Y_l^m(\theta, \phi)$</td> 741</tr> 742<tr> 743<td style="text-align:right">Energy eigenvalues <br/> $n = 1, 2, …$</td> 744<td style="text-align:left">$E_n = -\dfrac{e^2}{2a_0}\dfrac{1}{n^2} = -\dfrac{me^4}{2\hbar^2}\dfrac{1}{n^2} - \dfrac{R_H}{n^2}$</td> 745</tr> 746<tr> 747<td style="text-align:right">Rydberg’s constant</td> 748<td style="text-align:left">$R_H = 2.179 \times 10^{-18} \mathrm{J} = 13.6 \ \mathrm{eV}$</td> 749</tr> 750<tr> 751<td style="text-align:right">Bohr’s radius</td> 752<td style="text-align:left">$a_0 = \dfrac{\hbar^2}{me^2}$</td> 753</tr> 754<tr> 755<td style="text-align:right">Radial probability distribution</td> 756<td style="text-align:left">$P_{nl}(r) dr = r^2 R^2_{nl}(r) dr$</td> 757</tr> 758</tbody> 759</table> 760<h2 id="many-electron-and-proton-system">Many Electron and Proton System</h2> 761<h3 id="many-electron-atom">Many electron atom</h3> 762<table> 763<thead> 764<tr> 765<th style="text-align:right">Description</th> 766<th style="text-align:left">Equations</th> 767</tr> 768</thead> 769<tbody> 770<tr> 771<td style="text-align:right">Helium Schrodinger equation</td> 772<td style="text-align:left">$\left[ \underbrace{-\dfrac{\hbar^2}{2m}\nabla_1^2}_{\text{KE of }e^-_1} \overbrace{-\dfrac{\hbar^2}{2m}\nabla_2^2}^{\text{KE of }e^-_2} \underbrace{-\dfrac{2e^2}{\vert\mathbf{x}_1\vert}}_{e^-_1\text{-N attraction}} \overbrace{-\dfrac{2e^2}{\vert\mathbf{x}_2\vert}}^{e^-_2\text{-N attraction}} + \underbrace{\dfrac{e^2}{\vert\mathbf{x}_1 - \mathbf{x}_2\vert}}_{e^-_1 \text{-} e^-_2 \text{repulsion}} \right] \psi(\mathbf{x}_1, \mathbf{x}_2) = E \psi(\mathbf{x}_1, \mathbf{x}_2)$</td> 773</tr> 774<tr> 775<td style="text-align:right">Orbital approximation</td> 776<td style="text-align:left">$\psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) = \phi(\mathbf{x}_1)\phi(\mathbf{x}_2)\dots\phi(\mathbf{x}_n)$</td> 777</tr> 778<tr> 779<td style="text-align:right">Hartree orbital equations</td> 780<td style="text-align:left">$\left[ -\dfrac{\hbar^2 \nabla_i^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} + \sum\limits^N_{j=1, j\not= i} \displaystyle\int \dfrac{e^2 \phi_j^*(\mathbf{x}')\phi_j(\mathbf{x}')}{\vert\mathbf{x} - \mathbf{x}' \vert} d^3\mathbf{x}' \right] \phi_i(\mathbf{x}) = \varepsilon_{ii} \phi_i(\mathbf{x})$</td> 781</tr> 782</tbody> 783</table> 784<h3 id="spin">Spin</h3> 785<table> 786<thead> 787<tr> 788<th style="text-align:right">Description</th> 789<th style="text-align:left">Equations</th> 790</tr> 791</thead> 792<tbody> 793<tr> 794<td style="text-align:right">Components of $\hat{\mathbf{S}}$ does not commute</td> 795<td style="text-align:left">$[\hat{S}_i, \hat{S}_j] = i\hbar \hat{S}_k$</td> 796</tr> 797<tr> 798<td style="text-align:right">Components of $\hat{\mathbf{S}}$ commute with its magnitude</td> 799<td style="text-align:left">$[\hat{S}_i, S^2] = 0$</td> 800</tr> 801<tr> 802<td style="text-align:right">Eigenvalue of $\hat{\mathbf{S}}^2$</td> 803<td style="text-align:left">$\hat{\mathbf{S}}^2 \leftrightarrow \hbar^2 s(s+1)$</td> 804</tr> 805<tr> 806<td style="text-align:right">Eigenvalue of $\hat{S}_z$</td> 807<td style="text-align:left">$\hat{S}_z \leftrightarrow \hbar m_s$</td> 808</tr> 809</tbody> 810</table> 811<h4 id="electron-spin">Electron spin</h4> 812<table> 813<thead> 814<tr> 815<th style="text-align:right">Description</th> 816<th style="text-align:left">Equations</th> 817</tr> 818</thead> 819<tbody> 820<tr> 821<td style="text-align:right">Electron spin</td> 822<td style="text-align:left">$s = \frac{1}{2}$</td> 823</tr> 824<tr> 825<td style="text-align:right">Spin up function</td> 826<td style="text-align:left">$\alpha(m_s) = \begin{cases} 1 & m_s = +\frac{1}{2} \\ 0 & m_s = -\frac{1}{2} \end{cases}$</td> 827</tr> 828<tr> 829<td style="text-align:right">Spin down function</td> 830<td style="text-align:left">$\beta(m_s) = \begin{cases} 0 & m_s = +\frac{1}{2} \\ 1 & m_s = -\frac{1}{2} \end{cases}$</td> 831</tr> 832<tr> 833<td style="text-align:right">$\alpha$ is eigenfunction of $\hat{S_z}$</td> 834<td style="text-align:left">$\hat{S_z} \alpha = +\frac{1}{2}\hbar \alpha$</td> 835</tr> 836<tr> 837<td style="text-align:right">$\beta$ is eigenfunction of $\hat{S_z}$</td> 838<td style="text-align:left">$\hat{S_z} \beta = -\frac{1}{2}\hbar \beta$</td> 839</tr> 840<tr> 841<td style="text-align:right">$\alpha, \beta$ are eigenfunctions of $\hat{S^2}$</td> 842<td style="text-align:left">$\hat{S^2} \alpha = \hbar^2 s(s+1) \alpha = \frac{3}{4} \hbar^2 \alpha \newline \hat{S^2} \beta = \hbar^2 s(s+1) \beta = \frac{3}{4} \hbar^2 \beta$</td> 843</tr> 844<tr> 845<td style="text-align:right">Normalization</td> 846<td style="text-align:left">$\sum\limits_{m_s} \alpha^*\alpha = \sum\limits_{m_s} \beta^*\beta = 1$</td> 847</tr> 848<tr> 849<td style="text-align:right">Orthogonality</td> 850<td style="text-align:left">$\sum\limits_{m_s} \alpha^*\beta = \sum\limits_{m_s} \beta^*\alpha = 0$</td> 851</tr> 852</tbody> 853</table> 854<h3 id="identical-particles">Identical particles</h3> 855<table> 856<thead> 857<tr> 858<th style="text-align:right">Description</th> 859<th style="text-align:left">Equations</th> 860</tr> 861</thead> 862<tbody> 863<tr> 864<td style="text-align:right">Spin-spin permutation operator</td> 865<td style="text-align:left">$P_{ij} \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}
865_i, \dots, \mathbf{r}_j, \dots, \mathbf{r}_N) = \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_j, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N)$</td> 866</tr> 867<tr> 868<td style="text-align:right">Doing nothing</td> 869<td style="text-align:left">$P_{ij}P_{ij} = 1$</td> 870</tr> 871<tr> 872<td style="text-align:right">Symmetric eigenvalue</td> 873<td style="text-align:left">$\lambda = 1$</td> 874</tr> 875<tr> 876<td style="text-align:right">Anti-symmetric eigenvalue</td> 877<td style="text-align:left">$\lambda = -1$</td> 878</tr> 879<tr> 880<td style="text-align:right">Fermions (e.g. electron)</td> 881<td style="text-align:left">$\frac{1}{2}$-integer spin, anti-symmetric</td> 882</tr> 883<tr> 884<td style="text-align:right">Bosons</td> 885<td style="text-align:left">integer spin, symmetric</td> 886</tr> 887<tr> 888<td style="text-align:right">Pauli exclusion principle</td> 889<td style="text-align:left">$\psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = - \psi(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_i, \dots, \mathbf{r}_i, \dots, \mathbf{r}_N) = 0$</td> 890</tr> 891<tr> 892<td style="text-align:right">Slater determinant</td> 893<td style="text-align:left">$\Psi(\mathrm{x}_1, \mathrm{x}_2, \cdots, \mathrm{x}_N) = \dfrac{1}{\sqrt{N!}} \begin{vmatrix}\chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{vmatrix}$</td> 894</tr> 895<tr> 896<td style="text-align:right">Hartree-Fock orbital equations</td> 897<td style="text-align:left">$\left[ -\dfrac{\hbar^2\nabla^2}{2m} - \dfrac{Ze^2}{\vert\mathbf{x}\vert} \right] \phi_i(\mathbf{r}) + \displaystyle\sum_{j=1}^N \left[ \phi_i(\mathbf{r}) \int \dfrac{e^2 \phi^*_j(\mathbf{r}) \phi_j(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' - \phi_j(\mathbf{r}) \int \dfrac{e^2 \phi^*_j(\mathbf{r}) \phi_i(\mathbf{r})}{\vert\mathbf{x} - \mathbf{x}'\vert} d^3r' \right] = \varepsilon_i \phi_i(\mathbf{r})$</td> 898</tr> 899<tr> 900<td style="text-align:right">Molecular orbital by linear combination of atomic orbitals (MO-LCAO)</td> 901<td style="text-align:left">$\psi(\mathbf{x}) = c_1 \phi_1 (\mathbf{x}) + c_2 \phi_2 (\mathbf{x}) \newline \mathrm{MO} = c_1 (\mathrm{AO}) + c_2 (\mathrm{AO})$</td> 902</tr> 903<tr> 904<td style="text-align:right">Variational principle</td> 905<td style="text-align:left">$E = \dfrac{\langle \psi \vert H \vert \psi \rangle}{\langle \psi \vert \psi \rangle}$</td> 906</tr> 907</tbody> 908</table> 909</div><div class="post-footer" id="post-footer"> 910 <div class="post-info"> 911 <div class="post-info-line"> 912 <div class="post-info-mod"> 913 <span>Updated on 2021-12-23</span> 914 </div> 915 <div class="post-info-license"></div> 916 </div> 917 <div class="post-info-line"> 918 <div class="post-info-md"></div> 919 <div class="post-info-share"> 920 <span></span> 921 </div> 922 </div> 923 </div> 924 925 <div class="post-info-more"> 926 <section class="post-tags"></section> 927 <section> 928 <span><a href="javascript:void(0);" onclick="window.history.back();">Back</a></span> | <span><a href="/">Home</a></span> 929 </section> 930 </div> 931 932 <div class="post-nav"><a href="/chem/chem145/" class="prev" rel="prev" title="CHEM 145 Honors General Chemistry I"><i class="fas fa-angle-left fa-fw"></i>CHEM 145 Honors General Chemistry I</a></div> 933</div> 934</article> 935 936 937
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