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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&rsquo;s model    Description Equations     Hydrogen emission lines  $n_2 &gt; n_1$ $\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$   Bohr&rsquo;s radius $r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}$   Energy level in Bohr&rsquo;s model $E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}$   Emission of hydrogen atom  $n_2 &gt; n_1$ $\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$    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&rsquo;s formula $e^{i\theta} = \cos\theta &#43; i\sin\theta$   Solution of wave equation $\begin{aligned}\Psi(x, t) &amp;= A \sin(kx - \omega t &#43; \phi) \\ &amp;= \mathrm{Re}(Ae^{i(kx-\omega t &#43; \phi&#39;)})\end{aligned}$   Interfering traveling waves give standing wave $\begin{aligned}\Psi(x, t) &amp;= A[\sin(kx - \omega t) &#43; \sin(kx &#43; \omega t)] \\ &amp;= 2A \sin(kx)\cos(\omega t) \\ &amp;= \psi(x)\cos(\omega t) \end{aligned}$   Time-independent Schrodinger equation $-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} &#43; V(x)\psi(x) = E\psi(x)$   Time-dependent Schrodinger equation $-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x, t)}{\partial x^2} &#43; V(x, t)\Psi(x, t) = i\hbar\Psi(x, t)$   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 &hellip; 1." />
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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&rsquo;s model    Description Equations     Hydrogen emission lines  $n_2 &gt; n_1$ $\tilde{\nu} = \dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$   Bohr&rsquo;s radius $r = \dfrac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}$   Energy level in Bohr&rsquo;s model $E_n = -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2n^2}$   Emission of hydrogen atom  $n_2 &gt; n_1$ $\nu = \dfrac{m_e e^4}{8\varepsilon_0^2 h^3} \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)$    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&rsquo;s formula $e^{i\theta} = \cos\theta &#43; i\sin\theta$   Solution of wave equation $\begin{aligned}\Psi(x, t) &amp;= A \sin(kx - \omega t &#43; \phi) \\ &amp;= \mathrm{Re}(Ae^{i(kx-\omega t &#43; \phi&#39;)})\end{aligned}$   Interfering traveling waves give standing wave $\begin{aligned}\Psi(x, t) &amp;= A[\sin(kx - \omega t) &#43; \sin(kx &#43; \omega t)] \\ &amp;= 2A \sin(kx)\cos(\omega t) \\ &amp;= \psi(x)\cos(\omega t) \end{aligned}$   Time-independent Schrodinger equation $-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi(x)}{dx^2} &#43; V(x)\psi(x) = E\psi(x)$   Time-dependent Schrodinger equation $-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x, t)}{\partial x^2} &#43; V(x, t)\Psi(x, t) = i\hbar\Psi(x, t)$   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 &hellip; 1."/>
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84        <h1 class="single-title posts-title">CHEM 455 Physical Chemistry</h1>
85
86        <div class="post-meta">
87            <div class="post-meta-line"><i class="far fa-calendar-alt fa-fw"></i>&nbsp;<time datetime="2021-12-23">2021-12-23</time>&nbsp;</div>
88        </div>
89
90        <div class="details toc" id="toc-static"  kept="">
91                <div class="details-summary toc-title">
92                    <span>Contents</span>
93                    <span><i class="details-icon fas fa-angle-right"></i></span>
94                </div>
95                <div class="details-content toc-content" id="toc-content-static"><nav id="TableOfContents">
96  <ul>
97    <li><a href="#from-classical-to-quantum-mechanics">From Classical to Quantum Mechanics</a>
98      <ul>
99        <li><a href="#blackbody-radiation">Blackbody radiation</a></li>
100        <li><a href="#wave-particle-duality">Wave-particle duality</a></li>
101        <li><a href="#atomic-spectra-of-hydrogen-and-bohrs-model">Atomic spectra of hydrogen and Bohr&rsquo;s model</a></li>
102        <li><a href="#waves">Waves</a></li>
103      </ul>
104    </li>
105    <li><a href="#quantum-mechanical-postulates">Quantum-Mechanical Postulates</a>
106      <ul>
107        <li><a href="#operators">Operators</a></li>
108      </ul>
109    </li>
110    <li><a href="#simple-quantum-systems">Simple Quantum Systems</a>
111      <ul>
112        <li><a href="#stationary-states">Stationary states</a></li>
113        <li><a href="#particle-in-a-1d-box">Particle in a 1D box</a></li>
114        <li><a href="#particle-in-a-3d-box">Particle in a 3D box</a></li>
115        <li><a href="#finite-potential-well">Finite potential well</a></li>
116      </ul>
117    </li>
118    <li><a href="#commutators-and-uncertainty">Commutators and Uncertainty</a></li>
119    <li><a href="#spectroscopy">Spectroscopy</a>
120      <ul>
121        <li><a href="#dimer-model">Dimer model</a></li>
122        <li><a href="#vibration-quantum-harmonic-oscillator">Vibration: quantum harmonic oscillator</a></li>
123        <li><a href="#rotation-rigid-rotor">Rotation: rigid rotor</a>
124          <ul>
125            <li><a href="#classical-rigid-rotor">Classical rigid rotor</a></li>
126            <li><a href="#quantum-rigid-rotor">Quantum rigid rotor</a></li>
127          </ul>
128        </li>
129        <li><a href="#hydrogen-atom">Hydrogen atom</a></li>
130      </ul>
131    </li>
132    <li><a href="#many-electron-and-proton-system">Many Electron and Proton System</a>
133      <ul>
134        <li><a href="#many-electron-atom">Many electron atom</a></li>
135        <li><a href="#spin">Spin</a>
136          <ul>
137            <li><a href="#electron-spin">Electron spin</a></li>
138          </ul>
139        </li>
140        <li><a href="#identical-particles">Identical particles</a></li>
141      </ul>
142    </li>
143  </ul>
144</nav></div>
145            </div><div class="content" id="content"><h2 id="from-classical-to-quantum-mechanics">From Classical to Quantum Mechanics</h2>
146<h3 id="blackbody-radiation">Blackbody radiation</h3>
147<table>
148<thead>
149<tr>
150<th style="text-align:right">Description</th>
151<th style="text-align:left">Equations</th>
152</tr>
153</thead>
154<tbody>
155<tr>
156<td style="text-align:right">Energy quantization</td>
157<td style="text-align:left">$E = nh\nu$</td>
158</tr>
159<tr>
160<td style="text-align:right">Average energy of an oscillating dipole</td>
161<td style="text-align:left">$\langle E_{\text{osc}} \rangle = \dfrac{h\nu}{e^{h\nu / k_B T} - 1}$</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&rsquo;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/> $n_2 &gt; n_1$</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&rsquo;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&rsquo;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/> $n_2 &gt; n_1$</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&rsquo;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">$\begin{aligned}\Psi(x, t) &amp;= A \sin(kx - \omega t + \phi) \\ &amp;= \mathrm{Re}(Ae^{i(kx-\omega t + \phi')})\end{aligned}$</td>
263</tr>
264<tr>
265<td style="text-align:right">Interfering traveling waves give standing wave</td>
266<td style="text-align:left">$\begin{aligned}\Psi(x, t) &amp;= A[\sin(kx - \omega t) + \sin(kx + \omega t)] \\ &amp;= 2A \sin(kx)\cos(\omega t) \\ &amp;= \psi(x)\cos(\omega t) \end{aligned}$</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 &hellip;</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/> $n = 0, 1, 2, &hellip;$</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/> $n_x = 0, 1, 2, &hellip;$ <br/> $n_y = 0, 1, 2, &hellip;$ <br/> $n_z = 0, 1, 2, &hellip;$</td>
427<td style="text-align:left">$\begin{aligned}&amp;\psi_{n_x, n_y, n_z}(\mathbf{x}) \\ =&amp; \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\ =&amp; \sqrt{\dfrac{2}{L_x}}\sqrt{\dfrac{2}{L_y}}\sqrt{\dfrac{2}{L_z}} \sin\left(\dfrac{n_x \pi x}{L_x}\right)\sin\left(\dfrac{n_y \pi y}{L_y}\right)\sin\left(\dfrac{n_z \pi z}{L_z}\right)\end{aligned}$</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">$V(x) = \begin{cases}0 &amp; x\in [0, L] \\ V_0 &amp; \mathrm{elsewhere}\end{cases}$</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">$\begin{aligned}\sigma_A &amp;= \sqrt{\langle (A - \langle A \rangle^2 \rangle)} \\ &amp;= \sqrt{\langle A^2 \rangle - \langle A \rangle^2}\end{aligned}$</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">$\begin{aligned}\mathbf{H}_{\text{dimer}} &amp;= \mathbf{H}_{\text{free}} + \mathbf{H}_{\text{int}} \\ &amp;= \underbrace{\dfrac{\mathbf{\Pi}^2}{2M}}_{\text{COM coord}} + \underbrace{\dfrac{\mathbf{p}^2}{2\mu} + V(\vert\mathbf{x}\vert)}_{\text{rel coord}}\end{aligned}$</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/> $n = 0, 1, 2, &hellip;$</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/> $n = 0, 1, 2, &hellip;$</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/> $l = 0, 1, 2, &hellip;$</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/> $l = 0, 1, 2, &hellip;$</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/> $m = -l, &hellip;, 0, &hellip;, l$</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/> $n = 1, 2, &hellip;$</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/> $n = 1, 2, &hellip;$</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&rsquo;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&rsquo;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">$\alpha(m_s) = \begin{cases} 1 &amp; m_s = +\frac{1}{2} \\ 0 &amp; m_s = -\frac{1}{2} \end{cases}$</td>
827</tr>
828<tr>
829<td style="text-align:right">Spin down function</td>
830<td style="text-align:left">$\beta(m_s) = \begin{cases} 0 &amp; m_s = +\frac{1}{2} \\ 1 &amp; m_s = -\frac{1}{2} \end{cases}$</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">$\Psi(\mathrm{x}_1, \mathrm{x}_2, \cdots, \mathrm{x}_N) = \dfrac{1}{\sqrt{N!}} \begin{vmatrix}\chi_1(\mathbf{x}_1) &amp; \chi_2(\mathbf{x}_1) &amp; \cdots &amp; \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) &amp; \chi_2(\mathbf{x}_2) &amp; \cdots &amp; \chi_N(\mathbf{x}_2) \\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ \chi_1(\mathbf{x}_N) &amp; \chi_2(\mathbf{x}_N) &amp; \cdots &amp; \chi_N(\mathbf{x}_N) \end{vmatrix}$</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>
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