. . . . . . . . "\u0641\u064A \u0641\u064A\u0632\u064A\u0627\u0621 \u0627\u0644\u0645\u0627\u062F\u0629 \u0627\u0644\u0645\u0643\u062B\u0641\u0629\u060C \u062A\u0646\u0635 \u0646\u0638\u0631\u064A\u0629 \u0628\u0644\u0648\u062E \u0639\u0644\u0649 \u0623\u0646 \u062D\u0644\u0648\u0644 \u0645\u0639\u0627\u062F\u0644\u0629 \u0634\u0631\u0648\u062F\u0646\u063A\u0631 \u0641\u064A \u0627\u0644\u062C\u0647\u062F \u0627\u0644\u062F\u0648\u0631\u064A \u062A\u0623\u062E\u0630 \u0634\u0643\u0644 \u0645\u0648\u062C\u0629 \u0645\u0633\u062A\u0648\u064A\u0629 \u064A\u062A\u0645 \u062A\u0639\u062F\u064A\u0644\u0647\u0627 \u0628\u0648\u0627\u0633\u0637\u0629 \u062F\u0627\u0644\u0629 \u062F\u0648\u0631\u064A\u0629. \u0631\u064A\u0627\u0636\u064A\u064B\u0627: \u062D\u064A\u062B \u0647\u0648 \u0627\u0644\u0645\u0648\u0636\u0639\u060C \u0647\u064A \u062F\u0627\u0644\u0629 \u0627\u0644\u0645\u0648\u062C\u0629\u060C \u0647\u064A \u062F\u0627\u0644\u0629 \u062F\u0648\u0631\u064A\u0629 \u0644\u0647\u0627 \u0646\u0641\u0633 \u062A\u0648\u0627\u062A\u0631 \u0627\u0644\u0628\u0644\u0648\u0631\u0629\u060C \u0645\u062A\u062C\u0647 \u0627\u0644\u0645\u0648\u062C\u0629 \u0648\u0647\u0648 \u0646\u0627\u0642\u0644 \u0627\u0644\u0632\u062E\u0645 \u0627\u0644\u0628\u0644\u0648\u0631\u064A\u060C \u0647\u0648 \u0639\u062F\u062F \u0623\u0648\u064A\u0644\u0631\u060C \u0648 \u0647\u064A \u0627\u0644\u0648\u062D\u062F\u0629 \u0627\u0644\u062A\u062E\u064A\u0644\u064A\u0629. \u062A\u064F\u0639\u0631\u0641 \u0648\u0638\u0627\u0626\u0641 \u0647\u0630\u0627 \u0627\u0644\u0646\u0645\u0648\u0630\u062C \u0628\u062F\u0648\u0627\u0644 \u0628\u0644\u0648\u062E \u0623\u0648 \u062D\u0627\u0644\u0627\u062A \u0628\u0644\u0648\u062E\u060C \u0648\u062A\u0639\u0645\u0644 \u0643\u0623\u0633\u0627\u0633 \u0645\u0646\u0627\u0633\u0628 \u0644\u062F\u0627\u0644\u0629 \u0627\u0644\u0645\u0648\u062C\u0629 \u0623\u0648 \u062D\u0627\u0644\u0627\u062A \u0627\u0644\u0625\u0644\u0643\u062A\u0631\u0648\u0646\u0627\u062A \u0641\u064A \u0627\u0644\u0645\u0648\u0627\u062F \u0627\u0644\u0635\u0644\u0628\u0629 \u0627\u0644\u0628\u0644\u0648\u0631\u064A\u0629."@ar . "1123503723"^^ . . "In condensed matter physics, Bloch's theorem states that solutions to the Schr\u00F6dinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written Bloch function where is position, is the wave function, is a periodic function with the same periodicity as the crystal, the wave vector is the crystal momentum vector, is Euler's number, and is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids. Named after Swiss physicist Felix Bloch, the description of electrons in terms of Bloch functions, termed Bloch electrons (or less often Bloch Waves), underlies the concept of electronic band structures. These eigenstates are written with subscripts as , where is a discrete index, called the band index, which is present because there are many different wave functions with the same (each has a different periodic component ). Within a band (i.e., for fixed ), varies continuously with , as does its energy. Also, is unique only up to a constant reciprocal lattice vector , or, . Therefore, the wave vector can be restricted to the first Brillouin zone of the reciprocal lattice without loss of generality."@en . . "Funzioni di Bloch"@it . "Die Bloch-Funktion oder Bloch-Welle (nach Felix Bloch) ist eine allgemeine Form f\u00FCr die L\u00F6sung der station\u00E4ren Schr\u00F6dingergleichung f\u00FCr ein Teilchen in einem periodischen Potential, z. B. die Wellenfunktion eines Elektrons in einem kristallinen Festk\u00F6rper (Bloch-Elektron). Die Form dieser Wellenfunktionen wird durch das Bloch-Theorem festgelegt, welches ein Spezialfall des Floquet-Theorems ist:"@de . "Les ondes de Bloch, d'apr\u00E8s F\u00E9lix Bloch, sont les fonctions d'ondes d\u00E9crivant les \u00E9tats quantiques des \u00E9lectrons soumis \u00E0 un potentiel p\u00E9riodique. C'est notamment le cas du cristal parfait infini, les \u00E9lectrons sont soumis \u00E0 un potentiel p\u00E9riodique ayant la sym\u00E9trie de translation des atomes constituant le cristal."@fr . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0411\u043B\u043E\u0445\u0430"@ru . "Bloch-Funktion"@de . . . . . . . . . . "Una ona de Bloch o estat de Bloch (anomenat en honor de Felix Bloch) \u00E9s la funci\u00F3 d'ona d'una part\u00EDcula (normalment un electr\u00F3) col\u00B7locada en un potencial peri\u00F2dic. El teorema de Bloch postula que l'autofunci\u00F3 d'energia de tal sistema es pot escriure com el producte d'una funci\u00F3 d'ona plana i una funci\u00F3 peri\u00F2dica (funci\u00F3 peri\u00F2dica de Bloch) que t\u00E9 la mateixa periodicitat que el potencial:"@ca . "Onda de Bloch"@pt . . "Twierdzenie Blocha"@pl . . . "Proof with Character Theory"@en . . . . "Bloch's theorem"@en . . . "Una Onda de Bloch (tambi\u00E9n llamada Estado de Bloch, Funci\u00F3n de Bloch o Funci\u00F3n de onda de Bloch), llamada as\u00ED por el f\u00EDsico suizo Felix Bloch, es un tipo de funci\u00F3n de onda de una part\u00EDcula en un medio peri\u00F3dico, como un electr\u00F3n en un s\u00F3lido cristalino. Una funci\u00F3n de onda \u03C8 es una onda de Bloch si tiene la forma:\u200B donde r es la posici\u00F3n, u es una funci\u00F3n peri\u00F3dica con la misma periodicidad que el cristal, y k es un n\u00FAmero real, llamado el vector de onda del cristal. En otras palabras, si se multiplica una onda plana por una funci\u00F3n peri\u00F3dica, se obtiene una onda de Bloch."@es . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0411\u043B\u043E\u0445\u0430 \u2014 \u0432\u0430\u0436\u043D\u0430\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0430 \u0444\u0438\u0437\u0438\u043A\u0438 \u0442\u0432\u0451\u0440\u0434\u043E\u0433\u043E \u0442\u0435\u043B\u0430, \u0443\u0441\u0442\u0430\u043D\u0430\u0432\u043B\u0438\u0432\u0430\u044E\u0449\u0430\u044F \u0432\u0438\u0434 \u0432\u043E\u043B\u043D\u043E\u0432\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438 \u0447\u0430\u0441\u0442\u0438\u0446\u044B, \u043D\u0430\u0445\u043E\u0434\u044F\u0449\u0435\u0439\u0441\u044F \u0432 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u043E\u043C \u043F\u043E\u0442\u0435\u043D\u0446\u0438\u0430\u043B\u0435. \u041D\u0430\u0437\u0432\u0430\u043D\u0430 \u0432 \u0447\u0435\u0441\u0442\u044C \u0448\u0432\u0435\u0439\u0446\u0430\u0440\u0441\u043A\u043E\u0433\u043E \u0444\u0438\u0437\u0438\u043A\u0430 \u0424\u0435\u043B\u0438\u043A\u0441\u0430 \u0411\u043B\u043E\u0445\u0430. \u0412 \u043E\u0434\u043D\u043E\u043C\u0435\u0440\u043D\u043E\u043C \u0441\u043B\u0443\u0447\u0430\u0435 \u044D\u0442\u0443 \u0442\u0435\u043E\u0440\u0435\u043C\u0443 \u0447\u0430\u0441\u0442\u043E \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0442 \u0442\u0435\u043E\u0440\u0435\u043C\u043E\u0439 \u0424\u043B\u043E\u043A\u0435. \u0421\u0444\u043E\u0440\u043C\u0443\u043B\u0438\u0440\u043E\u0432\u0430\u043D\u0430 \u0432 1928 \u0433\u043E\u0434\u0443."@ru . . "\uBE14\uB85C\uD750 \uD30C(Bloch wave) \uB610\uB294 \uBE14\uB85C\uD750 \uC0C1\uD0DC(Bloch state)\uB780 \uC8FC\uAE30\uC801\uC778 \uD37C\uD150\uC15C \uC0C1\uC758 \uC785\uC790\uC5D0 \uB300\uD55C \uD30C\uB3D9 \uD568\uC218\uB2E4. \uC8FC\uAE30\uC801\uC778 \uD37C\uD150\uC15C\uC5D0\uC120 \uD30C\uB3D9\uD568\uC218\uB3C4 \uC8FC\uAE30\uC801\uC73C\uB85C \uB098\uD0C0\uB098\uAC8C \uB418\uB294\uB370 \uADF8 \uD615\uD0DC\uAC00 \uC678\uD53C\uC5D0 \uC8FC\uAE30\uC801\uC778 \uD568\uC218\uAC00 \uB4E4\uC5B4\uC788\uB294 \uD615\uD0DC\uB85C \uB418\uC5B4 \uC788\uB2E4\uB294 \uAC83\uC744 \uD3A0\uB9AD\uC2A4 \uBE14\uB85C\uD750\uAC00 \uBC1D\uD600\uB0B4\uC5C8\uB2E4. \uC774\uB97C \uBE14\uB85C\uD750 \uC815\uB9AC(Bloch theorem)\uB77C \uD55C\uB2E4."@ko . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0411\u043B\u043E\u0445\u0430 \u2014 \u0432\u0430\u0436\u043D\u0430\u044F \u0442\u0435\u043E\u0440\u0435\u043C\u0430 \u0444\u0438\u0437\u0438\u043A\u0438 \u0442\u0432\u0451\u0440\u0434\u043E\u0433\u043E \u0442\u0435\u043B\u0430, \u0443\u0441\u0442\u0430\u043D\u0430\u0432\u043B\u0438\u0432\u0430\u044E\u0449\u0430\u044F \u0432\u0438\u0434 \u0432\u043E\u043B\u043D\u043E\u0432\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438 \u0447\u0430\u0441\u0442\u0438\u0446\u044B, \u043D\u0430\u0445\u043E\u0434\u044F\u0449\u0435\u0439\u0441\u044F \u0432 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u043E\u043C \u043F\u043E\u0442\u0435\u043D\u0446\u0438\u0430\u043B\u0435. \u041D\u0430\u0437\u0432\u0430\u043D\u0430 \u0432 \u0447\u0435\u0441\u0442\u044C \u0448\u0432\u0435\u0439\u0446\u0430\u0440\u0441\u043A\u043E\u0433\u043E \u0444\u0438\u0437\u0438\u043A\u0430 \u0424\u0435\u043B\u0438\u043A\u0441\u0430 \u0411\u043B\u043E\u0445\u0430. \u0412 \u043E\u0434\u043D\u043E\u043C\u0435\u0440\u043D\u043E\u043C \u0441\u043B\u0443\u0447\u0430\u0435 \u044D\u0442\u0443 \u0442\u0435\u043E\u0440\u0435\u043C\u0443 \u0447\u0430\u0441\u0442\u043E \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0442 \u0442\u0435\u043E\u0440\u0435\u043C\u043E\u0439 \u0424\u043B\u043E\u043A\u0435. \u0421\u0444\u043E\u0440\u043C\u0443\u043B\u0438\u0440\u043E\u0432\u0430\u043D\u0430 \u0432 1928 \u0433\u043E\u0434\u0443."@ru . "\u30D6\u30ED\u30C3\u30DB\u306E\u5B9A\u7406"@ja . . "\u0411\u043B\u043E\u0301\u0445\u043E\u0432\u0441\u043A\u0430\u044F \u0432\u043E\u043B\u043D\u0430\u0301 (\u0432\u043E\u043B\u043D\u0430\u0301 \u0411\u043B\u043E\u0301\u0445\u0430) \u2014 \u043D\u0430\u0437\u0432\u0430\u043D\u043D\u0430\u044F \u0432 \u0447\u0435\u0441\u0442\u044C \u0424\u0435\u043B\u0438\u043A\u0441\u0430 \u0411\u043B\u043E\u0445\u0430 \u0432\u043E\u043B\u043D\u043E\u0432\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u0447\u0430\u0441\u0442\u0438\u0446\u044B (\u043E\u0431\u044B\u0447\u043D\u043E \u044D\u043B\u0435\u043A\u0442\u0440\u043E\u043D\u0430), \u0440\u0430\u0441\u043F\u043E\u043B\u043E\u0436\u0435\u043D\u043D\u043E\u0439 \u0432 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u043E\u043C \u043F\u043E\u0442\u0435\u043D\u0446\u0438\u0430\u043B\u0435. \u0421\u043E\u0441\u0442\u043E\u0438\u0442 \u0438\u0437 \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u043F\u043B\u043E\u0441\u043A\u043E\u0439 \u0432\u043E\u043B\u043D\u044B \u043D\u0430 \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u0443\u044E \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u0443\u044E \u0444\u0443\u043D\u043A\u0446\u0438\u044E (\u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u0447\u0430\u0441\u0442\u044C \u0431\u043B\u043E\u0445\u043E\u0432\u0441\u043A\u043E\u0439 \u0432\u043E\u043B\u043D\u043E\u0432\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438) unk(r), \u0438\u043C\u0435\u044E\u0449\u0443\u044E \u0442\u0443 \u0436\u0435 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u043D\u043E\u0441\u0442\u044C, \u0447\u0442\u043E \u0438 \u043F\u043E\u0442\u0435\u043D\u0446\u0438\u0430\u043B. \u0433\u0434\u0435 \u2014 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u0438\u0435 \u0444\u0443\u043D\u043A\u0446\u0438\u0438, k \u2014 \u0432\u043E\u043B\u043D\u043E\u0432\u043E\u0439 \u0432\u0435\u043A\u0442\u043E\u0440 \u0447\u0430\u0441\u0442\u0438\u0446\u044B."@ru . . "We evaluate the derivatives and \ngiven they are the coefficients of the following expansion in q where q is considered small with respect to k\n\nGiven are eigenvalues of \nWe can consider the following perturbation problem in q:\n\nPerturbation theory of the second order states that\n\nTo compute to linear order in q\n\nwhere the integrations are over a primitive cell or the entire crystal, given if the integral \n\nis normalized across the cell or the crystal.\n\nWe can simplify over q to obtain\n\nand we can reinsert the complete wave functions"@en . . . . "\u0641\u064A \u0641\u064A\u0632\u064A\u0627\u0621 \u0627\u0644\u0645\u0627\u062F\u0629 \u0627\u0644\u0645\u0643\u062B\u0641\u0629\u060C \u062A\u0646\u0635 \u0646\u0638\u0631\u064A\u0629 \u0628\u0644\u0648\u062E \u0639\u0644\u0649 \u0623\u0646 \u062D\u0644\u0648\u0644 \u0645\u0639\u0627\u062F\u0644\u0629 \u0634\u0631\u0648\u062F\u0646\u063A\u0631 \u0641\u064A \u0627\u0644\u062C\u0647\u062F \u0627\u0644\u062F\u0648\u0631\u064A \u062A\u0623\u062E\u0630 \u0634\u0643\u0644 \u0645\u0648\u062C\u0629 \u0645\u0633\u062A\u0648\u064A\u0629 \u064A\u062A\u0645 \u062A\u0639\u062F\u064A\u0644\u0647\u0627 \u0628\u0648\u0627\u0633\u0637\u0629 \u062F\u0627\u0644\u0629 \u062F\u0648\u0631\u064A\u0629. \u0631\u064A\u0627\u0636\u064A\u064B\u0627: \u062D\u064A\u062B \u0647\u0648 \u0627\u0644\u0645\u0648\u0636\u0639\u060C \u0647\u064A \u062F\u0627\u0644\u0629 \u0627\u0644\u0645\u0648\u062C\u0629\u060C \u0647\u064A \u062F\u0627\u0644\u0629 \u062F\u0648\u0631\u064A\u0629 \u0644\u0647\u0627 \u0646\u0641\u0633 \u062A\u0648\u0627\u062A\u0631 \u0627\u0644\u0628\u0644\u0648\u0631\u0629\u060C \u0645\u062A\u062C\u0647 \u0627\u0644\u0645\u0648\u062C\u0629 \u0648\u0647\u0648 \u0646\u0627\u0642\u0644 \u0627\u0644\u0632\u062E\u0645 \u0627\u0644\u0628\u0644\u0648\u0631\u064A\u060C \u0647\u0648 \u0639\u062F\u062F \u0623\u0648\u064A\u0644\u0631\u060C \u0648 \u0647\u064A \u0627\u0644\u0648\u062D\u062F\u0629 \u0627\u0644\u062A\u062E\u064A\u0644\u064A\u0629. \u062A\u064F\u0639\u0631\u0641 \u0648\u0638\u0627\u0626\u0641 \u0647\u0630\u0627 \u0627\u0644\u0646\u0645\u0648\u0630\u062C \u0628\u062F\u0648\u0627\u0644 \u0628\u0644\u0648\u062E \u0623\u0648 \u062D\u0627\u0644\u0627\u062A \u0628\u0644\u0648\u062E\u060C \u0648\u062A\u0639\u0645\u0644 \u0643\u0623\u0633\u0627\u0633 \u0645\u0646\u0627\u0633\u0628 \u0644\u062F\u0627\u0644\u0629 \u0627\u0644\u0645\u0648\u062C\u0629 \u0623\u0648 \u062D\u0627\u0644\u0627\u062A \u0627\u0644\u0625\u0644\u0643\u062A\u0631\u0648\u0646\u0627\u062A \u0641\u064A \u0627\u0644\u0645\u0648\u0627\u062F \u0627\u0644\u0635\u0644\u0628\u0629 \u0627\u0644\u0628\u0644\u0648\u0631\u064A\u0629. \u0633\u0645\u064A\u062A \u0639\u0644\u0649 \u0627\u0633\u0645 \u0627\u0644\u0641\u064A\u0632\u064A\u0627\u0626\u064A \u0627\u0644\u0633\u0648\u064A\u0633\u0631\u064A \u0641\u0644\u064A\u0643\u0633 \u0628\u0644\u0648\u062E\u060C \u0648\u0648\u0635\u0641 \u0627\u0644\u0625\u0644\u0643\u062A\u0631\u0648\u0646\u0627\u062A \u0645\u0646 \u062D\u064A\u062B \u062F\u0627\u0644\u0629 \u0628\u0644\u0648\u062E\u060C \u0627\u0644\u062A\u064A \u064A\u0637\u0644\u0642 \u0639\u0644\u064A\u0647\u0627 \u0625\u0644\u0643\u062A\u0631\u0648\u0646\u0627\u062A \u0628\u0644\u0648\u062E (\u0623\u0648 \u0641\u064A \u0643\u062B\u064A\u0631 \u0645\u0646 \u0627\u0644\u0623\u062D\u064A\u0627\u0646 \u0645\u0648\u062C\u0627\u062A \u0628\u0644\u0648\u062E)\u060C \u064A\u0643\u0645\u0646 \u0648\u0631\u0627\u0621 \u0645\u0641\u0647\u0648\u0645 \u0647\u064A\u0627\u0643\u0644 \u0627\u0644\u0646\u0637\u0627\u0642 \u0627\u0644\u0625\u0644\u0643\u062A\u0631\u0648\u0646\u064A\u0629. \u062A\u062A\u0645 \u0643\u062A\u0627\u0628\u0629 \u0647\u0630\u0647 \u0627\u0644\u062D\u0627\u0644\u0627\u062A \u0627\u0644\u0630\u0627\u062A\u064A\u0629 \u0645\u0639 \u0627\u0644\u0631\u0645\u0648\u0632 \u0627\u0644\u0641\u0631\u0639\u064A\u0629 \u0643\u0640\u060C \u062D\u064A\u062B \u0647\u0648 \u0645\u062D\u062F\u062F \u0627\u0644\u0646\u0637\u0627\u0642\u060C \u064A\u0633\u0645\u0649 \u0645\u0624\u0634\u0631 \u0627\u0644\u0646\u0637\u0627\u0642\u060C \u0648\u0647\u0648 \u0645\u0648\u062C\u0648\u062F \u0646\u0638\u0631\u064B\u0627 \u0644\u0648\u062C\u0648\u062F \u0627\u0644\u0639\u062F\u064A\u062F \u0645\u0646 \u0627\u0644\u062F\u0648\u0627\u0644 \u0627\u0644\u0645\u0648\u062C\u064A\u0629 \u0627\u0644\u0645\u062E\u062A\u0644\u0641\u0629 \u0628\u0646\u0641\u0633 (\u0644\u0643\u0644 \u0645\u0646\u0647\u0627 \u0645\u0643\u0648\u0646 \u062F\u0648\u0631\u064A \u0645\u062E\u062A\u0644\u0641 ). \u062F\u0627\u062E\u0644 \u0627\u0644\u0646\u0637\u0627\u0642 (\u0628\u0645\u0639\u0646\u0649 \u0622\u062E\u0631 )\u060C \u064A\u062E\u062A\u0644\u0641 \u0628\u0627\u0633\u062A\u0645\u0631\u0627\u0631 \u0645\u0639 \u060C \u0643\u0645\u0627 \u062A\u062E\u062A\u0644\u0641 \u0637\u0627\u0642\u062A\u0647\u0627. \u0623\u064A\u0636\u0627\u060C \u0641\u0631\u064A\u062F \u0641\u0642\u0637 \u062D\u062A\u0649 \u062B\u0628\u0627\u062A \u0645\u062A\u062C\u0647 \u0627\u0644\u0634\u0628\u064A\u0643\u0629 \u0627\u0644\u0645\u0642\u0644\u0648\u0628\u0629 \u060C \u0623\u0648\u060C . \u0644\u0630\u0644\u0643\u060C \u0645\u0648\u062C\u0647 \u0627\u0644\u0645\u0648\u062C\u0629 \u064A\u0645\u0643\u0646 \u062D\u0635\u0631\u0647 \u0641\u064A \u0623\u0648\u0644 \u0645\u0646\u0637\u0642\u0629 \u0628\u0631\u064A\u0644\u064A\u0648\u0646 \u0645\u0646 \u0627\u0644\u0634\u0628\u0643\u0629 \u0627\u0644\u0645\u062A\u0628\u0627\u062F\u0644\u0629 \u062F\u0648\u0646 \u0641\u0642\u062F \u0627\u0644\u0639\u0645\u0648\u0645\u064A\u0629."@ar . . "All Translations are unitary and Abelian.\nTranslations can be written in terms of unit vectors\n\nWe can think of these as commuting operators\n where \n\nThe commutativity of the operators gives three commuting cyclic subgroups which are infinite, 1-dimensional and abelian. All irreducible representations of Abelian groups are one dimensional.\n\nGiven they are one dimensional the matrix representation and the character are the same. The character is the representation over the complex numbers of the group or also the trace of the representation which in this case is a one dimensional matrix. \nAll these subgroups, given they are cyclic, they have characters which are appropriate roots of unity. In fact they have one generator which shall obey to , and therefore the character . Note that this is straightforward in the finite cyclic group case but in the countable infinite case of the infinite cyclic group there is a limit for where the character remains finite.\n\nGiven the character is a root of unity, for each subgroup the character can be then written as\n\n\nIf we introduce the Born\u2013von Karman boundary condition on the potential:\n\nwhere L is a macroscopic periodicity in the direction that can also be seen as a multiple of where \n\nThis substituting in the time independent Schr\u00F6dinger equation with a simple effective Hamiltonian\n\ninduces a periodicity with the wave function:\n\n\nAnd for each dimension a translation operator with a period L\n\n\nFrom here we can see that also the character shall be invariant by a translation of :\n\nand from the last equation we get for each dimension a periodic condition:\n\nwhere is an integer and \n\nThe wave vector identify the irreducible representation in the same manner as , and is a macroscopic periodic length of the crystal in direction . In this context, the wave vector serves as a quantum number for the translation operator.\n\nWe can generalize this for 3 dimensions\n\nand the generic formula for the wave function becomes:\n\ni.e. specializing it for a translation\n\nand we have proven Bloch\u2019s theorem.\n\nApart from the group theory technicalities this proof is interesting because it becomes clear how to generalize the Bloch theorem for groups that are not only translations.\n\nThis is typically done for Space groups which are a combination of a translation and a point group and it is used for computing the band structure, spectrum and specific heats of crystals given a specific crystal group symmetry like FCC or BCC and eventually an extra basis.\n\nIn this proof it is also possible to notice how is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.\n\nIn the generalized version of the Bloch theorem, the Fourier transform, i.e. the wave function expansion, gets generalized from a discrete Fourier transform which is applicable only for cyclic groups and therefore translations into a character expansion of the wave function where the characters are given from the specific finite point group.\n\nAlso here is possible to see how the characters can be treated as the fundamental building blocks instead of the irreducible representations themselves."@en . . . . . . "40969"^^ . . . "357657"^^ . "\u5728\u56FA\u4F53\u7269\u7406\u5B66\u4E2D\uFF0C\u5E03\u6D1B\u8D6B\u6CE2\uFF08Bloch wave\uFF09\u662F\u5468\u671F\u6027\u52BF\u573A\uFF08\u5982\u6676\u4F53\uFF09\u4E2D\u7C92\u5B50\uFF08\u4E00\u822C\u4E3A\u7535\u5B50\uFF09\u7684\u6CE2\u51FD\u6570\uFF0C\u53C8\u540D\u5E03\u6D1B\u8D6B\u6001\uFF08Bloch state\uFF09\u3002 \u5E03\u6D1B\u8D6B\u6CE2\u56E0\u5176\u63D0\u51FA\u8005\u7F8E\u7C4D\u745E\u58EB\u88D4\u7269\u7406\u5B66\u5BB6\u83F2\u5229\u514B\u65AF\u00B7\u5E03\u6D1B\u8D6B\u800C\u5F97\u540D\u3002 \u5E03\u6D1B\u8D6B\u6CE2\u7531\u4E00\u4E2A\u5E73\u9762\u6CE2\u548C\u4E00\u4E2A\u5468\u671F\u51FD\u6570 \uFF08\u5E03\u6D1B\u8D6B\u6CE2\u5305\uFF09\u76F8\u4E58\u5F97\u5230\u3002\u5176\u4E2D \u4E0E\u52BF\u573A\u5177\u6709\u76F8\u540C\u5468\u671F\u6027\u3002\u5E03\u6D1B\u8D6B\u6CE2\u7684\u5177\u4F53\u5F62\u5F0F\u4E3A\uFF1A \u5F0F\u4E2D \u4E3A\u6CE2\u5411\u91CF\u3002\u4E0A\u5F0F\u8868\u8FBE\u7684\u6CE2\u51FD\u6570\u79F0\u4E3A\u5E03\u6D1B\u8D6B\u51FD\u6570\u3002\u5F53\u52BF\u573A\u5177\u6709\u6676\u683C\u5468\u671F\u6027\u65F6\uFF0C\u5176\u4E2D\u7684\u7C92\u5B50\u6240\u6EE1\u8DB3\u7684\u6CE2\u52A8\u65B9\u7A0B\u7684\u89E3\u03C8\u5B58\u5728\u6027\u8D28\uFF1A \u8FD9\u4E00\u7ED3\u8BBA\u79F0\u4E3A\u5E03\u6D1B\u8D6B\u5B9A\u7406\uFF08Bloch's theorem\uFF09\uFF0C\u5176\u4E2D \u4E3A\u6676\u683C\u5468\u671F\u5411\u91CF\u3002\u53EF\u4EE5\u770B\u51FA\uFF0C\u5177\u6709\u4E0A\u5F0F\u6027\u8D28\u7684\u6CE2\u51FD\u6570\u53EF\u4EE5\u5199\u6210\u5E03\u6D1B\u8D6B\u51FD\u6570\u7684\u5F62\u5F0F\u3002 \u5E73\u9762\u6CE2\u6CE2\u5411\u91CF \uFF08\u53C8\u79F0\u201C\u5E03\u6D1B\u8D6B\u6CE2\u5411\u91CF\u201D\uFF0C\u5B83\u4E0E\u7EA6\u5316\u666E\u6717\u514B\u5E38\u6570\u7684\u4E58\u79EF\u5373\u4E3A\u7C92\u5B50\u7684\u6676\u4F53\u52A8\u91CF\uFF09\u8868\u5F81\u4E0D\u540C\u539F\u80DE\u95F4\u7535\u5B50\u6CE2\u51FD\u6570\u7684\u4F4D\u76F8\u53D8\u5316\uFF0C\u5176\u5927\u5C0F\u53EA\u5728\u4E00\u4E2A\u5012\u6613\u70B9\u9635\u5411\u91CF\u4E4B\u5185\u624D\u4E0E\u6CE2\u51FD\u6570\u6EE1\u8DB3\u4E00\u4E00\u5BF9\u5E94\u5173\u7CFB\uFF0C\u6240\u4EE5\u901A\u5E38\u53EA\u8003\u8651\u7B2C\u4E00\u5E03\u91CC\u6E0A\u533A\u5185\u7684\u6CE2\u5411\u91CF\uFF0C\u5373\u6240\u8C13\u201C\u7B80\u7EA6\u6CE2\u5411\u91CF\u201D\u3002\u5BF9\u4E00\u4E2A\u7ED9\u5B9A\u7684\u6CE2\u77E2\u548C\u52BF\u573A\u5206\u5E03\uFF0C\u7535\u5B50\u8FD0\u52A8\u7684\u859B\u5B9A\u8C14\u65B9\u7A0B\u5177\u6709\u4E00\u7CFB\u5217\u89E3\uFF0C\u79F0\u4E3A\u7535\u5B50\u7684\u80FD\u5E26\uFF0C\u5E38\u7528\u6CE2\u51FD\u6570\u7684\u4E0B\u6807n \u4EE5\u533A\u522B\u3002\u8FD9\u4E9B\u80FD\u5E26\u7684\u80FD\u91CF\u5728 \u7684\u5404\u4E2A\u5355\u503C\u533A\u5206\u754C\u5904\u5B58\u5728\u6709\u9650\u5927\u5C0F\u7684\u7A7A\u9699\uFF0C\u79F0\u4E3A\u80FD\u9699\u3002\u5728\u7B2C\u4E00\u5E03\u91CC\u6E0A\u533A\u4E2D\u6240\u6709\u80FD\u91CF\u672C\u5F81\u6001\u7684\u96C6\u5408\u6784\u6210\u4E86\u7535\u5B50\u7684\u80FD\u5E26\u7ED3\u6784\u3002\u5728\u5355\u7535\u5B50\u8FD1\u4F3C\u7684\u6846\u67B6\u5185\uFF0C\u5468\u671F\u6027\u52BF\u573A\u4E2D\u7535\u5B50\u8FD0\u52A8\u7684\u5B8F\u89C2\u6027\u8D28\u90FD\u53EF\u4EE5\u6839\u636E\u80FD\u5E26\u7ED3\u6784\u53CA\u76F8\u5E94\u7684\u6CE2\u51FD\u6570\u8BA1\u7B97\u51FA\u3002 \u4E0A\u8FF0\u7ED3\u679C\u7684\u4E00\u4E2A\u63A8\u8BBA\u4E3A\uFF1A\u5728\u786E\u5B9A\u7684\u5B8C\u6574\u6676\u4F53\u7ED3\u6784\u4E2D\uFF0C\u5E03\u6D1B\u8D6B\u6CE2\u5411\u91CF \u662F\u4E00\u4E2A\u5B88\u6052\u91CF\uFF08\u4EE5\u5012\u6613\u70B9\u9635\u5411\u91CF\u4E3A\u6A21\uFF09\uFF0C\u5373\u7535\u5B50\u6CE2\u7684\u7FA4\u901F\u5EA6\u4E3A\u5B88\u6052\u91CF\u3002\u6362\u8A00\u4E4B\uFF0C\u5728\u5B8C\u6574\u6676\u4F53\u4E2D\uFF0C\u7535\u5B50\u8FD0\u52A8\u53EF\u4EE5\u4E0D\u88AB\u683C\u70B9\u6563\u5C04\u5730\u4F20\u64AD\uFF08\u6240\u4EE5\u8BE5\u6A21\u578B\u53C8\u79F0\u4E3A\u8FD1\u81EA\u7531\u7535\u5B50\u8FD1\u4F3C\uFF09\uFF0C\u6676\u6001\u5BFC\u4F53\u7684\u7535\u963B\u4EC5\u4EC5\u6765\u81EA\u90A3\u4E9B\u7834\u574F\u4E86\u52BF\u573A\u5468\u671F\u6027\u7684\u6676\u4F53\u7F3A\u9677\u4EE5\u53CA\u7535\u5B50\u4E0E\u58F0\u5B50\u7684\u76F8\u4E92\u4F5C\u7528\u3002 \u4ECE\u859B\u5B9A\u8C14\u65B9\u7A0B\u51FA\u53D1\u53EF\u4EE5\u8BC1\u660E\uFF0C\u54C8\u5BC6\u987F\u7B97\u7B26\u4E0E\u5E73\u79FB\u7B97\u7B26\u7684\u4F5C\u7528\u6B21\u5E8F\u6EE1\u8DB3\u4EA4\u6362\u5F8B\uFF0C\u6240\u4EE5\u5468\u671F\u52BF\u573A\u4E2D\u7C92\u5B50\u7684\u672C\u5F81\u6CE2\u51FD\u6570\u603B\u662F\u53EF\u4EE5\u5199\u6210\u5E03\u6D1B\u8D6B\u51FD\u6570\u7684\u5F62\u5F0F\u3002\u66F4\u5E7F\u4E49\u5730\u8BF4\uFF0C\u672C\u5F81\u51FD\u6570\u6EE1\u8DB3\u7684\u7B97\u7B26\u4F5C\u7528\u5BF9\u79F0\u5173\u7CFB\u662F\u7FA4\u8BBA\u4E2D\u8868\u793A\u7406\u8BBA\u7684\u4E00\u4E2A\u7279\u4F8B\u3002 \u5E03\u6D1B\u8D6B\u6CE2\u7684\u6982\u5FF5\u7531\u83F2\u5229\u514B\u65AF\u00B7\u5E03\u6D1B\u8D6B\u57281928\u5E74\u7814\u7A76\u6676\u6001\u56FA\u4F53\u7684\u5BFC\u7535\u6027\u65F6\u9996\u6B21\u63D0\u51FA\u7684\uFF0C\u4F46\u5176\u6570\u5B66\u57FA\u7840\u5728\u5386\u53F2\u4E0A\u5374\u66FE\u7531\u4E54\u6CBB\u00B7\u5A01\u5EC9\u00B7\u5E0C\u5C14\uFF081877\u5E74\uFF09\uFF0C\uFF081883\u5E74\uFF09\u548C\u4E9A\u5386\u5C71\u5927\u00B7\u674E\u96C5\u666E\u8BFA\u592B\uFF081892\u5E74\uFF09\u7B49\u72EC\u7ACB\u5730\u63D0\u51FA\u3002\u56E0\u6B64\uFF0C\u7C7B\u4F3C\u6027\u8D28\u7684\u6982\u5FF5\u5728\u5404\u4E2A\u9886\u57DF\u6709\u7740\u4E0D\u540C\u7684\u540D\u79F0\uFF1A\u5E38\u5FAE\u5206\u65B9\u7A0B\u7406\u8BBA\u4E2D\u79F0\u4E3A\u5F17\u6D1B\u51EF\u7406\u8BBA\uFF08\u4E5F\u6709\u4EBA\u79F0\u201C\u674E\u96C5\u666E\u8BFA\u592B-\u5F17\u6D1B\u51EF\u5B9A\u7406\u201D\uFF09\uFF1B\u4E00\u7EF4\u5468\u671F\u6027\u6CE2\u52A8\u65B9\u7A0B\u5219\u6709\u65F6\u88AB\u79F0\u4E3A\u5E0C\u5C14\u65B9\u7A0B\u3002"@zh . . . . . . . "In fisica dello stato solido, le funzioni di Bloch sono le funzioni d'onda di singola particella, in genere un elettrone, in un potenziale periodico, come quello definito da un cristallo. Sono state introdotte nel 1928 dal fisico Felix Bloch, da cui prendono il nome; egli applic\u00F2 la teoria degli orbitali molecolari ai solidi metallici, considerandoli come un'unica particella con un'enormit\u00E0 di OM. Interessante \u00E8 il fatto che questa fu una delle prime applicazioni della OM, perfino antecedente all'applicazione alle molecole nella descrizione del legame covalente."@it . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0411\u043B\u043E\u0445\u0430"@uk . . . . . "Onda de Bloch"@es . . . . "\u0411\u043B\u043E\u0445\u0456\u0432\u0441\u044C\u043A\u0430 \u0445\u0432\u0438\u043B\u044F (\u0445\u0432\u0438\u043B\u044F \u0411\u043B\u043E\u0445\u0430) \u2014 \u043D\u0430\u0437\u0432\u0430\u043D\u0430 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0424\u0435\u043B\u0456\u043A\u0441\u0430 \u0411\u043B\u043E\u0445\u0430 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0447\u0430\u0441\u0442\u0438\u043D\u043A\u0438 (\u0437\u0430\u0437\u0432\u0438\u0447\u0430\u0439 \u0435\u043B\u0435\u043A\u0442\u0440\u043E\u043D\u0430), \u0440\u043E\u0437\u0442\u0430\u0448\u043E\u0432\u0430\u043D\u043E\u0457 \u0432 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u043E\u043C\u0443 \u043F\u043E\u0442\u0435\u043D\u0446\u0456\u0430\u043B\u0456. \u0421\u043A\u043B\u0430\u0434\u0430\u0454\u0442\u044C\u0441\u044F \u0437 \u0434\u043E\u0431\u0443\u0442\u043A\u0443 \u043F\u043B\u043E\u0441\u043A\u043E\u0457 \u0445\u0432\u0438\u043B\u0456 \u043D\u0430 \u0434\u0435\u044F\u043A\u0443 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0443 \u0444\u0443\u043D\u043A\u0446\u0456\u044E (\u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0430 \u0447\u0430\u0441\u0442\u0438\u043D\u0430 \u0431\u043B\u043E\u0445\u0456\u0432\u0441\u044C\u043A\u043E\u0457 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u043E\u0457 \u0444\u0443\u043D\u043A\u0446\u0456\u0457) unk(r), \u0449\u043E \u043C\u0430\u0454 \u0442\u0443 \u0436 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0456\u0441\u0442\u044C, \u0449\u043E \u0456 \u043F\u043E\u0442\u0435\u043D\u0446\u0456\u0430\u043B. \u0434\u0435 \u2014 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0456 \u0444\u0443\u043D\u043A\u0446\u0456\u0457, k \u2014 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0438\u0439 \u0432\u0435\u043A\u0442\u043E\u0440 \u0447\u0430\u0441\u0442\u0438\u043D\u043A\u0438."@uk . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0411\u043B\u043E\u0445\u0430 \u2014 \u043E\u0434\u043D\u0435 \u0456\u0437 \u043E\u0441\u043D\u043E\u0432\u043D\u0438\u0445 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u044C \u043A\u0432\u0430\u043D\u0442\u043E\u0432\u043E\u0457 \u0442\u0435\u043E\u0440\u0456\u0457 \u0456\u0434\u0435\u0430\u043B\u044C\u043D\u0438\u0445 \u043A\u0440\u0438\u0441\u0442\u0430\u043B\u0456\u0432, \u044F\u043A\u0435 \u0437\u0430\u0434\u0430\u0454 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0438\u0439 \u0432\u0438\u0433\u043B\u044F\u0434 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0438\u0445 \u0444\u0443\u043D\u043A\u0446\u0456\u0439 \u0435\u043B\u0435\u043A\u0442\u0440\u043E\u043D\u043D\u0438\u0445 \u0441\u0442\u0430\u043D\u0456\u0432 \u0443 \u0442\u0432\u0435\u0440\u0434\u043E\u043C\u0443 \u0442\u0456\u043B\u0456 \u0437 \u0442\u0440\u0430\u043D\u0441\u043B\u044F\u0446\u0456\u0439\u043D\u043E\u044E \u0441\u0438\u043C\u0435\u0442\u0440\u0456\u0454\u044E."@uk . . . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u0411\u043B\u043E\u0445\u0430 \u2014 \u043E\u0434\u043D\u0435 \u0456\u0437 \u043E\u0441\u043D\u043E\u0432\u043D\u0438\u0445 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u044C \u043A\u0432\u0430\u043D\u0442\u043E\u0432\u043E\u0457 \u0442\u0435\u043E\u0440\u0456\u0457 \u0456\u0434\u0435\u0430\u043B\u044C\u043D\u0438\u0445 \u043A\u0440\u0438\u0441\u0442\u0430\u043B\u0456\u0432, \u044F\u043A\u0435 \u0437\u0430\u0434\u0430\u0454 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0438\u0439 \u0432\u0438\u0433\u043B\u044F\u0434 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0438\u0445 \u0444\u0443\u043D\u043A\u0446\u0456\u0439 \u0435\u043B\u0435\u043A\u0442\u0440\u043E\u043D\u043D\u0438\u0445 \u0441\u0442\u0430\u043D\u0456\u0432 \u0443 \u0442\u0432\u0435\u0440\u0434\u043E\u043C\u0443 \u0442\u0456\u043B\u0456 \u0437 \u0442\u0440\u0430\u043D\u0441\u043B\u044F\u0446\u0456\u0439\u043D\u043E\u044E \u0441\u0438\u043C\u0435\u0442\u0440\u0456\u0454\u044E."@uk . "\u0411\u043B\u043E\u0445\u043E\u0432\u0441\u043A\u0430\u044F \u0432\u043E\u043B\u043D\u0430"@ru . . . . . . . . . . . . . . "Onde de Bloch"@fr . "In fisica dello stato solido, le funzioni di Bloch sono le funzioni d'onda di singola particella, in genere un elettrone, in un potenziale periodico, come quello definito da un cristallo. Sono state introdotte nel 1928 dal fisico Felix Bloch, da cui prendono il nome; egli applic\u00F2 la teoria degli orbitali molecolari ai solidi metallici, considerandoli come un'unica particella con un'enormit\u00E0 di OM. Interessante \u00E8 il fatto che questa fu una delle prime applicazioni della OM, perfino antecedente all'applicazione alle molecole nella descrizione del legame covalente."@it . . . . "\uBE14\uB85C\uD750 \uD30C(Bloch wave) \uB610\uB294 \uBE14\uB85C\uD750 \uC0C1\uD0DC(Bloch state)\uB780 \uC8FC\uAE30\uC801\uC778 \uD37C\uD150\uC15C \uC0C1\uC758 \uC785\uC790\uC5D0 \uB300\uD55C \uD30C\uB3D9 \uD568\uC218\uB2E4. \uC8FC\uAE30\uC801\uC778 \uD37C\uD150\uC15C\uC5D0\uC120 \uD30C\uB3D9\uD568\uC218\uB3C4 \uC8FC\uAE30\uC801\uC73C\uB85C \uB098\uD0C0\uB098\uAC8C \uB418\uB294\uB370 \uADF8 \uD615\uD0DC\uAC00 \uC678\uD53C\uC5D0 \uC8FC\uAE30\uC801\uC778 \uD568\uC218\uAC00 \uB4E4\uC5B4\uC788\uB294 \uD615\uD0DC\uB85C \uB418\uC5B4 \uC788\uB2E4\uB294 \uAC83\uC744 \uD3A0\uB9AD\uC2A4 \uBE14\uB85C\uD750\uAC00 \uBC1D\uD600\uB0B4\uC5C8\uB2E4. \uC774\uB97C \uBE14\uB85C\uD750 \uC815\uB9AC(Bloch theorem)\uB77C \uD55C\uB2E4."@ko . . . "El teorema de Bloch describe el movimiento de los electrones en un s\u00F3lido. Fue enunciado por el f\u00EDsico suizo Felix Bloch bas\u00E1ndose en la idea de que un s\u00F3lido posee una estructura microsc\u00F3pica peri\u00F3dica. A partir de esta hip\u00F3tesis, el teorema establece de qu\u00E9 manera deben ser las funciones de onda de los electrones, y permite tratar el movimiento de todos los electrones analizando \u00FAnicamente el movimiento de un solo electr\u00F3n."@es . "\u5728\u56FA\u4F53\u7269\u7406\u5B66\u4E2D\uFF0C\u5E03\u6D1B\u8D6B\u6CE2\uFF08Bloch wave\uFF09\u662F\u5468\u671F\u6027\u52BF\u573A\uFF08\u5982\u6676\u4F53\uFF09\u4E2D\u7C92\u5B50\uFF08\u4E00\u822C\u4E3A\u7535\u5B50\uFF09\u7684\u6CE2\u51FD\u6570\uFF0C\u53C8\u540D\u5E03\u6D1B\u8D6B\u6001\uFF08Bloch state\uFF09\u3002 \u5E03\u6D1B\u8D6B\u6CE2\u56E0\u5176\u63D0\u51FA\u8005\u7F8E\u7C4D\u745E\u58EB\u88D4\u7269\u7406\u5B66\u5BB6\u83F2\u5229\u514B\u65AF\u00B7\u5E03\u6D1B\u8D6B\u800C\u5F97\u540D\u3002 \u5E03\u6D1B\u8D6B\u6CE2\u7531\u4E00\u4E2A\u5E73\u9762\u6CE2\u548C\u4E00\u4E2A\u5468\u671F\u51FD\u6570 \uFF08\u5E03\u6D1B\u8D6B\u6CE2\u5305\uFF09\u76F8\u4E58\u5F97\u5230\u3002\u5176\u4E2D \u4E0E\u52BF\u573A\u5177\u6709\u76F8\u540C\u5468\u671F\u6027\u3002\u5E03\u6D1B\u8D6B\u6CE2\u7684\u5177\u4F53\u5F62\u5F0F\u4E3A\uFF1A \u5F0F\u4E2D \u4E3A\u6CE2\u5411\u91CF\u3002\u4E0A\u5F0F\u8868\u8FBE\u7684\u6CE2\u51FD\u6570\u79F0\u4E3A\u5E03\u6D1B\u8D6B\u51FD\u6570\u3002\u5F53\u52BF\u573A\u5177\u6709\u6676\u683C\u5468\u671F\u6027\u65F6\uFF0C\u5176\u4E2D\u7684\u7C92\u5B50\u6240\u6EE1\u8DB3\u7684\u6CE2\u52A8\u65B9\u7A0B\u7684\u89E3\u03C8\u5B58\u5728\u6027\u8D28\uFF1A \u8FD9\u4E00\u7ED3\u8BBA\u79F0\u4E3A\u5E03\u6D1B\u8D6B\u5B9A\u7406\uFF08Bloch's theorem\uFF09\uFF0C\u5176\u4E2D \u4E3A\u6676\u683C\u5468\u671F\u5411\u91CF\u3002\u53EF\u4EE5\u770B\u51FA\uFF0C\u5177\u6709\u4E0A\u5F0F\u6027\u8D28\u7684\u6CE2\u51FD\u6570\u53EF\u4EE5\u5199\u6210\u5E03\u6D1B\u8D6B\u51FD\u6570\u7684\u5F62\u5F0F\u3002 \u5E73\u9762\u6CE2\u6CE2\u5411\u91CF \uFF08\u53C8\u79F0\u201C\u5E03\u6D1B\u8D6B\u6CE2\u5411\u91CF\u201D\uFF0C\u5B83\u4E0E\u7EA6\u5316\u666E\u6717\u514B\u5E38\u6570\u7684\u4E58\u79EF\u5373\u4E3A\u7C92\u5B50\u7684\u6676\u4F53\u52A8\u91CF\uFF09\u8868\u5F81\u4E0D\u540C\u539F\u80DE\u95F4\u7535\u5B50\u6CE2\u51FD\u6570\u7684\u4F4D\u76F8\u53D8\u5316\uFF0C\u5176\u5927\u5C0F\u53EA\u5728\u4E00\u4E2A\u5012\u6613\u70B9\u9635\u5411\u91CF\u4E4B\u5185\u624D\u4E0E\u6CE2\u51FD\u6570\u6EE1\u8DB3\u4E00\u4E00\u5BF9\u5E94\u5173\u7CFB\uFF0C\u6240\u4EE5\u901A\u5E38\u53EA\u8003\u8651\u7B2C\u4E00\u5E03\u91CC\u6E0A\u533A\u5185\u7684\u6CE2\u5411\u91CF\uFF0C\u5373\u6240\u8C13\u201C\u7B80\u7EA6\u6CE2\u5411\u91CF\u201D\u3002\u5BF9\u4E00\u4E2A\u7ED9\u5B9A\u7684\u6CE2\u77E2\u548C\u52BF\u573A\u5206\u5E03\uFF0C\u7535\u5B50\u8FD0\u52A8\u7684\u859B\u5B9A\u8C14\u65B9\u7A0B\u5177\u6709\u4E00\u7CFB\u5217\u89E3\uFF0C\u79F0\u4E3A\u7535\u5B50\u7684\u80FD\u5E26\uFF0C\u5E38\u7528\u6CE2\u51FD\u6570\u7684\u4E0B\u6807n \u4EE5\u533A\u522B\u3002\u8FD9\u4E9B\u80FD\u5E26\u7684\u80FD\u91CF\u5728 \u7684\u5404\u4E2A\u5355\u503C\u533A\u5206\u754C\u5904\u5B58\u5728\u6709\u9650\u5927\u5C0F\u7684\u7A7A\u9699\uFF0C\u79F0\u4E3A\u80FD\u9699\u3002\u5728\u7B2C\u4E00\u5E03\u91CC\u6E0A\u533A\u4E2D\u6240\u6709\u80FD\u91CF\u672C\u5F81\u6001\u7684\u96C6\u5408\u6784\u6210\u4E86\u7535\u5B50\u7684\u80FD\u5E26\u7ED3\u6784\u3002\u5728\u5355\u7535\u5B50\u8FD1\u4F3C\u7684\u6846\u67B6\u5185\uFF0C\u5468\u671F\u6027\u52BF\u573A\u4E2D\u7535\u5B50\u8FD0\u52A8\u7684\u5B8F\u89C2\u6027\u8D28\u90FD\u53EF\u4EE5\u6839\u636E\u80FD\u5E26\u7ED3\u6784\u53CA\u76F8\u5E94\u7684\u6CE2\u51FD\u6570\u8BA1\u7B97\u51FA\u3002"@zh . . . . "If a wave function is an eigenstate of all of the translation operators , then is a Bloch state."@en . "Proof"@en . "\u5E03\u6D1B\u8D6B\u6CE2"@zh . . "\u0411\u043B\u043E\u0445\u0456\u0432\u0441\u044C\u043A\u0430 \u0445\u0432\u0438\u043B\u044F (\u0445\u0432\u0438\u043B\u044F \u0411\u043B\u043E\u0445\u0430) \u2014 \u043D\u0430\u0437\u0432\u0430\u043D\u0430 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0424\u0435\u043B\u0456\u043A\u0441\u0430 \u0411\u043B\u043E\u0445\u0430 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0430 \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0447\u0430\u0441\u0442\u0438\u043D\u043A\u0438 (\u0437\u0430\u0437\u0432\u0438\u0447\u0430\u0439 \u0435\u043B\u0435\u043A\u0442\u0440\u043E\u043D\u0430), \u0440\u043E\u0437\u0442\u0430\u0448\u043E\u0432\u0430\u043D\u043E\u0457 \u0432 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u043E\u043C\u0443 \u043F\u043E\u0442\u0435\u043D\u0446\u0456\u0430\u043B\u0456. \u0421\u043A\u043B\u0430\u0434\u0430\u0454\u0442\u044C\u0441\u044F \u0437 \u0434\u043E\u0431\u0443\u0442\u043A\u0443 \u043F\u043B\u043E\u0441\u043A\u043E\u0457 \u0445\u0432\u0438\u043B\u0456 \u043D\u0430 \u0434\u0435\u044F\u043A\u0443 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0443 \u0444\u0443\u043D\u043A\u0446\u0456\u044E (\u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0430 \u0447\u0430\u0441\u0442\u0438\u043D\u0430 \u0431\u043B\u043E\u0445\u0456\u0432\u0441\u044C\u043A\u043E\u0457 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u043E\u0457 \u0444\u0443\u043D\u043A\u0446\u0456\u0457) unk(r), \u0449\u043E \u043C\u0430\u0454 \u0442\u0443 \u0436 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0456\u0441\u0442\u044C, \u0449\u043E \u0456 \u043F\u043E\u0442\u0435\u043D\u0446\u0456\u0430\u043B. \u0434\u0435 \u2014 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0456 \u0444\u0443\u043D\u043A\u0446\u0456\u0457, k \u2014 \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0438\u0439 \u0432\u0435\u043A\u0442\u043E\u0440 \u0447\u0430\u0441\u0442\u0438\u043D\u043A\u0438. \u0417\u0433\u0456\u0434\u043D\u043E \u0437 \u0442\u0435\u043E\u0440\u0435\u043C\u043E\u044E \u0411\u043B\u043E\u0445\u0430, \u0432 \u0442\u0430\u043A\u043E\u043C\u0443 \u0432\u0438\u0433\u043B\u044F\u0434\u0456 \u043C\u043E\u0436\u043D\u0430 \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u0438 \u0432\u0441\u0435 \u0432\u043B\u0430\u0441\u043D\u0456 \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u043E\u0457 \u0441\u0438\u0441\u0442\u0435\u043C\u0438. \u0412\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u0456 \u0457\u043C \u0432\u043B\u0430\u0441\u043D\u0456 \u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u0435\u043D\u0435\u0440\u0433\u0456\u0457 En(k) = En(k + K) \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0456 \u043F\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u0430\u0445 \u043E\u0431\u0435\u0440\u043D\u0435\u043D\u043E\u0457 \u0491\u0440\u0430\u0442\u043A\u0438 k. \u041E\u0441\u043A\u0456\u043B\u044C\u043A\u0438 \u0440\u0456\u0432\u043D\u0456 \u0435\u043D\u0435\u0440\u0433\u0456\u0457, \u0449\u043E \u0432\u0456\u0434\u043D\u043E\u0441\u044F\u0442\u044C\u0441\u044F \u0434\u043E \u043A\u043E\u043D\u043A\u0440\u0435\u0442\u043D\u043E\u0433\u043E \u0456\u043D\u0434\u0435\u043A\u0441\u0443 n, \u0437\u043C\u0456\u043D\u044E\u044E\u0442\u044C\u0441\u044F \u043D\u0435\u043F\u0435\u0440\u0435\u0440\u0432\u043D\u043E \u043F\u043E \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0438\u043C \u0432\u0435\u043A\u0442\u043E\u0440\u0430\u043C k, \u043A\u0430\u0436\u0443\u0442\u044C \u043F\u0440\u043E \u0435\u043D\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u043D\u0443 \u0437\u043E\u043D\u0456 \u0437 \u0456\u043D\u0434\u0435\u043A\u0441\u043E\u043C n. \u0422\u0430\u043A \u044F\u043A \u0432\u043B\u0430\u0441\u043D\u0456 \u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u0435\u043D\u0435\u0440\u0433\u0456\u0457 \u043F\u0440\u0438 \u0437\u0430\u0434\u0430\u043D\u043E\u043C\u0443 n \u043F\u0435\u0440\u0456\u043E\u0434\u0438\u0447\u043D\u0456 \u043F\u043E k, \u0442\u043E \u0445\u0432\u0438\u043B\u044C\u043E\u0432\u0438\u0439 \u0432\u0435\u043A\u0442\u043E\u0440 \u043C\u043E\u0436\u0435 \u0431\u0443\u0442\u0438 \u0437\u0430\u0434\u0430\u043D\u0438\u0439 \u043B\u0438\u0448\u0435 \u0437 \u0442\u043E\u0447\u043D\u0456\u0441\u0442\u044E \u0434\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u0456\u0432 \u043E\u0431\u0435\u0440\u043D\u0435\u043D\u043E\u0457 \u0491\u0440\u0430\u0442\u043A\u0438, \u0432\u0441\u0435 \u0440\u0456\u0437\u043D\u0456 \u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F En(k) \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u044E\u0442\u044C \u0432\u0435\u043A\u0442\u043E\u0440\u0430\u043C k \u0437 \u043F\u0435\u0440\u0448\u043E\u0457 \u0437\u043E\u043D\u0438 \u0411\u0440\u0456\u043B\u043B\u044E\u0435\u043D\u0430 \u043E\u0431\u0435\u0440\u043D\u0435\u043D\u043E\u0457 \u0491\u0440\u0430\u0442\u043A\u0438, \u0456 \u0440\u043E\u0437\u0433\u043B\u044F\u0434\u0443 \u043F\u0456\u0434\u043B\u044F\u0433\u0430\u044E\u0442\u044C \u0441\u0430\u043C\u0435 \u0432\u043E\u043D\u0438."@uk . . "El teorema de Bloch describe el movimiento de los electrones en un s\u00F3lido. Fue enunciado por el f\u00EDsico suizo Felix Bloch bas\u00E1ndose en la idea de que un s\u00F3lido posee una estructura microsc\u00F3pica peri\u00F3dica. A partir de esta hip\u00F3tesis, el teorema establece de qu\u00E9 manera deben ser las funciones de onda de los electrones, y permite tratar el movimiento de todos los electrones analizando \u00FAnicamente el movimiento de un solo electr\u00F3n."@es . . . . . "Una ona de Bloch o estat de Bloch (anomenat en honor de Felix Bloch) \u00E9s la funci\u00F3 d'ona d'una part\u00EDcula (normalment un electr\u00F3) col\u00B7locada en un potencial peri\u00F2dic. El teorema de Bloch postula que l'autofunci\u00F3 d'energia de tal sistema es pot escriure com el producte d'una funci\u00F3 d'ona plana i una funci\u00F3 peri\u00F2dica (funci\u00F3 peri\u00F2dica de Bloch) que t\u00E9 la mateixa periodicitat que el potencial: Els autovalors d'energia corresponents s\u00F3n \u03F5n(k) = \u03F5n(k + K), peri\u00F2dic amb una periodicitat K d'un vector de xarxa rec\u00EDproca. Les energies associades amb l'\u00EDndex n varien cont\u00EDnuament amb el vector d'ona k i formen una banda d'energia identificada per un \u00EDndex de banda n. Els autovalors per una n donada s\u00F3n peri\u00F2dics en k; tots els valors diferents de \u03F5n(k) ocorren per k-valors dins de la primera zona de Brillouin de la xarxa rec\u00EDproca. De fet, el teorema de Bloch \u00E9s una conseq\u00FC\u00E8ncia directa de la simetria translacional dels cristalls, la qual cosa significa que el cristall \u00E9s invariant sota un moviment translacional de la forma , on s\u00F3n enters i s\u00F3n els vectors de xarxa primitius. Si denota l'operaci\u00F3 de translaci\u00F3 que pot ser aplicada a una funci\u00F3 d'ona en la direcci\u00F3 de la forma , on s\u00F3n enters, es pot veure que l'operaci\u00F3 forma un grup amb la mateixa llei de combinaci\u00F3 que . Com que el sistema cristal\u00B7l\u00ED \u2013i, per tant, el seu hamiltoni\u00E0\u2013 \u00E9s invariant despr\u00E9s de tals translacions, l'operador de translaci\u00F3 ha de ser commutatiu amb l'operador hamiltoni\u00E0, per la qual cosa poden ser diagonalitzats simult\u00E0niament. D'aquesta manera, cada funci\u00F3 pr\u00F2pia del hamiltoni\u00E0 pot ser una funci\u00F3 pr\u00F2pia de l'operador de translaci\u00F3. Per mantenir la funci\u00F3 d'ona normalitzada de manera correcta, l'autovalor per a l'operador de translaci\u00F3 ha de ser de la forma , on \u00E9s una funci\u00F3 del vector de translaci\u00F3 . Aplicant aquestes dues translacions i consecutivament a una funci\u00F3 d'ona, es pot mostrar . Per tant, la funci\u00F3 es pot escriure com el producte escalar dels vectors de translaci\u00F3 i un vector a causa de la linealitat de . En aquesta l\u00EDnia s'ha dedu\u00EFt que una funci\u00F3 pr\u00F2pia de l'operador hamiltoni\u00E0 d'un sistema amb simetria translacional discreta (tal com un cristall) \u00E9s sempre una funci\u00F3 pr\u00F2pia dels operadors de translaci\u00F3 discrets sim\u00E8trics amb l'autovalor . En altres paraules, cada autovalor del hamiltoni\u00E0 forma la base per una representaci\u00F3 unidimensional del grup d'operacions de translaci\u00F3 especificades per la xarxa de Bravais i el vector es pot considerar una etiqueta per la representaci\u00F3 irreductible."@ca . . . . . . . "Teorema de Bloch"@ca . . "\u0411\u043B\u043E\u0445\u0456\u0432\u0441\u044C\u043A\u0430 \u0445\u0432\u0438\u043B\u044F"@uk . "\u91CF\u5B50\u529B\u5B66\u3084\u7269\u6027\u7269\u7406\u5B66\u306B\u304A\u3051\u308B\u30D6\u30ED\u30C3\u30DB\u306E\u5B9A\u7406\uFF08\u30D6\u30ED\u30C3\u30DB\u306E\u3066\u3044\u308A\u3001\u82F1: Bloch's theorem\uFF09\u3068\u306F\u3001\u30CF\u30DF\u30EB\u30C8\u30CB\u30A2\u30F3\u304C\u7A7A\u9593\u7684\u306A\u5468\u671F\u6027\uFF08\u4E26\u9032\u5BFE\u79F0\u6027\uFF09\u3092\u3082\u3064\u5834\u5408\u306B\u3001\u305D\u306E\u56FA\u6709\u95A2\u6570\u304C\u6E80\u305F\u3059\u6027\u8CEA\u3092\u8868\u3057\u305F\u5B9A\u7406\u306E\u3053\u3068\u30021928\u5E74\u306B\u3001\u30D5\u30A7\u30EA\u30C3\u30AF\u30B9\u30FB\u30D6\u30ED\u30C3\u30DB\u306B\u3088\u3063\u3066\u5C0E\u51FA\u3055\u308C\u305F\u3002 \u7D50\u6676\u306F\u57FA\u672C\u683C\u5B50\u30D9\u30AF\u30C8\u30EB\u3060\u3051\u4E26\u9032\u3059\u308B\u3068\u81EA\u5206\u81EA\u8EAB\u3068\u91CD\u306A\u308A\u5408\u3046\u305F\u3081\u3001\u4E26\u9032\u5BFE\u79F0\u6027\u3092\u6301\u3064\u3002\u3088\u3063\u3066\u7D50\u6676\u306E\u30A8\u30CD\u30EB\u30AE\u30FC\u30D0\u30F3\u30C9\u3092\u8A08\u7B97\u3059\u308B\u969B\u306B\u30D6\u30ED\u30C3\u30DB\u306E\u5B9A\u7406\u306F\u91CD\u8981\u3068\u306A\u308B\u3002"@ja . . . . . "Proof of Lemma"@en . . . . . . . . . . "Ona de Bloch"@ca . "Assume that we have a wave function which is an eigenstate of all the translation operators. As a special case of this, \n \nfor j = 1, 2, 3, where Cj are three numbers which do not depend on r. It is helpful to write the numbers Cj in a different form, by choosing three numbers \u03B81, \u03B82, \u03B83 with :\n \nAgain, the \u03B8j are three numbers which do not depend on r. Define , where bj are the reciprocal lattice vectors . Finally, define \n\nThen\n\nThis proves that u has the periodicity of the lattice. Since , that proves that the state is a Bloch state."@en . . . . . "Teorema de Bloch"@es . . "We remain with"@en . . "Uma onda de Bloch (ou estado Bloch), em homenagem o f\u00EDsico su\u00ED\u00E7o Felix Bloch, \u00E9 um tipo de fun\u00E7\u00E3o de onda para uma part\u00EDcula em um ambiente de repeti\u00E7\u00E3o peri\u00F3dica, mais comumente um el\u00E9tron em um cristal."@pt . . . . . "\u91CF\u5B50\u529B\u5B66\u3084\u7269\u6027\u7269\u7406\u5B66\u306B\u304A\u3051\u308B\u30D6\u30ED\u30C3\u30DB\u306E\u5B9A\u7406\uFF08\u30D6\u30ED\u30C3\u30DB\u306E\u3066\u3044\u308A\u3001\u82F1: Bloch's theorem\uFF09\u3068\u306F\u3001\u30CF\u30DF\u30EB\u30C8\u30CB\u30A2\u30F3\u304C\u7A7A\u9593\u7684\u306A\u5468\u671F\u6027\uFF08\u4E26\u9032\u5BFE\u79F0\u6027\uFF09\u3092\u3082\u3064\u5834\u5408\u306B\u3001\u305D\u306E\u56FA\u6709\u95A2\u6570\u304C\u6E80\u305F\u3059\u6027\u8CEA\u3092\u8868\u3057\u305F\u5B9A\u7406\u306E\u3053\u3068\u30021928\u5E74\u306B\u3001\u30D5\u30A7\u30EA\u30C3\u30AF\u30B9\u30FB\u30D6\u30ED\u30C3\u30DB\u306B\u3088\u3063\u3066\u5C0E\u51FA\u3055\u308C\u305F\u3002 \u7D50\u6676\u306F\u57FA\u672C\u683C\u5B50\u30D9\u30AF\u30C8\u30EB\u3060\u3051\u4E26\u9032\u3059\u308B\u3068\u81EA\u5206\u81EA\u8EAB\u3068\u91CD\u306A\u308A\u5408\u3046\u305F\u3081\u3001\u4E26\u9032\u5BFE\u79F0\u6027\u3092\u6301\u3064\u3002\u3088\u3063\u3066\u7D50\u6676\u306E\u30A8\u30CD\u30EB\u30AE\u30FC\u30D0\u30F3\u30C9\u3092\u8A08\u7B97\u3059\u308B\u969B\u306B\u30D6\u30ED\u30C3\u30DB\u306E\u5B9A\u7406\u306F\u91CD\u8981\u3068\u306A\u308B\u3002"@ja . "Uma onda de Bloch (ou estado Bloch), em homenagem o f\u00EDsico su\u00ED\u00E7o Felix Bloch, \u00E9 um tipo de fun\u00E7\u00E3o de onda para uma part\u00EDcula em um ambiente de repeti\u00E7\u00E3o peri\u00F3dica, mais comumente um el\u00E9tron em um cristal."@pt . . "Les ondes de Bloch, d'apr\u00E8s F\u00E9lix Bloch, sont les fonctions d'ondes d\u00E9crivant les \u00E9tats quantiques des \u00E9lectrons soumis \u00E0 un potentiel p\u00E9riodique. C'est notamment le cas du cristal parfait infini, les \u00E9lectrons sont soumis \u00E0 un potentiel p\u00E9riodique ayant la sym\u00E9trie de translation des atomes constituant le cristal."@fr . . . . "\u0411\u043B\u043E\u0301\u0445\u043E\u0432\u0441\u043A\u0430\u044F \u0432\u043E\u043B\u043D\u0430\u0301 (\u0432\u043E\u043B\u043D\u0430\u0301 \u0411\u043B\u043E\u0301\u0445\u0430) \u2014 \u043D\u0430\u0437\u0432\u0430\u043D\u043D\u0430\u044F \u0432 \u0447\u0435\u0441\u0442\u044C \u0424\u0435\u043B\u0438\u043A\u0441\u0430 \u0411\u043B\u043E\u0445\u0430 \u0432\u043E\u043B\u043D\u043E\u0432\u0430\u044F \u0444\u0443\u043D\u043A\u0446\u0438\u044F \u0447\u0430\u0441\u0442\u0438\u0446\u044B (\u043E\u0431\u044B\u0447\u043D\u043E \u044D\u043B\u0435\u043A\u0442\u0440\u043E\u043D\u0430), \u0440\u0430\u0441\u043F\u043E\u043B\u043E\u0436\u0435\u043D\u043D\u043E\u0439 \u0432 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u043E\u043C \u043F\u043E\u0442\u0435\u043D\u0446\u0438\u0430\u043B\u0435. \u0421\u043E\u0441\u0442\u043E\u0438\u0442 \u0438\u0437 \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u044F \u043F\u043B\u043E\u0441\u043A\u043E\u0439 \u0432\u043E\u043B\u043D\u044B \u043D\u0430 \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u0443\u044E \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u0443\u044E \u0444\u0443\u043D\u043A\u0446\u0438\u044E (\u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u0430\u044F \u0447\u0430\u0441\u0442\u044C \u0431\u043B\u043E\u0445\u043E\u0432\u0441\u043A\u043E\u0439 \u0432\u043E\u043B\u043D\u043E\u0432\u043E\u0439 \u0444\u0443\u043D\u043A\u0446\u0438\u0438) unk(r), \u0438\u043C\u0435\u044E\u0449\u0443\u044E \u0442\u0443 \u0436\u0435 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u043D\u043E\u0441\u0442\u044C, \u0447\u0442\u043E \u0438 \u043F\u043E\u0442\u0435\u043D\u0446\u0438\u0430\u043B. \u0433\u0434\u0435 \u2014 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u0438\u0435 \u0444\u0443\u043D\u043A\u0446\u0438\u0438, k \u2014 \u0432\u043E\u043B\u043D\u043E\u0432\u043E\u0439 \u0432\u0435\u043A\u0442\u043E\u0440 \u0447\u0430\u0441\u0442\u0438\u0446\u044B. \u0421\u043E\u0433\u043B\u0430\u0441\u043D\u043E \u0442\u0435\u043E\u0440\u0435\u043C\u0435 \u0411\u043B\u043E\u0445\u0430, \u0432 \u0442\u0430\u043A\u043E\u043C \u0432\u0438\u0434\u0435 \u043C\u043E\u0436\u043D\u043E \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u044C \u0432\u0441\u0435 \u0441\u043E\u0431\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0435 \u0444\u0443\u043D\u043A\u0446\u0438\u0438 \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u044B. \u0421\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044E\u0449\u0438\u0435 \u0438\u043C \u0441\u043E\u0431\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0435 \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u044F \u044D\u043D\u0435\u0440\u0433\u0438\u0438 En(k) = En(k + K) \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u043D\u044B \u043F\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u0430\u043C \u043E\u0431\u0440\u0430\u0442\u043D\u043E\u0439 \u0440\u0435\u0448\u0451\u0442\u043A\u0438 K. \u041F\u043E\u0441\u043A\u043E\u043B\u044C\u043A\u0443 \u0443\u0440\u043E\u0432\u043D\u0438 \u044D\u043D\u0435\u0440\u0433\u0438\u0438, \u043E\u0442\u043D\u043E\u0441\u044F\u0449\u0438\u0435\u0441\u044F \u043A \u043A\u043E\u043D\u043A\u0440\u0435\u0442\u043D\u043E\u043C\u0443 \u0438\u043D\u0434\u0435\u043A\u0441\u0443 n, \u0438\u0437\u043C\u0435\u043D\u044F\u044E\u0442\u0441\u044F \u043D\u0435\u043F\u0440\u0435\u0440\u044B\u0432\u043D\u043E \u043F\u043E \u0432\u043E\u043B\u043D\u043E\u0432\u044B\u043C \u0432\u0435\u043A\u0442\u043E\u0440\u0430\u043C k, \u0433\u043E\u0432\u043E\u0440\u044F\u0442 \u043E\u0431 \u044D\u043D\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0437\u043E\u043D\u0435 \u0441 \u0438\u043D\u0434\u0435\u043A\u0441\u043E\u043C n. \u0422\u0430\u043A \u043A\u0430\u043A \u0441\u043E\u0431\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0435 \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u044F \u044D\u043D\u0435\u0440\u0433\u0438\u0438 \u043F\u0440\u0438 \u0437\u0430\u0434\u0430\u043D\u043D\u043E\u043C n \u043F\u0435\u0440\u0438\u043E\u0434\u0438\u0447\u043D\u044B \u043F\u043E k, \u0442\u043E \u0432\u043E\u043B\u043D\u043E\u0432\u043E\u0439 \u0432\u0435\u043A\u0442\u043E\u0440 \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u0437\u0430\u0434\u0430\u043D \u043B\u0438\u0448\u044C \u0441 \u0442\u043E\u0447\u043D\u043E\u0441\u0442\u044C\u044E \u0434\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043E\u0432 \u043E\u0431\u0440\u0430\u0442\u043D\u043E\u0439 \u0440\u0435\u0448\u0451\u0442\u043A\u0438, \u0432\u0441\u0435 \u0440\u0430\u0437\u043B\u0438\u0447\u043D\u044B\u0435 \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u044F En(k) \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044E\u0442 \u0432\u0435\u043A\u0442\u043E\u0440\u0430\u043C k \u0438\u0437 \u043F\u0435\u0440\u0432\u043E\u0439 \u0437\u043E\u043D\u044B \u0411\u0440\u0438\u043B\u043B\u044E\u044D\u043D\u0430 \u043E\u0431\u0440\u0430\u0442\u043D\u043E\u0439 \u0440\u0435\u0448\u0451\u0442\u043A\u0438, \u0438 \u0440\u0430\u0441\u0441\u043C\u043E\u0442\u0440\u0435\u043D\u0438\u044E \u043F\u043E\u0434\u043B\u0435\u0436\u0430\u0442 \u0438\u043C\u0435\u043D\u043D\u043E \u043E\u043D\u0438."@ru . "Twierdzenie Blocha \u2013 jedno z podstawowych twierdze\u0144 w fizyce cia\u0142a sta\u0142ego, m\u00F3wi\u0105ce o og\u00F3lnej postaci rozwi\u0105zania r\u00F3wnania Schr\u00F6dingera dla periodycznego potencja\u0142u. Autorem tego twierdzenia jest Felix Bloch."@pl . . "Lemma"@en . . "The second order term\n\nAgain with \n\nEliminating and we have the theorem"@en . . . . . . . . "\uBE14\uB85C\uD750 \uD30C"@ko . . . . . "Una Onda de Bloch (tambi\u00E9n llamada Estado de Bloch, Funci\u00F3n de Bloch o Funci\u00F3n de onda de Bloch), llamada as\u00ED por el f\u00EDsico suizo Felix Bloch, es un tipo de funci\u00F3n de onda de una part\u00EDcula en un medio peri\u00F3dico, como un electr\u00F3n en un s\u00F3lido cristalino. Una funci\u00F3n de onda \u03C8 es una onda de Bloch si tiene la forma:\u200B donde r es la posici\u00F3n, u es una funci\u00F3n peri\u00F3dica con la misma periodicidad que el cristal, y k es un n\u00FAmero real, llamado el vector de onda del cristal. En otras palabras, si se multiplica una onda plana por una funci\u00F3n peri\u00F3dica, se obtiene una onda de Bloch. Las ondas de Bloch son importantes debido al Teorema de Bloch, que afirma que el autoestado de energ\u00EDa de un electr\u00F3n en un cristal puede ser descrito con ondas de Bloch. M\u00E1s precisamente, indica que la funci\u00F3n de onda de un electr\u00F3n en un cristal tiene una base que est\u00E1 formada solamente por autoestados de energ\u00EDa de Bloch. Este hecho da lugar a la teor\u00EDa de bandas. Los autoestados de las funciones de onda se escriben con sub\u00EDndices, como \u03C8n k, donde n es un valor discreto, llamado \u00EDndice de energ\u00EDa de banda, y que diferencia a las m\u00FAltiples ondas de Bloch con el mismo k (cada una con un componente peri\u00F3dico u distinto). Dentro de una banda (es decir, para un determinado n), \u03C8n k var\u00EDa de manera continua con k, as\u00ED como su energ\u00EDa."@es . . . "Twierdzenie Blocha \u2013 jedno z podstawowych twierdze\u0144 w fizyce cia\u0142a sta\u0142ego, m\u00F3wi\u0105ce o og\u00F3lnej postaci rozwi\u0105zania r\u00F3wnania Schr\u00F6dingera dla periodycznego potencja\u0142u. Autorem tego twierdzenia jest Felix Bloch."@pl . "Die Bloch-Funktion oder Bloch-Welle (nach Felix Bloch) ist eine allgemeine Form f\u00FCr die L\u00F6sung der station\u00E4ren Schr\u00F6dingergleichung f\u00FCr ein Teilchen in einem periodischen Potential, z. B. die Wellenfunktion eines Elektrons in einem kristallinen Festk\u00F6rper (Bloch-Elektron). Die Form dieser Wellenfunktionen wird durch das Bloch-Theorem festgelegt, welches ein Spezialfall des Floquet-Theorems ist: Die Periodizit\u00E4t des Potentials \u00FCbertr\u00E4gt sich also auf und damit auf die Aufenthaltswahrscheinlichkeit des betrachteten Teilchens im Potential. F\u00FCr ein Elektron in so einem Energieeigenzustand ist daher die Aufenthaltswahrscheinlichkeit in jeder Elementarzelle gleich gro\u00DF und zeigt den gleichen r\u00E4umlichen Verlauf. In einem kristallinen Festk\u00F6rper ist die Periodizit\u00E4t gegeben durch das Kristallgitter, ist ein Gittervektor. Ist das Potential zeitunabh\u00E4ngig, kann als reell angesetzt werden."@de . "In condensed matter physics, Bloch's theorem states that solutions to the Schr\u00F6dinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written Bloch function where is position, is the wave function, is a periodic function with the same periodicity as the crystal, the wave vector is the crystal momentum vector, is Euler's number, and is the imaginary unit."@en . "\u0646\u0638\u0631\u064A\u0629 \u0628\u0644\u0648\u062E"@ar . . . . . .