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<p><span>Solutions to physical problems with time-dependent potentials are quite rare. In most cases, we resort to time-dependent perturbation theory to obtain an approximate answer, valid only at early times and for small potentials. </span></p><p><span>One of the few exactly solvable problems is the two-level system with the Hamiltonian:
</span></p>
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<script type="math/tex; mode=display" id="MathJax-Element-11">\begin{align*}
H =
\begin{pmatrix}
\frac{\hbar}{2}\omega_0 & \frac{\hbar}{2}\omega_1 e^{-i\omega t}\\
\frac{\hbar}{2}\omega_1 e^{i\omega t} & -\frac{\hbar}{2}\omega_0\\
\end{pmatrix}
\end{align*}
</script>
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</div>
<p><span>This can be seen as coupling the magnetic moment </span>
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x="8561" y="0"></use><use xlink:href="#E4-MJMATHI-6E" x="8906" y="0"></use><use xlink:href="#E4-MJMAIN-28" x="9506" y="0"></use><use xlink:href="#E4-MJMATHI-3C9" x="9895" y="0"></use><use xlink:href="#E4-MJMATHI-74" x="10517" y="0"></use><use xlink:href="#E4-MJMAIN-29" x="10878" y="0"></use><use xlink:href="#E4-MJMAIN-2C" x="11267" y="0"></use><g transform="translate(11711,0)"><use xlink:href="#E4-MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4-MJMAIN-30" x="1073" y="-213"></use></g><use xlink:href="#E4-MJMAIN-29" x="12924" y="0"></use></g></svg></span><script type="math/tex"> \vec{B} = (B \space cos(\omega t),B \space sin(\omega t),B_0) </script><span>. As time carries on, the probability distribution of a particle initially in the spin-up state </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.807ex" height="5.846ex" viewBox="0 -1509.8 5944.7 2517" role="img" focusable="false" style="vertical-align: -2.339ex;"><defs><path stroke-width="0" id="E5-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E5-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 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219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E6-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E6-MJMAIN-2217" x="510" y="513"></use><use xlink:href="#E6-MJMAIN-3D" x="1092" y="0"></use><g transform="translate(1870,0)"><g transform="translate(397,0)"><rect stroke="none" width="1805" height="60" x="0" y="220"></rect><g transform="translate(60,412)"><use transform="scale(0.707)" 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inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.971ex" height="2.577ex" viewBox="0 -806.1 3862.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E8-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="0" id="E8-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 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750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E9-MJMAIN-2193" d="M473 86Q483 86 483 67Q483 63 483 61T483 56T481 53T480 50T478 48T474 47T470 46T464 44Q428 35 391 14T316 -55T264 -168Q264 -170 263 -173T262 -180T261 -184Q259 -194 251 -194Q242 -194 238 -176T221 -121T180 -49Q169 -34 155 -21T125 2T95 20T67 33T44 42T27 47L21 49Q17 53 17 67Q17 87 28 87Q33 87 42 84Q158 52 223 -45L230 -55V312Q230 391 230 482T229 591Q229 662 231 676T243 693Q244 694 251 694Q264 692 270 679V-55L277 -45Q307 1 353 33T430 76T473 86Z"></path><path stroke-width="0" id="E9-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E9-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E9-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E9-MJMATHI-50" x="0" y="0"></use><use xlink:href="#E9-MJMAIN-28" x="751" y="0"></use><use xlink:href="#E9-MJMAIN-2193" x="1140" y="0"></use><use xlink:href="#E9-MJMAIN-29" x="1640" y="0"></use><use xlink:href="#E9-MJMAIN-3D" x="2306" y="0"></use><use xlink:href="#E9-MJMAIN-31" x="3362" y="0"></use></g></svg></span><script type="math/tex">P(\downarrow) = 1</script><span> -- there can be a zero probability to find the particle in its initial state!</span></p><p><span>Our simulation displays the proability over time of a particle being in a spin-up or spin-down state.</span></p></div>