Quantum Analysis on Time Behavior of a Lengthening Pendulum

<div>The probability density <svg height="15.0208pt" id="M126" style="vertical-align:-3.429399pt" version="1.1" viewbox="-0.0498162 -11.5914 20.4436 15.0208" width="20.4436pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M162 -163V703H101V-163H162Z" id="g113-9"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="263" vert-adv-y="263"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z" id="g113-245"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="654" vert-adv-y="654"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.923,0)"><path d="M162 -163V703H101V-163H162Z" id="g113-9"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="263" vert-adv-y="263"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,15.342,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg> given in (<a href="https://static.hindawi.com/articles/physri/volume-2016/1696105/figures/#EEq42">36</a>) as a function of angle <svg height="9.49473pt" id="M127" style="vertical-align:-0.2063999pt" version="1.1" 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92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z" id="g113-117"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="341" vert-adv-y="341"></glyph.data></g></svg> for <svg height="9.49473pt" id="M129" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 28.0195 9.49473" width="28.0195pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L210 518H371L383 557H218L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L143 557H62L51 518H135L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z" id="g113-95"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="516" 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We see that the wave packet oscillates around the center like a classical one.</div>

Physics Research International

Quantum Analysis on Time Behavior of a Lengthening Pendulum

Figure 2