This paper addresses gaps identified in pedagogical studies of how misunderstanding of De Broglie waves affects later coursework and presents a heuristic for understanding the De Broglie frequency of composite. De Broglie&#x2019;s little known derivation is reviewed with a new illustration based on his description. Simple techniques for reference frame independent analysis of a moving double slit electron interference experiment are not previously found in any literature and cement the concepts. Points of similarity and difference between De Broglie and Schr&#xf6;dinger waves are explained. The necessity of momentum, energy, and wavelength changes in the electrons in order for them to be vertically displaced in their own reference frame is shown to be required to make the double slit analysis work. A relativistic kinematic analysis of De Broglie frequency is provided showing how the higher De Broglie frequency of moving particles is consistent with Special Relativity and time dilation and that it demonstrates a natural system which obeys Einstein&#x2019;s clock synchronization convention of simultaneity and no other. Students will be better prepared to identify practical approaches to solving problems and to think about fundamental questions.

Robert L. Shuler

Common Pedagogical Issues with De Broglie Waves: Moving Double Slits, Composite Mass, and Clock Synchronization

Transparent conducting Ga:ZnO (GZO) and Al:ZnO (AZO) layers have been deposited by spin coating on glass substrates using crystalline nanoparticles redispersed in 1-propanol. The coatings have been sintered in air at 600°C for 15 min and then postannealed in a reducing atmosphere at 400°C for 90 min. The effect of Ga and Al doping on the structural, morphological, optical, and electrical properties of the obtained thin films was investigated. Both films were found to be crystalline with a hexagonal structure. A single step spin coated layer 52–56 nm thick is obtained. To increase the thickness and lower the obtained sheet resistance multilayers coatings have been used. The visible transmission of both layers is high (<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.70224pt" id="M1" height="9.50777pt" version="1.1" viewBox="-0.0498162 -8.80553 36.5491 9.50777" width="36.5491pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-85" d="M655 676H628C608 656 600 650 571 650H156C118 650 114 651 102 676H77C63 615 45 548 24 485L60 486C81 531 97 562 114 580C134 603 148 611 252 611H308L217 127C201 41 196 38 105 32L98 0H384L393 32C300 38 288 41 304 127L393 611H466C553 611 571 604 582 580C592 559 597 530 595 485L630 489C636 548 645 624 655 676Z"/></g><g transform="matrix(.013,0,0,-0.013,12.064,0)"><path id="g54-92" d="M530 234V290L81 530V468L467 263V261L81 57V-6L530 234Z"/></g><g transform="matrix(.013,0,0,-0.013,23.683,0)"><path id="g50-57" d="M253 635C142 635 71 555 71 471C71 401 113 353 178 316C144 294 106 267 91 252C70 231 46 202 46 157C46 51 132 -12 241 -12C328 -12 442 52 442 168C442 257 379 305 311 343C354 373 380 398 388 407C406 428 417 458 417 487C417 568 348 635 253 635ZM240 600C288 600 340 563 340 481C340 423 313 387 279 360C207 394 148 427 148 499C148 549 182 600 240 600ZM252 23C188 23 129 73 129 163C129 217 162 266 211 298C289 260 360 217 360 144C360 69 312 23 252 23Z"/></g><g transform="matrix(.013,0,0,-0.013,30.014,0)"><path id="g50-49" d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"/></g></svg>%). The influence of the sintering temperature and the optimum doping concentration was investigated. Five layers synthesized with doping ratio of 1 mol.% and sintered at 600°C and then submitted to reducing treatment in forming gas exhibited a minimum resistivity value of 7.4 × 10<sup>−2</sup> Ω·cm for GZO layer and 1.45 Ω·cm for AZO coating.

A. Alkahlout

A Comparative Study of Spin Coated Transparent Conducting Thin Films of Gallium and Aluminum Doped ZnO Nanoparticles

<i>Problem</i>. The explanation of nuclear properties from the strong force upwards has been elusive.<i> Approach</i>. Design methods were used to develop conceptual mechanics for the bonding arrangements between nucleons, based on the covert structures for the proton and neutron as defined by the Cordus theory, a type of nonlocal hidden-variable design with discrete fields.<i> Findings</i>. Nuclear bonding arises from the synchronous interaction between the discrete fields of the proton and neutron. This results in not one but multiple types of bond, cis- and transphasic, and assembly of chains and bridges of nucleons into a nuclear polymer. The synchronous interaction constrains the relative orientation of nucleons, and hence the nuclear polymer takes only certain spatial layouts. The stability of nuclides is entirely predicted by morphology of the nuclear polymer and the cis-/transphasic nature of the bonds. The theory successfully explains the qualitative stability characteristics of all hydrogen and helium nuclides.<i> Originality</i>. Novel contributions include the concept of a nuclear polymer and its mechanics; an explanation of the stability, instability, or nonexistence of nuclides starting from the strong/synchronous force; explanation of the role of the neutron. The theory opens a new field of mechanics by which nucleon interactions may be understood.

Dirk J. Pons

Arion D. Pons

Aiden J. Pons

Nuclear Polymer Explains the Stability, Instability, and Nonexistence of Nuclides

By three independent hints it will be demonstrated that still at present there is a substantial lack of theoretical understanding of the CMB phenomenon. One point, as we show, is that at the phase of the recombination era one cannot assume complete thermodynamic equilibrium conditions but has to face both deviations in the velocity distributions of leptons and baryons from a Maxwell-Boltzmann distribution and automatically correlated deviations of photons from a Planck law. Another point is that at the conventional understanding of the CMB evolution in an expanding universe one has to face growing CMB temperatures with growing look-back times. We show, however, here that the expected CMB temperature increases would be prohibitive to star formation in galaxies at redshifts higher than <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.29774pt" id="M1" height="8.64656pt" version="1.1" viewBox="-0.0498162 -8.34882 27.7846 8.64656" width="27.7846pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-123" d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"/></g><g transform="matrix(.013,0,0,-0.013,10.107,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,21.37,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g></svg> where nevertheless the cosmologically most relevant supernovae have been observed. The third point in our present study has to do with the assumption of a constant vacuum energy density which is required by the present <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M2" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.05107 8.68572" width="9.05107pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-140" d="M671 0V28C608 36 593 47 562 130C495 309 432 486 368 665L336 656C270 481 202 311 134 139C98 47 84 39 20 28V0H241V28C167 38 156 47 178 115C217 236 276 383 327 530H329C375 398 437 223 484 92C499 47 489 37 424 28V0H671Z"/></g></svg>CDM-cosmology. Our studies here rather lead to the conclusion that cosmic vacuum energy density scales with the inverse square of the cosmic expansion scale <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M3" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 44.8654 11.5564" width="44.8654pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.013,0,0,-0.013,11.783,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,23.046,0)"><use xlink:href="#g113-83"/></g><g transform="matrix(.013,0,0,-0.013,31.197,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.013,0,0,-0.013,35.695,0)"><path id="g113-117" d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"/></g><g transform="matrix(.013,0,0,-0.013,40.128,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>. Thus we come to the conclusion that with the interpretation of the present-day high quality CMB data still needs to be considered carefully.

Hans-J&#xf6;rg Fahr

Michael Sokaliwska

Remaining Problems in Interpretation of the Cosmic Microwave Background

A series of titanium-based coatings (50&#x2013;550&#x2009;<i>&#x3BC;</i>m thick) on an aluminium substrate was deposited via multichamber detonation sprayer with different barrel lengths (300, 400, and 500, all in mm). The titanium-based coatings obtained in these three experiments were examined by optical microscopy, scanning electron microscopy, and X-ray phase analysis. The hardness tests were carried out by the Vickers method with a test load of 50&#x2009;g. The multichamber detonation sprayer with a barrel length of 500&#x2009;mm produced the dense layers of coating with hardness of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3625803pt" id="M1" height="9.06134pt" version="1.1" viewBox="-0.0498162 -8.69876 59.8308 9.06134" width="59.8308pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.0135,0,0,-0.0135,6.503,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g><g transform="matrix(.0135,0,0,-0.0135,13.006,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g><g transform="matrix(.0135,0,0,-0.0135,19.51,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.0135,0,0,-0.0135,29.041,0)"><path id="g117-37" d="M535 290V340H323V533H265V340H52V290H265V84H323V290H535ZM535 -22V28H52V-22H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,40.022,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.0135,0,0,-0.0135,46.525,0)"><path id="g113-54" d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"/></g><g transform="matrix(.0135,0,0,-0.0135,53.028,0)"><use xlink:href="#g113-49"/></g></svg>&#x2009;HV<sub>0.05</sub> and porosity of 0.24&#x25;, the best result in the experiments.

M. Kovaleva

Yu. Tyurin

N. Vasilik

O. Kolisnichenko

M. Prozorova

M. Arseenko

V. Sirota

I. Pavlenko

Structure and Microhardness of Titanium-Based Coatings Formed by Multichamber Detonation Sprayer

The solution of the field equations of the conformal theory of gravitation with Dirac scalar field in Cartan-Weyl spacetime at the very early Universe is obtained. In this theory dark energy (described by an effective cosmological constant) is a function of the Dirac scalar field <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M1" height="13.2498pt" version="1.1" viewBox="-0.0498162 -9.6779 8.00062 13.2498" width="8.00062pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-224" d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"/></g></svg>. This solution describes the exponential decreasing of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M2" height="13.2498pt" version="1.1" viewBox="-0.0498162 -9.6779 8.00062 13.2498" width="8.00062pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-224"/></g></svg> at the inflation stage and has a limit to a constant value of the dark energy at large time. This can give a way to solving the fundamental cosmological constant problem as a consequence of the fields dynamics in the early Universe.

O. V. Babourova

B. N. Frolov

Dark Energy as a Cosmological Consequence of Existence of the Dirac Scalar Field in Nature

In the framework of proposed theory of spacetime deformation/distortion, we have a way to deform the spacetime through a nontrivial choice of the distortion-complex, displaying different connections, which may reveal different post-Riemannian spacetime structures as corollary. We extend this theory to address, in particular, the gauge model of the most general metric-affine gravity carrying both nontrivial torsion and nonmetricity. This model is constructed in the framework of the first order Lagrangian expressed in terms of the gauge potentials and their first derivatives. The equations of the standard theory, which have no propagating modes for torsion, can be equivalently replaced in modified framework by the modified equations, which in the limit of reducing the affine group leads to the modified Einstein-Cartan theory with dynamical torsion and beyond. In testing the modified framework for various particular cases, we use the Lagrange multipliers for extinguishing nonmetricity and torsion.

Gagik Ter-Kazarian

Modified Theories of Gravitation behind the Spacetime Deformation

We aimed to evaluate the ground-state and excite-state energy eigenvalue (En), wave function, and the time-independent correlation function of the atomic density fluctuation of a particle under the harmonics oscillator Cosine asymmetric potential (Saad et al. 2013). Instead of using the 6-point kernel of 4 Green&#x2019;s function (Cherroret and Skipetrov, 2008), averaged over disorder, we use the numerical shooting method (NSM) to solve the Schr&#xf6;dinger equation of quantum mechanics system with Cosine asymmetric potential. Since our approach does not use complicated formulas, it requires much less computational effort when compared to the Green functions techniques (Cherroret and Skipetrov, 2008). We show that the idea of the program of evaluating time-independent correlation function of atomic density is underdamped motion for the Cosine asymmetric potential from the numerical shooting method of this problem. Comparison of the time-independent correlation function obtained from numerical shooting method by Boonchui and Hutem (2012) and correlation function experiment by Kasprzak et al. (2008). We show the intensity of atomic density fluctuation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M1" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 121.552 12.0395" width="121.552pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><path id="g113-226" d="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"/></g><g transform="matrix(.0135,0,0,-0.0135,11.693,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.0135,0,0,-0.0135,18.495,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,23.182,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/></g><g transform="matrix(.0135,0,0,-0.0135,30.756,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.0135,0,0,-0.0135,39.229,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,51.796,-.231)"><path id="g119-25" d="M470 619L451 641C422 610 391 595 348 595C316 595 288 601 248 612C197 628 172 633 138 633C97 633 43 615 0 555L19 532C54 562 86 578 130 578C166 578 189 571 222 562C267 549 301 540 337 540C381 540 434 557 470 619Z"/></g><g transform="matrix(.0135,0,0,-0.0135,50.967,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0135,0,0,-0.0135,57.793,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,62.481,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0135,0,0,-0.0135,70.055,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,77.77,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,90.39,-.231)"><path id="g119-113" d="M658 624L641 649C599 612 554 595 493 595C444 595 402 603 342 615C268 634 232 641 184 641C129 641 57 621 0 557L17 532C67 566 113 586 176 586C230 586 265 577 316 565C380 550 430 540 482 540C540 540 611 558 658 624Z"/></g><g transform="matrix(.0135,0,0,-0.0135,88.751,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g><g transform="matrix(.0135,0,0,-0.0135,99.481,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,104.168,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0135,0,0,-0.0135,111.742,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,116.43,0)"><use xlink:href="#g113-42"/></g></svg> in harmonics oscillator Cosine asymmetric potential by numerical shooting method.

Artit Hutem

Piyarut Moonsri

Evaluated Excited-State Time-Independent Correlation Function and Eigenfunction of the Harmonics Oscillator Cosine Asymmetric Potential via Numerical Shooting Method

We consider a particle in a spatial symmetric/asymmetric potential driven by time periodic bichromatic AC fields of ratchet type. The associated time-dependent Schr&#xF6;dinger equation is conveniently tackled with the Floquet theory. We next proceed to investigate the ratchet effect induced by the driver, comparing the symmetric with the asymmetric cases. It turns out that the current in the asymmetric case is stronger than that of the symmetric one. Besides, we also investigate the case where the driver is a delta kicked acting on our spatial potential with more emphasis on its chaotic behaviour. Here we check that the current emerges as the phase space is mixed and that the system with asymmetric spatial potential becomes more chaotic than the symmetric one at low kicking strength.

Nkongho Ayuketang Arreyndip

Kenfack Anatole

Comparing the Ratchet Effects of Cold Atoms in Periodically Symmetric and Asymmetric Optical Potentials

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