مقاله پژوهشی: حل معادلۀ دیراک برای هامیلتونی شبه‌هرمیتی و گذار از ترازهای انرژی

نوع مقاله : مقاله پژوهشی

نویسندگان

1 استادیار، گروه فیزیک، دانشکده علوم پایه، دانشگاه شاهد، تهران، ایران

2 دانشجوی کارشناسی ارشد، گروه فیزیک، دانشکدة علوم پایه، دانشگاه شاهد، تهران، ایران.

3 دانشجوی کارشناسی ارشد، گروه فیزیک، دانشکدة علوم پایه، دانشگاه شاهد، تهران، ایران

چکیده

در این مقاله، معادلۀ نسبیّتی دیراک در یک بعد، برای ذرّه­ای در یک میدان الکترومغناطیسی خارجی با تقریب جرم مؤثّر وابسته به مکان، و در نبود پتانسیل برداری، مورد بررسی قرار می­گیرد. با حذف مؤلّفۀ اسپینور پایین و ترکیب جفت معادلات، معادلۀ شرودینگرگونه­ای برای مؤلّفۀ اسپینورِ بالا به دست می­آید.  با استفاده از تبدیلات کانونیک و معرّفی دو عملگر دیفرانسیلی مرتبۀ اوّل هرمیتی و پاد هرمیتی، فرمول­بندی مناسبی برای هامیلتونی­های شبه‌هرمیتی با تقارنِ پاریته-وارونی زمان (PT) نتیجه می‌شود. مقایسة معادلۀ حاصل از هامیلتونی شبه‌هرمیتی با معادلۀ غیر نسبیّتی شرودینگر، به یک فرمول­بندی کلّی برای پتانسیل­های موهومی غیرهرمیتی حل­پذیر یک‌بعدی با طیف انرژی حقیقی منجر خواهد شد. همچنین با به‌کارگیری این رهیافت، پتانسیل­های موهومی پاشل- تلر و اسکارف 2  با طیف انرژی حقیقی در معادلۀ دیراک با چارچوب جرم مؤثّر وابسته به مکان و تقارن PT  بررسی و کاربرد آنها بیان می‌گردد. به ازای برخی پارامترها، شاهد پدیدۀ گذار (عبور) از ترازهای انرژی خواهیم بود. در واقع به تعبیری می‌توان گفت که ترازهای انرژی از طیف انرژی ناپدید می­شوند. همچنین برای مثال‌های ذکر شده، نمودارهای پتانسیل نیز رسم شده­است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Research Paper: Solution of the Dirac Equation for Pseudo-Hermitian Hamiltonian and Energy-levels Crossing

نویسندگان [English]

  • Zahra Bakhshi 1
  • Fereshte Soleimani 2
  • Sare Khoshdooni 3
1 Assistant Professor, Department of Physics, Faculty of Basic Sciences, Shahed University, Tehran, Iran
2 Master Student, Department of Physics, Faculty of Basic Sciences, Shahed University, Tehran, Iran
3 Master Student, Department of Physics, Faculty of Basic Sciences, Shahed University, Tehran, Iran
چکیده [English]

In this paper, the relativistic Dirac equation in one dimension is investigated for a particle in an external electromagnetic field, with the property of position-dependent effective mass approximation (PDEM), in the absence of vector potential. By removing the lower spinor component and combining the pair of equations, a Schrödinger-like equation is obtained for the upper spinor component. Using canonical transformations and introducing two first-order Hermitian and anti-Hermitian differential operators, a formalism for pseudo-hermitic Hamiltonians with parity-time reversal symmetry (PT) has been obtained. Comparing the equation derived from pseudo-Hermitian Hamiltonian with the non-relativistic Schrödinger equation leads to a general formalism for one-dimensional solvable imaginary non-Hermitian potentials with real energy spectra. Also, using this process, the complex potentials of Pöschl-Teller and Scarf II with real energy spectra in Dirac equation with PDEM approximationand PT symmetry have been investigated and their application has been expressed. For some particular parameters we will see the phenomenon of energy-levels crossing. In fact, it means that energy levels disappear from the spectrum. Also, for the mentioned examples, potential figures are drawn.

کلیدواژه‌ها [English]

  • Dirac equation
  • Schrödinger Equation
  • imaginary potential
  • Pseudo-Hermitian Hamiltonian
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