PDM-Charged Quantum Particles Moving in Magnetic Fields

Abstract

The classical and quantum mechanical correspondence for constant mass settings is used, along with some nonlocal point transformation (NPT), to find the position-dependent mass (PDM) classical and quantum Hamiltonians. Consequently, the PDM-momentum operator is constructed. The same recipe is followed to identify the form of the PDM-minimal coupling of electromagnetic interactions for the classical and quantum PDM-Hamiltonians. Using azimuthally symmetrized cylindrical coordinates, some PDM-charged particles moving in magnetic (constant or position-dependent (PD)) and Aharonov-Bohm (AB) flux fields along with some interaction potentials are considered. Their separability and solvability under PDM-settings is also reported. Systems of PDM-charged particles moving in three fields: constant magnetic, AB-flux, and pseudoharmonic oscillator potential, or generalized Killingbeck potential fields are solved for different radial cylindrical PDM settings. Spectral signatures of the one-dimensional z-dependent Schr¨odinger part on the overall eigenvalues and eigenfunctions, are reported using two z-dependent potential models (infinite potential well and Morse-type potentials). PDM-charged particles moving in an inverse power-law-type radial PD-magnetic fields are considered. Under such settings, the exact solutions of almost-quasi-free PDM-charged particles (i.e., no interaction potential) endowed with two type of radial cylindrical PDM settings are obtained. Furthermore, a Yukawa-type PDM-charged particle with a specific PDM setting moving in PD-magnetic and AB-flux fields along with Yukawa plus Kratzer type potential force fields is analyzed (using the Nikiforov-Uvarov (NU) method) to come out with exact solutions of the system. Exact or conditionally exact eigenvalues and eigenfunctions are analytically obtained. Keywords: position-dependent mass Hamiltonian, point transformation, PDM - momentum operator, PDM minimal-coupling, cylindrical coordinates, constant magnetic and position-dependent magnetic fields, Aharonov-Bohm flux field, almost-quasi-free PDM-charged particles, pseudo-harmonic osillator and Killingbeck potentials, Yukawa-plus-Kratzer potential, Nikiforov-Uvarov exact solvability.

OZ: Klasik ve Kuantum Mekaniksel tekˆabuliyet ve Yerel Olmayan Nokta D¨on¨us¸ ¨um (YOND) y¨ontemi kullanarak Pozisyon Ba˘gımlı K¨utle (PBK) Hamilton ve momentum operat¨orleri tespit edilmis¸tir. Bu y¨ontemle minimal kuplajlı elektromagnetik etkiles¸meler ic¸in Klasik ve Kuantum PBK-Hamilton fonksiyonları bulunmus¸tur. Eksensel simetrik silindirik koordinatlar kullanarak s¸arjlı partik¨ullerin, sabit (veya pozisyon ba˘gımlı (PB)), magnetik Aharanov–Bohm (AB) akısı ve etkiles¸im potansiyelleri ic¸indeki hareketleri ele alınmıs¸tır. PBK durumunda hareket denklemleri ayrım ve c¸o¨zu¨m ac¸ısından incelenmis¸tir. U¨ c¸ durum ele alınmıs¸tır: Sabit magnetik alan, AB-akısı, sahte harmonik hareket potansiyeli veya genellenmis¸ Killingbeck potansiyel alanlarında radyal silindirik c¸ ¨oz¨umler verilmis¸tir. Tek boyutlu, z–ba˘gımlı Schr¨odinger denkleminin spektral is¸aretlerinden, iki farklı potansiyel (sonsuz kuyu ve Morse–gibi) ic¸in uygun de˘ger ve fonksiyonları elde edilmis¸tir. Burada PBK partik¨ulleri ic¸in ters–¨ustel etkiles¸im alanları durumunda radyal PB-magnetik alan ele alınmıs¸tır. Yaklas¸ık, sahte serbest g¨or¨un¨uml¨u, s¸arjlı PBK durumunda kesin silindirik c¸ ¨oz¨umler verilmis¸tir. ˙Ilˆaveten Yukawa tipi s¸arjlı PBK, PB–magnetik, AB–akılı ve Kratzer tipi potansiyel katkılı alanlar ic¸in Nikiforov–Uvarov (NU) y¨ontemi sayesinde c¸ ¨oz¨umler bulunmus¸tur. Kesin veya s¸artlı uygun de˘ger fonksiyonları elde edilmis¸tir. Anahtar Kelimeler: pozisyon ba˘gımlı k¨utle Hamilton fonksiyonu, momentum operat¨or¨u, minimal kuplaj, silindirik koordinatlar, sabit ve pozisyon ba˘gımlı magnetik alan, Aharanov – Bohm akı alanı, yaklas¸ık serbest pozisyon ba˘gımlı s¸arj partik¨ul, sahte–harmonik titres¸im, Killingbech, Yukawa – Kratzer potansiyelleri, Nikiforov – Uvarov kesin c¸ ¨oz¨uml¨ul¨u˘g¨u.