Kajian Osilator Dirac dengan Solusi Non-Relativistik Tiga Dimensi dan Relativistik Dua Dimensi

Roihan, Akh. Zaki (2025) Kajian Osilator Dirac dengan Solusi Non-Relativistik Tiga Dimensi dan Relativistik Dua Dimensi. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Tugas akhir ini menganalisis osilator Dirac dengan solusi non-relativistik pada kasus tiga dimensi dan relativistik untuk kasus dua dimensi. Solusi non-relativistik menunjukkan osilator Dirac terdiri atas tiga kontribusi utama, yaitu energi dari osilator harmonik tiga dimensi biasa, koreksi konstanta sebesar 3/2ℏω, serta suku tambahan berupa interaksi kopling spin-orbit L·S. Pada kasus dua dimensi digunakan dua pendekatan, yaitu pendekatan langsung persamaan diferensial dengan separasi variabel dan pendekatan operator. Dalam pendekatan langsung, persamaan Dirac diselesaikan dengan mensubstitusi komponen spinor, menghasilkan energi eigen, yaitu E=±mc^2\sqrt{1+(2ℏω/mc^2)(n_c+n_y∓1/2)}, dengan n_x, n_y sebagai bilangan kuantum osilator. Pendekatan operator memanfaatkan operator anihilasi dan kreasi baru, yaitu (a_l,a_l^†), menghasilkan energi eigen E=±mc^2\sqrt{1+4ℏω/mc^2}n_l, dimana n_l adalah bilangan kuantum tunggal baru. Perbandingan kedua pendekatan menunjukkan bahwa pendekatan operator lebih ringkas karena memanfaatkan simetri gabungan antara arah x dan y, sehingga momentum sudut tidak muncul secara eksplisit. Sebaliknya, pendekatan langsung memberikan pemisahan yang lebih eksplisit terhadap kontribusi dari masing-masing arah serta terhadap struktur momentum sudut. Meskipun demikian, hasil dari pendekatan operator masih perlu dikaji ulang, terutama dalam hal struktur degenerasi energi, yang belum muncul secara eksplisit dalam pendekatan operator. Hal ini berbeda dengan pendekatan persamaan diferensial, yang telah menunjukkan struktur degenerasi energi yang sesuai.
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This thesis analyzes the Dirac oscillator with non-relativistic solutions in the three-dimensional case and relativistic solutions in the two-dimensional case. The non-relativistic solution shows that the Dirac oscillator consists of three main contributions: the energy of the usual three-dimensional harmonic oscillator, a constant correction term of 3/2ℏω, and an additional term representing the spin-orbit coupling interaction L·S. In the two-dimensional case, two approaches are employed, namely the direct differential equation approach with variable separation, and the operator approach. In the direct approach, the Dirac equation is solved by substituting the spinor components, yielding the energy eigenvalues E=±mc^2\sqrt{1+(2ℏω/mc^2)(n_c+n_y∓1/2)}, where (n_x,n_y) are the oscillator quantum numbers. The operator approach utilizes new annihilation and creation operators, namely (a_l,a_l^†), and yields the energy eigenvalues E=±mc^2\sqrt{1+4ℏω/mc^2}n_l, where n_l is a new single quantum number. A comparison of the two approaches shows that the operator approach is more compact, as it exploits the combined symmetry of the x and y directions, resulting in the angular momentum not appearing explicitly. In contrast, the direct approach provides a more explicit separation of the contributions from each direction as well as the angular momentum structure. Nevertheless, the results from the operator approach still require further examination, particularly regarding the energy degeneracy structure, which does not appear explicitly in the operator approach. This is in contrast to the differential equation approach, which has shown a proper energy degeneracy structure.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Operator Anihilasi-Kreasi, Osilator Dirac, Osilator Harmonik, Annihilation-Creation Operators, Dirac Oscillator, Harmonic Oscillator
Subjects:	Q Science > Q Science (General) > Q180.55.M38 Mathematical models Q Science > QA Mathematics > QA371 Differential equations--Numerical solutions Q Science > QA Mathematics > QA401 Mathematical models.
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Physics > 45201-(S1) Undergraduate Thesis
Depositing User:	Akh. Zaki Roihan
Date Deposited:	28 Jul 2025 06:47
Last Modified:	28 Jul 2025 06:47
URI:	http://repository.its.ac.id/id/eprint/121885

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