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Science Essence Journal

การคำนวณคาเบองตนของพลงงานสถานะพนของสสารประเภทเฟอรมออนในสองมตทประกอบดวยไฮโดรเจนหนงและสองอะตอม (Basic Calculations of Ground-State Energy of Two Dimensional Fermionic Matter Consisting of One and Two Hydrogen Atoms)

สิริ สิรินิลกุล

Published Jun 20, 2012• 0 citations
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Abstract

การคำนวณเบื้องต้นของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออนในสองมิติที่ประกอบด้วยอะตอมไฮโดรเจน ได้จากการพิจารณาฟังก์ชันดีเทอร์มิเนน ที่สอดคล้องกับฟังก์ชันคลื่นนอร์มัลไลซ์และฟังก์ชันสปินนอร์มัลไลซ์ พลังงานสถานะพื้นสองค่าที่คำนวณได้ ได้แก่พลังงานสถานพื้นแบบแม่นตรงของอะตอมไฮโดรเจนจำนวนหนึ่งอะตอมเท่ากับ ซึ่งพิจารณาภายใต้อันตรกิริยาแบบคูลอมป์ และค่าขอบเขตบนของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออน ที่ประกอบด้วยอะตอมไฮโดรเจนจำนวนสองอะตอมเท่ากับ The basic calculations of the ground-state energy of two dimensional fermionic matter consisting of hydrogen atom have been shown by considering the determinantal function with normalized wavefunction and normalized spin functions. Two ground-state energies are derived. The exact ground-state energy, based on considering the one hydrogen atom wavefunction in the ground-state under Coulomb interaction in two dimensions, is . The upper bound of the ground-state energy, based on considering 2 hydrogen atoms with infinitely separated 2 clusters, each in its ground-state, with nuclear charges each having one electron, is .

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การคำนวณเบื้องต้นของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออนในสองมิติที่ประกอบด้วยอะตอมไฮโดรเจน ได้จากการพิจารณาฟังก์ชันดีเทอร์มิเนน ที่สอดคล้องกับฟังก์ชันคลื่นนอร์มัลไลซ์และฟังก์ชันสปินนอร์มัลไลซ์ พลังงานสถานะพื้นสองค่าที่คำนวณได้ ได้แก่พลังงานสถานพื้นแบบแม่นตรงของอะตอมไฮโดรเจนจำนวนหนึ่งอะตอมเท่ากับ ซึ่งพิจารณาภายใต้อันตรกิริยาแบบคูลอมป์ และค่าขอบเขตบนของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออน ที่ประกอบด้วยอะตอมไฮโดรเจนจำนวนสองอะตอมเท่ากับ The basic calculations of the ground-state energy of two dimensional fermionic matter consisting of hydrogen atom have been shown by considering the determinantal function with normalized wavefunction and normalized spin functions. Two ground-state energies are derived. The exact ground-state energy, based on considering the one hydrogen atom wavefunction in the ground-state under Coulomb interaction in two dimensions, is . The upper bound of the ground-state energy, based on considering 2 hydrogen atoms with infinitely separated 2 clusters, each in its ground-state, with nuclear charges each having one electron, is .
ขอบเขตล่างที่ชัดเจนของพลังงานสถานะพื้นของสสารที่เป็นกลางประเภทเฟอร์มิออนในสองมิติภายใต้อันตรกิริยาคูลอมบ์โดยที่ประจุบวกถูกกำหนดให้อยู่กับที่ คือ ในหน่วยไรด์เบิร์ก ได้จากการพิจารณาขอบเขตล่างของพลังงานจลน์ในรูปยกกำลังอินทิกรัลของ เมื่อ คือความหนาแน่นของอนุภาค ยิ่งไปกว่านั้น ยังพบว่าพลังงานที่สถานะพื้นของสสารแปรผกผันกับตัวคูณสปิน The rigorous lower bound for the ground state energy, in Rydberg unit, in two dimensions of neutral matter of fermionic types with Coulomb interactions with fixed positive charges is possessed by considering, in process, lower bound for the kinetic energy as some power of an integral of where is the particle density and, moreover, it is the inverse proportion with the spin multiplicity.
The ground state of a two-dimensional charged quantum system is studied using the method of correlated-basis functions. For particles obeying the Bose-Einstein statistics we compute the ground-state energy variationally for all particle densities using the Bijl-Dingle-Jastrow trial wave function. The Bose ground-state energy and wave function so determined are used next to study the Fermi system. We compute the Fermi ground-state energy using a cluster expansion approach, which is again valid for all particle densities, and find that the ground state is paramagnetic. While these calculations provide variational upper bounds to the ground-state energies, we also establish independent lower bounds to the ground-state energies.
We derive an approximate analytic formula for the ground-state energy of the charged anyon gas. Our approach is based on the harmonically confined two-dimensional (2D) Coulomb anyon gas and a regularization procedure for vanishing confinement. To take into account the fractional statistics and Coulomb interaction we introduce a function, which depends on both the statistics and density parameters ($\ensuremath{\nu}$ and rs{r}_{s}rs​, respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large rs{r}_{s}rs​ (the 2D Wigner crystal), and to the Hartree-Fock (HF) energy of the spin-polarized 2D electron gas, and the dense 2D Coulomb Bose gas at very small rs{r}_{s}rs​. The latter is calculated by use of the Bogoliubov approximation. Applied to the boson system $(\ensuremath{\nu}=0)$ our results are very close to recent results from Monte Carlo (MC) calculations. For spin-polarized electron systems $(\ensuremath{\nu}=1)$ our comparison leads to a critical judgment concerning the density range, to which the HF approximation and MC simulations apply. In dependence on $\ensuremath{\nu}$, our analytic formula yields ground-state energies, which monotonously increase from the bosonic to the fermionic side if rs>1{r}_{s}>1rs​>1. For ${r}_{s}\ensuremath{\leqslant}1$ it shows a nonmonotonous behavior indicating a breakdown of the assumed continuous transformation of bosons into fermions by variation of the parameter $\ensuremath{\nu}$.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.