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.
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 .
Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density,
ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is
2πℏ2aρ/m where
a is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange
N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.
We establish a new iterative method for computing rigorous lower bounds for the ground-state energies of all atoms and ions. The accuracy of the resulting bounds depends on two ingredients: the quality of the initial energy estimate and the quality of the used upper bound for the maximum number of bound electrons,
Nc(Z), as a function of the nuclear charge
Z>0. In particular, this new method allows us to establish the explicit lower bound
E(N,Z)≥−0.8710N1/3Z2 for the ground-state energy
E(N,Z) of the atom/ion with
N∈N electrons and nuclear charge
Z∈(0,N], thereby considerably improving the best known constant
A=(3/2)1/3≈1.1447 in the estimate
E(N,Z)≥−AN1/3Z2 for
0<Z≤N and coming much closer to the best possible constant given by the Thomas-Fermi asymptotics
limN→∞∣E(N,N)∣N−7/3=−ETF(1)≈0.7687 derived in Adv. Math., 23(1) by Lieb and Simon. To demonstrate the potential and the limits of our method, we also compute explicit lower bounds for the ground-state energies
E(N,N) of all atoms with
2≤N≤86 and we compare them with the bounds obtained in J. Chem. Phys., 62(8) by Hertel, Lieb, Thirring and the bounds obtained by Seco in his dissertation. Finally, we also provide a uniqueness result for the expectation values of the kinetic, potential and interaction energies on the ground-state space.