ANOMALOUS BEHAVIOR OF IDEAL FERMI GAS BELOW 2D: THE "IDEAL QUANTUM DOT" AND THE PAUL EXCLUSION PRINCIPLE

2009 ◽  
Vol 23 (20n21) ◽  
pp. 4121-4128 ◽  
Author(s):  
M. GRETHER ◽  
M. DE LLANO ◽  
M. HOWARD LEE
Keyword(s):  
Quantum Dot ◽  
Chemical Potential ◽  
Pauli Exclusion Principle ◽  
Fermi Gas ◽  
Anomalous Behavior ◽  
Absolute Temperature ◽  
Exclusion Principle ◽  
Ideal Quantum ◽  
Spatial Dimensionality ◽  
The Ideal

A physical interpretation is given to a curious "hump" that develops in the chemical potential as a function of absolute temperature in an ideal Fermi gas for any spatial dimensionality d < 2, integer or not, in contrast with the more familiar monotonic decrease for all d ≥ 2. The hump height increases without limit as d decreases to zero. The divergence at d = 0 is shown to be a clear manifestation of the Pauli Exclusion Principle whereby two spinless fermions cannot sit on top of each other in configuration space. The hump itself is thus an obvious precursor of this manifestation, otherwise well understood in momentum space. It also constitutes an "ideal quantum dot" when d = 0.

scholarly journals ON THE DERIVATION OF THERMODYNAMIC QUANTITIES OF IDEAL FERMI GAS IN HARMONIC TRAP

2017 ◽  
Vol 126 (1B) ◽  
Author(s):  
PHẠM NGUYỄN THÀNH VINH
Keyword(s):  
Heat Capacity ◽  
Total Energy ◽  
Chemical Potential ◽  
Three Dimensional ◽  
Fermi Gas ◽  
Integral Function ◽  
Harmonic Trap ◽  
Thermodynamic Quantities ◽  
Comprehensive Study ◽  
The Ideal

In this paper, we provide comprehensive study of the thermodynamic quantities of the ideal Fermi gas confined in a three-dimensional harmonic trap by using the properties of Fermi – Dirac integral function both analytically and numerically. The dependences of the chemical potential, total energy and heat capacity on the temperature are obtained via the appropriately approximated analytic formulae. Afterwards, the results are compared with the exact numerical ones in order to evaluate the applicability of these formulae.

scholarly journals Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field

2020 ◽  
Author(s):  
Hajo Leschke ◽  
Alexander V. Sobolev ◽  
Wolfgang Spitzer
Keyword(s):  
Magnetic Field ◽  
Ground State ◽  
Constant Magnetic Field ◽  
Chemical Potential ◽  
Fermi Gas ◽  
Von Neumann Entropy ◽  
Zero Field ◽  
Quantum Gas ◽  
Asymptotic Growth ◽  
The Ideal

AbstractWe consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane $${{\mathbb {R}}}^2$$ R 2 perpendicular to an external constant magnetic field of strength $$B>0$$ B > 0 . We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential $$\mu \ge B$$ μ ≥ B (in suitable physical units). For this (pure) state we define its local entropy $$S(\Lambda )$$ S ( Λ ) associated with a bounded (sub)region $$\Lambda \subset {{\mathbb {R}}}^2$$ Λ ⊂ R 2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $$\Lambda $$ Λ of finite area $$|\Lambda |$$ | Λ | . In this setting we prove that the leading asymptotic growth of $$S(L\Lambda )$$ S ( L Λ ) , as the dimensionless scaling parameter $$L>0$$ L > 0 tends to infinity, has the form $$L\sqrt{B}|\partial \Lambda |$$ L B | ∂ Λ | up to a precisely given (positive multiplicative) coefficient which is independent of $$\Lambda $$ Λ and dependent on B and $$\mu $$ μ only through the integer part of $$(\mu /B-1)/2$$ ( μ / B - 1 ) / 2 . Here we have assumed the boundary curve $$\partial \Lambda $$ ∂ Λ of $$\Lambda $$ Λ to be sufficiently smooth which, in particular, ensures that its arc length $$|\partial \Lambda |$$ | ∂ Λ | is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case $$B=0$$ B = 0 , where an additional logarithmic factor $$\ln (L)$$ ln ( L ) is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space $$\text{ L}^2({{\mathbb {R}}}^2)$$ L 2 ( R 2 ) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies. As opposed to the case $$B=0$$ B = 0 , the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.

scholarly journals ON THE DERIVATION OF THERMODYNAMIC QUANTITIES OF IDEAL FERMI GAS IN HARMONIC TRAP

2017 ◽  
Vol 126 (1B) ◽  
pp. 117 ◽  
Author(s):  
Pham Nguyen Thanh Vinh
Keyword(s):  
Heat Capacity ◽  
Total Energy ◽  
Chemical Potential ◽  
Three Dimensional ◽  
Fermi Gas ◽  
Integral Function ◽  
Harmonic Trap ◽  
Thermodynamic Quantities ◽  
Comprehensive Study ◽  
The Ideal

In this paper, we provide comprehensive study of the thermodynamic quantities of the ideal Fermi gas confined in a three-dimensional harmonic trap by using the properties of Fermi – Dirac integral function both analytically and numerically. The dependences of the chemical potential, total energy and heat capacity on the temperature are obtained via the appropriately approximated analytic formulae. Afterwards, the results are compared with the exact numerical ones in order to evaluate the applicability of these formulae.

Hardness and HOMO-LUMO Gap Probed by the Helium Atom Pushing the Molecular Surface of the First-Row Hydride Molecules

2003 ◽  
Vol 68 (12) ◽  
pp. 2344-2354 ◽  
Author(s):  
Edyta Małolepsza ◽  
Lucjan Piela
Keyword(s):  
Helium Atom ◽  
Pauli Exclusion Principle ◽  
Structure Change ◽  
Molecular Surface ◽  
Exclusion Principle ◽  
Probe Position ◽  
Repulsion Energy ◽  
Ionic Dissociation ◽  
Probe Site ◽  
Homo Lumo

A molecular surface defined as an isosurface of the valence repulsion energy may be hard or soft with respect to probe penetration. As a probe, the helium atom has been chosen. In addition, the Pauli exclusion principle makes the electronic structure change when the probe pushes the molecule (at a fixed positions of its nuclei). This results in a HOMO-LUMO gap dependence on the probe site on the isosurface. A smaller gap at a given probe position reflects a larger reactivity of the site with respect to the ionic dissociation.

Identical Particles and Second Quantization: Occupation Number Representation

2018 ◽  
Author(s):  
Norman J. Morgenstern Horing
Keyword(s):  
Occupation Number ◽  
Annihilation Operator ◽  
Pauli Exclusion Principle ◽  
Exclusion Principle ◽  
Number Representation ◽  
Second Quantization ◽  
Identical Particles ◽  
Fundamental Properties ◽  
Minimum Uncertainty ◽  
Creation Operators

Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.

scholarly journals Fermi gas approach to general rank theories and quantum curves

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Naotaka Kubo
Keyword(s):  
Matrix Models ◽  
Critical Role ◽  
Three Dimensional ◽  
Fermi Gas ◽  
Fermi Gases ◽  
Partition Functions ◽  
Density Matrices ◽  
General Rank ◽  
Chern Simons ◽  
The Ideal

Abstract It is known that matrix models computing the partition functions of three-dimensional $$ \mathcal{N} $$ N = 4 superconformal Chern-Simons theories described by circular quiver diagrams can be written as the partition functions of ideal Fermi gases when all the nodes have equal ranks. We extend this approach to rank deformed theories. The resulting matrix models factorize into factors depending only on the relative ranks in addition to the Fermi gas factors. We find that this factorization plays a critical role in showing the equality of the partition functions of dual theories related by the Hanany-Witten transition. Furthermore, we show that the inverses of the density matrices of the ideal Fermi gases can be simplified and regarded as quantum curves as in the case without rank deformations. We also comment on four nodes theories using our results.

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