广义不确定性原理下广义外势中n维理想费米气体的热力学性质
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摘要
采用Thomas-Fermi的半经典近似,研究了广义不确定性原理下广义外势中n维理想费米气体的热力学性质.解析计算出平均粒子数、内能和热容等热力学量,给出了低温条件下上述热力学量及化学势、费米能和基态能的解析表达式以及考虑广义不确定性原理的修正项;在低温条件下,数值分析了外势与广义不确定性原理对铜电子气体及电子密度更高的电子系统热力学性质的影响,发现:1)考虑广义不确定性原理时,外势对电子系统的影响很大,使广义不确定性原理的修正项增加了6~11个数量级.2)粒子数密度越大、粒子质量越小,广义不确定性原理的影响越大.3)广义不确定性原理导致内能随温度的增加先增大,当温度升到某一数值时(对三维谐振势中的铜电子气体,T/T_(F0)~0.22)时,增值为0,温度再增加内能减少;热容随温度的增加减少;化学势、费米能和基态能随温度的升高而增大.
With the semi-classical(Thomas-Fermi)approximation,the thermodynamic properties of a Fermi gas in generalized external potential are studied under the generalized uncertainty principle(GUP).The total particle number,internal energy and heat capacity of the Fermi system are calculated.Then,analytical expressions of the total particle number,internal energy,heat capacity,chemical potential,Fermi energy,ground state energy and amendments resulted by GUP are obtained at low temperatures.The influences of both the generalized uncertainty principle and external potential on the thermodynamic properties of copper electron gas and other electronic systems with higher electron density are studied numerically at low temperature.We find:1)External potential has a significant impact on electronic system when the generalized uncertainty principle be considered.The amendments of the generalized uncertainty principle have increased by 6-11 orders of magnitude.2)When the number density of particle is bigger and the quality of particle is smaller,the influence of generalized uncertainty principle is bigger.3)When the generalized uncertainty principle is taken into account,the chemical potential,Fermi energy and the ground state energy increase with the increase of temperature,while the heat capacity decreases.When the temperature was low than 0.22 times T_(F0),the internal energy increased with the increase of temperature,but which became to reduce when temperature was high than 0.22 times T_(F0) for copper electron gas.
With the semi-classical(Thomas-Fermi)approximation,the thermodynamic properties of a Fermi gas in generalized external potential are studied under the generalized uncertainty principle(GUP).The total particle number,internal energy and heat capacity of the Fermi system are calculated.Then,analytical expressions of the total particle number,internal energy,heat capacity,chemical potential,Fermi energy,ground state energy and amendments resulted by GUP are obtained at low temperatures.The influences of both the generalized uncertainty principle and external potential on the thermodynamic properties of copper electron gas and other electronic systems with higher electron density are studied numerically at low temperature.We find:1)External potential has a significant impact on electronic system when the generalized uncertainty principle be considered.The amendments of the generalized uncertainty principle have increased by 6-11 orders of magnitude.2)When the number density of particle is bigger and the quality of particle is smaller,the influence of generalized uncertainty principle is bigger.3)When the generalized uncertainty principle is taken into account,the chemical potential,Fermi energy and the ground state energy increase with the increase of temperature,while the heat capacity decreases.When the temperature was low than 0.22 times T_(F0),the internal energy increased with the increase of temperature,but which became to reduce when temperature was high than 0.22 times T_(F0) for copper electron gas.
引文
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[3]Garay L J.Quantum gravity and minimum length[J].Int J Mod Phys A,1995,10(02):145-165.
[4]Scardigli F.Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment[J].Phys Lett B,1999,452(1):39-44.
[5]Chang L N,Minic D,Okamura N,et al.Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations[J].Phys Rev D,2001,65(12):397-398.
[6]Chang L N,Minic D,Takeuchi T,et al.The effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem[J].Phys Rev D,2002,65(12):397-398.
[7]Li Xiang.Black hole entropy without brick walls[J].Phys Rev D,1995,52(4):9-13.
[8]Pedram P,Amirfakhrian M,Shababi H.On the(2+1)-dimensional Dirac equation in a constant magnetic field with a minimal length uncertainty[J].Int J Mod Phys D,2015,24(02):1550016.
[9]Fityo V T.Statistical physics in deformed spaces with minimal length[J].Phys Lett A,2007,372(37):5872-5877.
[10]Panella O.Casimir effect in minimal length theories based on a generalized uncertainty principle[J].Phys Rev D,2012,85(4):125-127.
[11]Brau F,Buisseret F.Minimal length uncertainty relation and gravitational quantum well[J].Phys Rev D,2006,74(3):307-307.
[12]Pedram P.New approach to nonperturbative quantum mechanics with minimal length uncertainty[J].Phys Rev D,2011,85(2):926-927.
[13]Vakili B,Gorji M A.Thermostatistics with minimal length uncertainty relation[J].J Sta Mech,2012,2012(10):P10013.
[14]李鹤龄,王娟娟,杨斌,等.广义不确定性原理下费米气体低温热力学性质[J].物理学报,2015,64(8):80502-080502.
[15]Regal C A,Greiner M,Jin D S.Observation of resonance condensation of fermionic atom pairs[J].Phys Rev Lett,2004,92(4):040403.
[16]Jochim S,Bartenstein M,Altmeyer A,et al.Bose-Einstein condensation of molecules[J].Science,2003,302(5653):2101-2103.
[17]Li M Z,Yan Z J,Chen J C,et al.Thermodynamic properties of an ideal Fermi gas in an external potential with U=brt in any dimensional space[J].Phys Rev A,1998,58(58):1445-1447.
[18]苏国珍,陈丽璇.弱相互作用费米气体的热力学性质[J].物理学报,2005,53(4):984-990.
[19]门福殿.弱磁场中弱相互作用费米气体的热力学性质[J].物理学报,2006,55(4):1622-1627.
[20]Chou T T,Yang C N,Yu L H.Bose-Einstein condensation of atoms in a trap[J].Phys Rev A,1996,53(6):4257-4259.
[21]Huang K.Statistical mechanics[M].New York:Wiley,1963:272-276
[22]Pathria R K.Statistical Mechanics[M].London:Pergamon Press,1977.
[23]赵仁,张丽春,李怀繁.广义测不准关系与三维BTZ黑洞熵[J].物理学报,2009,58(04):2193-2196.
[24]Quesne C,Tkachuk V M.More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum[J].J Phys A:Math General,2004,37(371):10095-10113.
[25]Dehmelt H.A single atomic particle forever floating at rest in free space:new value for electron radius[J].Phys Scr,1988,22(T22):102-110.