Chapter 14. Ideal Bose gas Equation of state


 Stephanie Short
 1 years ago
 Views:
Transcription
1 Chapter 14 Ideal Bose gas In this chapter, we shall study the thermodynamic properties of a gas of noninteracting bosons. We will show that the symmetrization of the wavefunction due to the indistinguishability of particles has important consequences on the behavior of the system. The most important consequence of the quantum mechanical symmetrization is the Bose Einstein condensation, which is in this sense a special phase transition as it occurs in a system of noninteracting particles. We shall consider, as an example, a gas of photons and a gas of phonons Equation of state We consider a gas of noninteracting bosons in a volume at temperature T and chemical potential µ. The system is allowed to interchange particles and energy with the surroundings. The appropriate ensemble to treat this manybody system is the grand canonical ensemble. Our bosons are nonrelativistic particles with spin s, whose oneparticle energies ε( k) include only the kinetic energy term and are given by ε( k) = ε(k) = h2 k 2 2m. Since the lowest oneparticle energy ε 0 is zero, ε 0 = ε(0) = 0, the chemical potential must fulfill the condition: as we discussed in chapter 12. < µ < 0, We consider a situation when the gas is in a box with volume = L x L y L z and subject to periodic boundary conditions. In the thermodynamical limit (N,, with 177
2 178 CHAPTER 14. IDEAL BOSE GAS n = N = const), the sums over the wavevector k can be replaced by integrals as in the case of the Fermi gas. However, here we have to be careful when µ happens to approach the value 0. In order to see what kind of trouble we then might get into, let us calculate the ground state occupation. The expectation value of the occupation number is given by < ˆn r >= f + (ε r ) = 1 e β(εr µ) 1. For βµ << 1, the occupation of the lowest energy state ε(0) = 0 can be approximated as 1 < ˆn 0ms >= e βµ 1 = 1 1 βµ βµ, which means that < ˆn 0ms > can assume any macroscopic value. On the other hand, the density of states Ω(E), which has the same expression as for fermionic systems (eq. 13.8), Ω(E) E 0 as E 0. This is where we are going to encounter a problem: if we replace r by de Ω(E), we will get that the ground state has zero weight even though, as we have just shown, it can be macroscopically occupied. In the case of a fermionic system, we would not have this problem due to the Pauli principle: < ˆn r >: 0 < ˆn r > 1. In order to avoid the problem in the bosonic system, we treat the ground state separately. The grand canonical potential Ω(T,,z) (do not confuse it with the DOS Ω(E)) can be written then as βω(t,,z) = (2s+1) (2π) 34π + (2s+1)ln(1 z) }{{} term corresponding to the ground state ε(0) = 0 With the substitution β x = hk 2m and the definition of the thermal de Broglie wavelength 2πβ h 2 λ = m, expression (14.1) becomes βω(t,,z) = 2s+1 4 λ 3 π We use the Taylor series expansion of the ln, 0 ln(1 y) = 0 dk k 2 ln ( 1 ze βε(k)). (14.1) ( dxx 2 ln 1 ze x2) +(2s+1)ln(1 z). (14.2) n=1 y n n ( y < 1),
3 14.1. EQUATION OF STATE 179 for the integral: We define the functions and 0 ( dxx 2 ln 1 ze x2) π = 4 g 5/2 (z) = 4 π g 3/2 (z) = z d dz g 5/2(z) = 0 n=1 z n n 5/2. ( dxx 2 ln 1 ze x2) = n=1 z n n 3/2. n=1 z n n 5/2 With these definitions, the grand canonical potential (eq. (14.2) takes the form βω(t,,z) = 2s+1 λ 3 g 5/2 (z)+(2s+1)ln(1 z). (14.3) Note that except for the extra term on the righthand side, it has the same form as the expression for the Fermi gas. From Ω = P, we get βp = 2s+1 g λ 3 5/2 (z) 2s+1 ln(1 z). (14.4) We express the fugacity z in terms of the density of particles n: n = < ˆN > ( ) = z z βp T, = 2s+1 g λ 3 3/2 (z)+ 2s+1 z 1 z (14.5) and then, by combining eq. (14.4) and (14.5), we obtain the thermal equation of state. The term n 0 = 2s+1 z 1 z in eq. (14.5) describes the contribution of the ground state to the particle density n as can be seen from with 1 < ˆn 0m s >= 1 1 z 1 1 = 1 m s = s, s+1,...,+s. z 1 z n 0 2s+1, n 0 can be macroscopically large, which corresponds to the BoseEinstein condensation.
4 180 CHAPTER 14. IDEAL BOSE GAS The caloric equation of state can be obtained by calculating the internal energy U: ( ) ( ) U = β lnz(t,z,) = β βω(t,,z) By considering eq. (14.3), one obtains z, U = 3 2 k BT 2s+1 λ 3 g 5/2 (z). (14.6) This expression is the same as the one for the Fermi gas when f 5/2 (z) is substituted by g 5/2 (z). Combining eq. (14.6) and eq. (14.5), one can derive the caloric equation of state. Now, we combine (14.4) and eq. (14.6) to get z, U = 3 2 P k BT(2s+1)ln(1 z). Note that the expression for the Bose gas has an extra term 3 2 k BT(2s+1)ln(1 z), which was absent in the case of the Fermi gas as well as in the case of the ideal gas of particles Classical limit We first investigate the classical limit (nondegenerate Bose gas) corresponding to low particle densities and high temperatures. In this limit, z = e βµ << 1 so that 1 < ˆn r >= z 1 e βεr 1 << 1 ze βεr and the BoseEinstein distribution function reduces to the MaxwellBoltzmann one. All energy levels are only weakly occupied. The probability of double occupancy is practically zero. It is therefore to be expected that in this limit the differences between Bose, Fermiand Boltzmann statistics are negligible. As z << 1, it is sufficient to retain only the first two terms of the series for g 5/2 (z) and g 3/2 (z): g 5/2 (z) z + z2 2 5/2, g 3/2 (z) z + z2 2 3/2.
5 14.2. CLASSICAL LIMIT 181 With that, the particle density takes the following form: n 2s+1 ( z 1+ z ) + 2s+1 λ 3 2 3/2 2s+1 λ 3 z ( 1+ z 2 3/2 ) In the zeroth approximation, we thus have that and therefore z 1 z + 2s+1 z(1 z). ( ) nλ 3 (2s+1)z (0) 1+ λ3 z (0) nλ 3 (2s+1) ( 1+ λ3 As in the case of the Fermi gas, here the classical limit corresponds to nλ 3 << 1, i.e., small particle densities and small thermal de Broglie wavelengths. The condition λ 3 << 1 allows us to neglect the correcting term coming from the explicit consideration of the ground state contribution so that we can write z (0) nλ3 2s+1, which is the same result as we had for fermions. The equation of state reduces then to the equation of state for classical particles: ). βp = 2s+1 nλ 3 λ 3 2s+1, P = < ˆN > k B T. In order to obtain the next order corrections, we make use of the equation n 2s+1 ( z 1+ z ) λ 3 2 3/2 where we neglect the term corresponding to the ground state contribution, which can be rewritten in terms of z (0) as ) nλ 3 2s+1 z(0) z (1+ (1) z(1) 2 3/2 and then as z (1) z (0) ( 1+ z(1) 2 3/2 ). (14.7)
6 182 CHAPTER 14. IDEAL BOSE GAS In the righthand side of eq. (14.7), we approximate z (1) by z (0) : ) z (1) z(0) z (1 (0) z(0). 1+ z(0) 2 3/2 2 3/2 We substitute this expression in the pressure equation for the ideal Bose gas, βp 2s+1 λ 3 g 5/2 (z) 2s+1 λ 3 z (1), where the ground state contribution has been neglected. This gives us P =< ˆN > k B T [ 1 ] nλ (2s+1) The last term in this expression are the quantum corrections. Note that if we consider now, for comparison, the equation of state for a real gas (the van der Waals equation, for instance), we would observe that similar additive corrections appear with respect to the ideal case due to the interaction between particles. Here, however, the additive terms originate from the indistinguishability principle and not from the interaction among particles. The correcting term for the ideal Fermi gas was positive and therefore contributed as a repulsion among particles. For the Bose gas, the additive term is negative and therefore contributes as an attraction among particles. Quantitatively, the quantum corrections are much smaller than terms coming from the interaction among particles BoseEinstein condensation We consider now the limit of high particle densities and low temperatures(quantum limit), where one finds important qualitative differences between bosons, fermions and classical particles. For bosons, we have the restriction < µ 0, which determines the range of value that z assumes: 0 < z < 1. The functions g α (z) = n=1 z n n α
7 14.3. BOSEEINSTEIN CONDENSATION 183 are related to one another through the recursive formula: g α 1 (z) = z d dz g α(z). In the interval 0 z 1, they are positive and monotonically growing functions of z (see Fig. 14.1). Figure 14.1: g α family. For z = 1, g α coincide with the Riemann function ζ, defined as 1 ζ(n) = p = 1 dy yn 1 n (1 2 1 n )Γ(n) e y +1. Thus, p=1 ( ) 5 g 5/2 (1) = ζ = 1.342, 2 ( ) 3 g 3/2 (1) = ζ = and g 1/2 (z) diverges at z = 1. From the expression for the particle density for bosons n, n = 2s+1 g λ 3 3/2 (z)+ 2s+1 z z 1, we have that the particle density of the (2s+1)degenerate ground state ε(k = 0) = 0, defined as n 0 = 2s+1 z z 1, can be written in the following form: n 0 = n 2s+1 λ 3 g 3/2 (z). Since, as can be observed in Fig. 14.1, g 3/2 (z) for 0 < z 1 varies in the interval [0, 2.612], there should be temperatures T and densities n where n > 2s+1 λ 3 g 3/2 (1). 0
8 184 CHAPTER 14. IDEAL BOSE GAS In this case, n 0 > 0, that is, a finite fraction of bosons occupies the ground state. When βµ 0, n 0 becomes macroscopically large. This phenomenon is called BoseEinstein condensation. This macroscopic state would not be that spectacular if it occured only for temperatures k B T < ε 1 ε(k = 0) = ε 1, (14.8) where ε 1 is the first excited state. For macroscopic systems, this condition is fulfilled for temperatures T < K so that at T = 0 all particles are in the ground state. What is so spectacular about the BoseEinstein condensation is that it sets in at much higher temperatures than those imposed by eq. (14.8). In fact, the phase transition to the condensed state occurs when nλ 3 = (2s+1)g 3/2 (1), whichdefinesthecritical valueforthedebrogliewavelengthλ c foragivenparticledensity n: λ 3 c = 2s+1 ( 2π h 2 ) 3/2 n g 3/2(1) = mk B T c and hence the critical temperature T c at which the phase transition occurs: k B T c = 2π h2 m ( ) 2/3 n. (2s+1)g 3/2 (1) For a given temperature T, we can find the critical density n c : n c (T) = 2s+1 ( ) 3/2 mkb T g λ 3 3/2 (1) = (2s+1) 2π h 2 g 3/2(1). Since the particle density is proportional to the inverse the average interparticle distance r to the third power, n 1 r 3, we can argue that the BoseEinstein condensation occurs when the thermal de Broglie wavelength λ approaches r : λ r. Comparison of the critical temperature T c of the ideal Bose gas with the Fermi temperature T F of the ideal Fermi gas Consider s = 1/2 fermions with mass m f and s = 0 bosons with mass m b, both system with equal densities n f = n b = n. The T F /T c is then given as T F T c = 1 4π (3π2 g 3/2 (1)) 2/3m b m f 1.45 m b m f.
9 14.3. BOSEEINSTEIN CONDENSATION 185 For m b m f 1, both temperatures are of the same order of magnitude. Let us consider now a gas of conduction electrons and a gas of 4 He atoms as representative fermionic and bosonic systems. In this case, m b m f , T f 10 4 K and therefore T c 3.13 K. This value of T c is much bigger that what one would expect for a normal occupation of the ground state, T K. In this example, we demonstrate that the BoseEinstein condensation is a phase transition. Fugacity In order to understand the phenomenon, let us represent n, n = 2s+1 g λ 3 3/2 (z)+ 2s+1 z 1 z, graphically. For that, we plot the function y 1 (z) and the constant y 1 (z) = g 3/2 (z)+ λ3 z (1 z) y 2 = nλ3 2s+1 on the same graph. The intersection between these two curves provides z as a function of T and n. With this graphical representation, we can obtain z(t,n) at a given.
10 186 CHAPTER 14. IDEAL BOSE GAS The solution z b corresponds to the condensate region since there ( ) nλ 3 > g 3/2 (1). 2s+1 In the thermodynamic limit (N,, n = const) the term coming from the ground state contribution λ 3 z (1 z) has to remain finite. This implies that (1 z) has to remain finite, i.e., 1 z has to behave as 1. Therefore, for, we can approximate the fugacity as nλ 3 solution of z = 2s+1 = g nλ 3 3/2(z) for 2s+1 < g 3/2(1), nλ 3 1 for 2s+1 g 3/2(1). Then, since n 0 n 0 = n 2s+1 λ 3 g 3/2 (z), n 0 if nλ 3 2s+1 < g 3/2(1), ( ) 3/2 n 0 2s+1 1 n nλ g 3/2(1) = 1 λ3 c T 3 λ = 1 nλ 3 if 3 2s+1 g 3/2(1). For 0 < T < T c, we have a mixture of two phases (see Fig. 14.2): the condensate phase formed by the macroscopic fraction of N 0 bosons out of N occupying the lowest energy level: [ ( ) ] 3/2 T N 0 = N 1 and the gas phase formed by the rest of the particles N 1 which are in excited ( k 0) states: ( ) 3/2 T N 1 = N N 0 = N. T c T c T c
11 14.4. THERMODYNAMICS FOR THE IDEAL BOSE GAS 187 Figure 14.2: Occupation density of the ground state of the ideal Bose gas as a function of temperature. At T = 0, N 0 = N, N 1 = 0. The BoseEinstein condensation takes place at T = T c Thermodynamics for the ideal Bose gas With the BoseEinstein condensation and the concept of a mixed state of gas and condensate, we have a strong analogy with the gasliquid phase transition that we discussed when studying properties of a real gas. As in that case, the abrupt change of n 0 at T c leads to discontinuities in the thermodynamic potentials of the ideal Bose gas, which have different analytical expressions above and below the critical point (T c, n c ). Thermal equation of state We consider the thermodymanic limit (N,, n = const). In the equation for pressure for the Bose gas given as the second term approaches zero: βp = 2s+1 g λ 3 5/2 (z) 2s+1 ln(1 z), 2s+1 ln(1 z) 0 for z < 1, i.e., n < n c. For the condensation region (n n c ), where z 1, the above limit is not trivial. Here we make use of the fact that as discussed previously, and also get 2s+1 (1 z) 1, ln(1 z) 0.
12 188 CHAPTER 14. IDEAL BOSE GAS Summarizing, we get the following equation for pressure: 2s+1 g λ 3 5/2 (z) for n < n c, βp = 2s+1 g λ 3 5/2 (1) for n > n c. In the condensation region, the pressure is independent of the volume and particle number; it only depends on temperature. This was also the case in the gasliquid phase transition. Considering this analogy, the BoseEinstein condensation can be described as a firstorder phase transition. The BoseEinstein condensation was predicted by Satyendra Bose and Albert Einstein in It took almost 70 years to have an experimental corroboration of this phenomenon with the ultracold gas systems. Previous experiments had been done with 4 He as well as with hydrogen.
Quantum Grand Canonical Ensemble
Chapter 16 Quantum Grand Canonical Ensemble How do we proceed quantum mechanically? For fermions the wavefunction is antisymmetric. An N particle basis function can be constructed in terms of singleparticle
More informationPhysics 112 The Classical Ideal Gas
Physics 112 The Classical Ideal Gas Peter Young (Dated: February 6, 2012) We will obtain the equation of state and other properties, such as energy and entropy, of the classical ideal gas. We will start
More informationBoseEinstein Condensate: A New state of matter
BoseEinstein Condensate: A New state of matter KISHORE T. KAPALE June 24, 2003 BOSEEINSTEIN CONDENSATE: A NEW STATE OF MATTER 1 Outline Introductory Concepts Bosons and Fermions Classical and Quantum
More informationStatistical Mechanics Notes. Ryan D. Reece
Statistical Mechanics Notes Ryan D. Reece August 11, 2006 Contents 1 Thermodynamics 3 1.1 State Variables.......................... 3 1.2 Inexact Differentials....................... 5 1.3 Work and Heat..........................
More informationPhonon II Thermal Properties
Phonon II Thermal Properties Physics, UCF OUTLINES Phonon heat capacity Planck distribution Normal mode enumeration Density of states in one dimension Density of states in three dimension Debye Model for
More informationCHAPTER 4. Cluster expansions
CHAPTER 4 Cluster expansions The method of cluster expansions allows to write the grandcanonical thermodynamic potential as a convergent perturbation series, where the small parameter is related to the
More informationThermal and Statistical Physics Department Exam Last updated November 4, L π
Thermal and Statistical Physics Department Exam Last updated November 4, 013 1. a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F =
More information84 Quantum Theory of ManyParticle Systems ics [Landau and Lifshitz (198)] then yields the thermodynamic potential so that one can rewrite the statist
Chapter 6 Bosons and fermions at finite temperature After reviewing some statistical mechanics in Sec. 6.1, the occupation number representation is employed to derive some standard results for noninteracting
More informationThermal & Statistical Physics Study Questions for the Spring 2018 Department Exam December 6, 2017
Thermal & Statistical Physics Study Questions for the Spring 018 Department Exam December 6, 017 1. a. Define the chemical potential. Show that two systems are in diffusive equilibrium if 1. You may start
More informationPhysics 3700 Introduction to Quantum Statistical Thermodynamics Relevant sections in text: Quantum Statistics: Bosons and Fermions
Physics 3700 Introduction to Quantum Statistical Thermodynamics Relevant sections in text: 7.1 7.4 Quantum Statistics: Bosons and Fermions We now consider the important physical situation in which a physical
More information+ 1. which gives the expected number of Fermions in energy state ɛ. The expected number of Fermions in energy range ɛ to ɛ + dɛ is then dn = n s g s
Chapter 8 Fermi Systems 8.1 The Perfect Fermi Gas In this chapter, we study a gas of noninteracting, elementary Fermi particles. Since the particles are noninteracting, the potential energy is zero,
More informationPhys Midterm. March 17
Phys 7230 Midterm March 17 Consider a spin 1/2 particle fixed in space in the presence of magnetic field H he energy E of such a system can take one of the two values given by E s = µhs, where µ is the
More informationGrandcanonical ensembles
Grandcanonical ensembles As we know, we are at the point where we can deal with almost any classical problem (see below), but for quantum systems we still cannot deal with problems where the translational
More informationWorkshop on Bose Einstein Condensation IMS  NUS Singapore
Workshop on Bose Einstein Condensation IMS  NUS Singapore 1/49 Boselike condensation in halfbose halffermi statistics and in Fuzzy BoseFermi Statistics Mirza Satriawan Research Group on Theoretical
More information1) K. Huang, Introduction to Statistical Physics, CRC Press, 2001.
Chapter 1 Introduction 1.1 Literature 1) K. Huang, Introduction to Statistical Physics, CRC Press, 2001. 2) E. M. Lifschitz and L. P. Pitajewski, Statistical Physics, London, Landau Lifschitz Band 5. 3)
More information8.333: Statistical Mechanics I Re: 2007 Final Exam. Review Problems
8.333: Statistical Mechanics I Re: 007 Final Exam Review Problems The enclosed exams (and solutions) from the previous years are intended to help you review the material. ******** Note that the first parts
More informationLowdimensional Bose gases Part 1: BEC and interactions
Lowdimensional Bose gases Part 1: BEC and interactions Hélène Perrin Laboratoire de physique des lasers, CNRSUniversité Paris Nord Photonic, Atomic and Solid State Quantum Systems Vienna, 2009 Introduction
More informationElements of Statistical Mechanics
Dr. Y. Aparna, Associate Prof., Dept. of Physics, JNTU College of ngineering, JNTU  H, lements of Statistical Mechanics Question: Discuss about principles of MaxwellBoltzmann Statistics? Answer: Maxwell
More informationUNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II
UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II January 20, 2017 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.
More informationInsigh,ul Example. E i = n i E, n i =0, 1, 2,..., 8. N(n 0,n 1,n 2,..., n 8 )= n 1!n 2!...n 8!
STATISTICS Often the number of particles in a system is prohibitively large to solve detailed state (from Schrodinger equation) or predict exact motion (using Newton s laws). Bulk properties (pressure,
More informationQuantum Field Theory
Quantum Field Theory PHYSP 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S1 A proper description of particle physics
More informationSet 3: Thermal Physics
Set 3: Thermal Physics Equilibrium Thermal physics describes the equilibrium distribution of particles for a medium at temperature T Expect that the typical energy of a particle by equipartition is E kt,
More informationSeptember 6, 3 7:9 WSPC/Book Trim Size for 9in x 6in book96 7 Quantum Theory of ManyParticle Systems Eigenstates of Eq. (5.) are momentum eigentates.
September 6, 3 7:9 WSPC/Book Trim Size for 9in x 6in book96 Chapter 5 Noninteracting Fermi gas The consequences of the Pauli principle for an assembly of fermions that is localized in space has been discussed
More informationHigh order corrections to density and temperature of fermions and bosons from quantum fluctuations and the CoMDα Model
High order corrections to density and temperature of fermions and bosons from quantum fluctuations and the CoMDα Model Hua Zheng, 1,2 Gianluca Giuliani, 1 Matteo Barbarino, 1 and Aldo Bonasera 1,3 1 Cyclotron
More information2m + U( q i), (IV.26) i=1
I.D The Ideal Gas As discussed in chapter II, microstates of a gas of N particles correspond to points { p i, q i }, in the 6Ndimensional phase space. Ignoring the potential energy of interactions, the
More informationMicrocanonical ensemble model of particles obeying BoseEinstein and FermiDirac statistics
Indian Journal o Pure & Applied Physics Vol. 4, October 004, pp. 749757 Microcanonical ensemble model o particles obeying BoseEinstein and FermiDirac statistics Y K Ayodo, K M Khanna & T W Sakwa Department
More informationIV. Classical Statistical Mechanics
IV. Classical Statistical Mechanics IV.A General Definitions Statistical Mechanics is a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom. As discussed
More informationThe free electron gas: Sommerfeld model
The free electron gas: Sommerfeld model Valence electrons: delocalized gas of free electrons (selectrons of alkalies) which is not perturbed by the ion cores (ionic radius of Na + equals 0.97 Å, internuclear
More informationUNIVERSITY COLLEGE LONDON. University of London EXAMINATION FOR INTERNAL STUDENTS. For The Following Qualifications:
UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications: B. Sc. M. Sci. Physics 2B28: Statistical Thermodynamics and Condensed Matter Physics
More informationINDIAN INSTITUTE OF TECHNOLOGY GUWAHATI Department of Physics MID SEMESTER EXAMINATION Statistical Mechanics: PH704 Solution
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI Department of Physics MID SEMESTER EXAMINATION Statistical Mechanics: PH74 Solution. There are two possible point defects in the crystal structure, Schottky and
More informationSection 4 Statistical Thermodynamics of delocalised particles
Section 4 Statistical Thermodynamics of delocalised particles 4.1 Classical Ideal Gas 4.1.1 Indistinguishability Since the partition function is proportional to probabilities it follows that for composite
More informationDensity and temperature of fermions and bosons from quantum fluctuations
Density and temperature of fermions and bosons from quantum fluctuations Hua Zheng and Aldo Bonasera 1 1 Laboratori Nazionali del Sud, INFN, via Santa Sofia, 6, 951 Catania, Italy In recent years, the
More informationBasic Concepts and Tools in Statistical Physics
Chapter 1 Basic Concepts and Tools in Statistical Physics 1.1 Introduction Statistical mechanics provides general methods to study properties of systems composed of a large number of particles. It establishes
More informationThe GrossPitaevskii Equation A NonLinear Schrödinger Equation
The GrossPitaevskii Equation A NonLinear Schrödinger Equation Alan Aversa December 29, 2011 Abstract Summary: The GrossPitaevskii equation, also called the nonlinear Schrödinger equation, describes
More informationTheorie der Kondensierten Materie I WS 16/ Chemical potential in the BCS state (40 Punkte)
Karlsruhe Institute for Technology Institut für Theorie der Kondensierten Materie Theorie der Kondensierten Materie I WS 6/7 Prof Dr A Shnirman Blatt PD Dr B Narozhny, T Ludwig Lösungsvorschlag Chemical
More information8.044 Lecture Notes Chapter 9: Quantum Ideal Gases
8.044 Lecture Notes Chapter 9: Quantum Ideal Gases Lecturer: McGreevy 9. Range of validity of classical ideal gas...................... 92 9.2 Quantum systems with many indistinguishable particles............
More informationIntermission: Let s review the essentials of the Helium Atom
PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the
More informationModern Physics for Scientists and Engineers International Edition, 4th Edition
Modern Physics for Scientists and Engineers International Edition, 4th Edition http://optics.hanyang.ac.kr/~shsong 1. THE BIRTH OF MODERN PHYSICS 2. SPECIAL THEORY OF RELATIVITY 3. THE EXPERIMENTAL BASIS
More informationFrom Last Time Important new Quantum Mechanical Concepts. Atoms and Molecules. Today. Symmetry. Simple molecules.
Today From Last Time Important new Quantum Mechanical Concepts Indistinguishability: Symmetries of the wavefunction: Symmetric and Antisymmetric Pauli exclusion principle: only one fermion per state Spin
More informationLecture 8. The Second Law of Thermodynamics; Energy Exchange
Lecture 8 The Second Law of Thermodynamics; Energy Exchange The second law of thermodynamics Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reading for
More informationPhase Transitions. µ a (P c (T ), T ) µ b (P c (T ), T ), (3) µ a (P, T c (P )) µ b (P, T c (P )). (4)
Phase Transitions A homogeneous equilibrium state of matter is the most natural one, given the fact that the interparticle interactions are translationally invariant. Nevertheless there is no contradiction
More informationBachelor Thesis. BoseEinstein condensates in magnetic microtraps with attractive atomatom interactions. Florian Atteneder
Bachelor Thesis BoseEinstein condensates in magnetic microtraps with attractive atomatom interactions Florian Atteneder 33379 Supervisor: Ao.Univ.Prof. Mag. Dr. Ulrich Hohenester University of Graz
More informationList of Comprehensive Exams Topics
List of Comprehensive Exams Topics Mechanics 1. Basic Mechanics Newton s laws and conservation laws, the virial theorem 2. The Lagrangian and Hamiltonian Formalism The Lagrange formalism and the principle
More information2. Thermodynamics. Introduction. Understanding Molecular Simulation
2. Thermodynamics Introduction Molecular Simulations Molecular dynamics: solve equations of motion r 1 r 2 r n Monte Carlo: importance sampling r 1 r 2 r n How do we know our simulation is correct? Molecular
More informationChapter 7: Quantum Statistics
Part II: Applications SDSMT, Physics 2014 Fall 1 Introduction Photons, E.M. Radiation 2 Blackbody Radiation The Ultraviolet Catastrophe 3 Thermal Quantities of Photon System Total Energy Entropy 4 Radiation
More informationSommerfeldDrude model. Ground state of ideal electron gas
SommerfeldDrude model Recap of Drude model: 1. Treated electrons as free particles moving in a constant potential background. 2. Treated electrons as identical and distinguishable. 3. Applied classical
More informationLecture 32: The Periodic Table
Lecture 32: The Periodic Table (source: What If by Randall Munroe) PHYS 2130: Modern Physics Prof. Ethan Neil (ethan.neil@colorado.edu) Announcements Homework #9 assigned, due next Wed. at 5:00 PM as usual.
More informationLecture 5. HartreeFock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 HartreeFock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particlenumber representation: General formalism The simplest starting point for a manybody state is a system of
More information1 Bose condensation and Phase Rigidity
1 Bose condensation and Phase Rigidity 1.1 Overview and reference literature In this part of the course will explore various ways of describing the phenomenon of BoseEinstein Condensation (BEC), and also
More informationSection 10: Many Particle Quantum Mechanics Solutions
Physics 143a: Quantum Mechanics I Section 10: Many Particle Quantum Mechanics Solutions Spring 015, Harvard Here is a summary of the most important points from this week (with a few of my own tidbits),
More informationFermionic Algebra and Fock Space
Fermionic Algebra and Fock Space Earlier in class we saw how harmonicoscillatorlike bosonic commutation relations [â α,â β ] = 0, [ ] â α,â β = 0, [ ] â α,â β = δ α,β (1) give rise to the bosonic Fock
More informationHW posted on web page HW10: Chap 14 Concept 8,20,24,26 Prob. 4,8. From Last Time
HW posted on web page HW10: Chap 14 Concept 8,20,24,26 Prob. 4,8 From Last Time Philosophical effects in quantum mechanics Interpretation of the wave function: Calculation using the basic premises of quantum
More informationThe goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq
Chapter. The microcanonical ensemble The goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq } = A that give
More informationAtoms, Molecules and Solids. From Last Time Superposition of quantum states Philosophy of quantum mechanics Interpretation of the wave function:
Essay outline and Ref to main article due next Wed. HW 9: M Chap 5: Exercise 4 M Chap 7: Question A M Chap 8: Question A From Last Time Superposition of quantum states Philosophy of quantum mechanics Interpretation
More informationBefore we go to the topic of hole, we discuss this important topic. The effective mass m is defined as. 2 dk 2
Notes for Lecture 7 Holes, Electrons In the previous lecture, we learned how electrons move in response to an electric field to generate current. In this lecture, we will see why the hole is a natural
More informationUnitary Fermi gas in the ɛ expansion
Unitary Fermi gas in the ɛ expansion Yusuke Nishida Department of Physics, University of Tokyo December 006 PhD Thesis Abstract We construct systematic expansions around four and two spatial dimensions
More informationElectrons in metals PHYS208. revised Go to Topics Autumn 2010
Go to Topics Autumn 010 Electrons in metals revised 0.1.010 PHYS08 Topics 0.1.010 Classical Models The Drude Theory of metals Conductivity  static electric field Thermal conductivity Fourier Law WiedemannFranz
More informationCalculating Band Structure
Calculating Band Structure Nearly free electron Assume plane wave solution for electrons Weak potential V(x) Brillouin zone edge Tight binding method Electrons in local atomic states (bound states) Interatomic
More informationRealization of BoseEinstein Condensation in dilute gases
Realization of BoseEinstein Condensation in dilute gases Guang Bian May 3, 8 Abstract: This essay describes theoretical aspects of BoseEinstein Condensation and the first experimental realization of
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 3 The BornOppenheimer approximation C.K. Skylaris Learning outcomes Separate molecular Hamiltonians to electronic and nuclear parts according to the BornOppenheimer
More informationThe following experimental observations (between 1895 and 1911) needed new quantum ideas:
The following experimental observations (between 1895 and 1911) needed new quantum ideas: 1. Spectrum of Black Body Radiation: Thermal Radiation 2. The photo electric effect: Emission of electrons from
More informationSingle Particle State AB AB
LECTURE 3 Maxwell Boltzmann, Femi, and Bose Statistics Suppose we have a gas of N identical point paticles in a box of volume V. When we say gas, we mean that the paticles ae not inteacting with one anothe.
More informationDownloaded from UvADARE, the institutional repository of the University of Amsterdam (UvA)
Downloaded from UvADARE, the institutional repository of the University of Amsterdam (UvA) http://hdl.handle.net/11245/2.99140 File ID Filename Version uvapub:99140 319071.pdf unknown SOURCE (OR PART
More information3. Introductory Nuclear Physics 1; The Liquid Drop Model
3. Introductory Nuclear Physics 1; The Liquid Drop Model Each nucleus is a bound collection of N neutrons and Z protons. The mass number is A = N + Z, the atomic number is Z and the nucleus is written
More informationElementary Lectures in Statistical Mechanics
George DJ. Phillies Elementary Lectures in Statistical Mechanics With 51 Illustrations Springer Contents Preface References v vii I Fundamentals: Separable Classical Systems 1 Lecture 1. Introduction 3
More informationPhysics 607 Final Exam
Physics 607 Final Exam Please show all significant steps clearly in all problems. 1. Let E,S,V,T, and P be the internal energy, entropy, volume, temperature, and pressure of a system in thermodynamic equilibrium
More informationFermi Condensates ULTRACOLD QUANTUM GASES
Fermi Condensates Markus Greiner, Cindy A. Regal, and Deborah S. Jin JILA, National Institute of Standards and Technology and University of Colorado, and Department of Physics, University of Colorado,
More informationLecture 6: FluctuationDissipation theorem and introduction to systems of interest
Lecture 6: FluctuationDissipation theorem and introduction to systems of interest In the last lecture, we have discussed how one can describe the response of a wellequilibriated macroscopic system to
More informationKinetic theory. Collective behaviour of large systems Statistical basis for the ideal gas equation Deviations from ideality
Kinetic theory Collective behaviour of large systems Statistical basis for the ideal gas equation Deviations from ideality Learning objectives Describe physical basis for the kinetic theory of gases Describe
More informationBEC Vortex Matter. Aaron Sup October 6, Advisor: Dr. Charles Hanna, Department of Physics, Boise State University
BEC Vortex Matter Aaron Sup October 6, 006 Advisor: Dr. Charles Hanna, Department of Physics, Boise State University 1 Outline 1. Bosons: what are they?. BoseEinstein Condensation (BEC) 3. Vortex Formation:
More informationto satisfy the large number approximations, W W sys can be small.
Chapter 12. The canonical ensemble To discuss systems at constant T, we need to embed them with a diathermal wall in a heat bath. Note that only the system and bath need to be large for W tot and W bath
More informationDefinite Integral and the Gibbs Paradox
Acta Polytechnica Hungarica ol. 8, No. 4, 0 Definite Integral and the Gibbs Paradox TianZhi Shi College of Physics, Electronics and Electrical Engineering, HuaiYin Normal University, HuaiAn, JiangSu, China,
More informationSolution to the exam in TFY4230 STATISTICAL PHYSICS Friday december 19, 2014
NTNU Page 1 of 10 Institutt for fysikk Fakultet for fysikk, informatikk og matematikk This solution consists of 10 pages. Problem 1. Cold Fermi gases Solution to the exam in TFY4230 STATISTICAL PHYSICS
More informationDirect observation of quantum phonon fluctuations in ultracold 1D Bose gases
Laboratoire Charles Fabry, Palaiseau, France Atom Optics Group (Prof. A. Aspect) Direct observation of quantum phonon fluctuations in ultracold 1D Bose gases Julien Armijo* * Now at Facultad de ciencias,
More informationComparing and Improving Quark Models for the Triply Bottom Baryon Spectrum
Comparing and Improving Quark Models for the Triply Bottom Baryon Spectrum A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science degree in Physics from the
More informationReference for most of this talk:
Cold fermions Reference for most of this talk: W. Ketterle and M. W. Zwierlein: Making, probing and understanding ultracold Fermi gases. in Ultracold Fermi Gases, Proceedings of the International School
More informationChapter 6: The Electronic Structure of the Atom Electromagnetic Spectrum. All EM radiation travels at the speed of light, c = 3 x 10 8 m/s
Chapter 6: The Electronic Structure of the Atom Electromagnetic Spectrum V I B G Y O R All EM radiation travels at the speed of light, c = 3 x 10 8 m/s Electromagnetic radiation is a wave with a wavelength
More informationPhase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. HansHenning Klauss. Institut für Festkörperphysik TU Dresden
Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality HansHenning Klauss Institut für Festkörperphysik TU Dresden 1 References [1] Stephen Blundell, Magnetism in Condensed
More informationIntroduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ
More informationName Class Date. Chapter: Arrangement of Electrons in Atoms
Assessment Chapter Test A Chapter: Arrangement of Electrons in Atoms In the space provided, write the letter of the term that best completes each sentence or best answers each question. 1. Which of the
More information5. GrossPitaevskii theory
5. GrossPitaevskii theory Outline N noninteracting bosons N interacting bosons, manybody Hamiltonien Meanfield approximation, order parameter GrossPitaevskii equation Collapse for attractive interaction
More informationare divided into sets such that the states belonging
Gibbs' Canonical in Ensemble Statistical By Takuzo Distribution Mechanics. Law SAKAI (Read Feb. 17,1940) 1. librium The distribution state is obtained law for an assembly various methods. of molecules
More informationChapter 1. The Properties of Gases Fall Semester Physical Chemistry 1 (CHM2201)
Chapter 1. The Properties of Gases 2011 Fall Semester Physical Chemistry 1 (CHM2201) Contents The Perfect Gas 1.1 The states of gases 1.2 The gas laws Real Gases 1.3 Molecular interactions 1.4 The van
More informationCHEMISTRY Topic #1: Bonding What Holds Atoms Together? Spring 2012 Dr. Susan Lait
CHEMISTRY 2000 Topic #1: Bonding What Holds Atoms Together? Spring 2012 Dr. Susan Lait Why Do Bonds Form? An energy diagram shows that a bond forms between two atoms if the overall energy of the system
More informationVan der Waals equation: eventbyevent fluctuations, quantum statistics and nuclear matter
Van der Waals equation: eventbyevent fluctuations, quantum statistics and nuclear matter Volodymyr Vovchenko In collaboration with Dmitry Anchishkin, Mark Gorenstein, and Roman Poberezhnyuk based on:
More informationInteracting Fermi Gases
Interacting Fermi Gases Mike Hermele (Dated: February 11, 010) Notes on Interacting Fermi Gas for Physics 7450, Spring 010 I. FERMI GAS WITH DELTAFUNCTION INTERACTION Since it is easier to illustrate
More information8 Wavefunctions  Schrödinger s Equation
8 Wavefunctions  Schrödinger s Equation So far we have considered only free particles  i.e. particles whose energy consists entirely of its kinetic energy. In general, however, a particle moves under
More informationvan Quantum tot Molecuul
10 HC10: Molecular and vibrational spectroscopy van Quantum tot Molecuul Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ j.rojo@vu.nl Molecular and Vibrational Spectroscopy Based
More informationarxiv:nuclth/ v1 24 Aug 2005
Test of modified BCS model at finite temperature V. Yu. Ponomarev, and A. I. Vdovin Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 498 Dubna, Russia Institut für Kernphysik,
More informationChE 503 A. Z. Panagiotopoulos 1
ChE 503 A. Z. Panagiotopoulos 1 STATISTICAL MECHANICAL ENSEMLES 1 MICROSCOPIC AND MACROSCOPIC ARIALES The central question in Statistical Mechanics can be phrased as follows: If particles (atoms, molecules,
More informationLecture 12: Phonon heat capacity
Lecture 12: Phonon heat capacity Review o Phonon dispersion relations o Quantum nature of waves in solids Phonon heat capacity o Normal mode enumeration o Density of states o Debye model Review By considering
More informationMesoscopic NanoElectroMechanics of Shuttle Systems
* Mesoscopic NanoElectroMechanics of Shuttle Systems Robert Shekhter University of Gothenburg, Sweden Lecture1: Mechanically assisted singleelectronics Lecture2: Quantum coherent nanoelectromechanics
More informationCondensed matter physics FKA091
Condensed matter physics FKA091 Ermin Malic Department of Physics Chalmers University of Technology Henrik Johannesson Department of Physics University of Gothenburg Teaching assistants: Roland Jago &
More informationarxiv:quantph/ v1 7 Dec 1996
Trapped Fermi gases D.A. Butts, D.S. Rokhsar, Department of Physics, University of California, Berkeley, CA 947207300 Volen Center for Complex Systems, Brandeis University, Waltham, MA 02254 (December
More informationPhysics 221B Spring 2018 Notes 30 The ThomasFermi Model
Copyright c 217 by Robert G. Littlejohn Physics 221B Spring 218 Notes 3 The ThomasFermi Model 1. Introduction The ThomasFermi model is a relatively crude model of multielectron atoms that is useful
More informationEnergy Level Energy Level Diagrams for Diagrams for Simple Hydrogen Model
Quantum Mechanics and Atomic Physics Lecture 20: Real Hydrogen Atom /Identical particles http://www.physics.rutgers.edu/ugrad/361 physics edu/ugrad/361 Prof. Sean Oh Last time Hydrogen atom: electron in
More informationUnderstand the basic principles of spectroscopy using selection rules and the energy levels. Derive Hund s Rule from the symmetrization postulate.
CHEM 5314: Advanced Physical Chemistry Overall Goals: Use quantum mechanics to understand that molecules have quantized translational, rotational, vibrational, and electronic energy levels. In a large
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More informationThe Charged Liquid Drop Model Binding Energy and Fission
The Charged Liquid Drop Model Binding Energy and Fission 103 This is a simple model for the binding energy of a nucleus This model is also important to understand fission and how energy is obtained from
More information