Chapter 2 Principles of Quantum Statistics In this chapter we will study one of the most important assumptions of the quantum the- ory, the symmetrization postulate, and its consequence for the statistical properties of an ensemble of particles at thermal equilibrium. The three representative distribution func- tions, Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions, are introduced here. Then, we will describe the other important assumption of the quantum theory, the non-commutability of conjugate observables, and its consequence for the uncertainty product in determining a certain pair of observables. The discussions in this chapter pro- vide the foundation of thermal noise and quantum noise. Most of the discussions in this chapter follow the excellent texts on stastical mechanics by F. Rief [1] and on quantum mechanics by C. Cohen-Tannoudji et al. [2]. 2.1 Symmetrization Postulate of Quantum Mechanics 2.1.1 Quantum indistinguishability Let us consider a following thought experiment. Two identical quantum particles are incident upon a 50%{50% particle beam splitter and we measure the output by two particle counters. 1 or 2 ? 50-50% beam splitter Figure 2.1: Collision of two identical particles. To analyze this problem, we label the two particles as particle 1 and particle 2 based on 1 the click event in the detectors. One possible input state is j1;R; 2;Li ¢ ¢ ¢ particle 1 is in the right input port and particle 2 is in the left input port. Since identical quantum particles are \indistinguishable", the following input state is equally possible : j1;L; 2;Ri ¢ ¢ ¢ particle 1 is in the left input port and particle 2 is in the right input port. Most general input state is thus constructed as the linear superposition of the two : c0j1;R; 2;Li + c1j1;L; 2;Ri 2 2 (2.1) (jc0j + jc1j = 1 if the two states are orthogonal.) As shown below, an experimental result is dependent on the choice of the two c-numbers c0 and c1. Therefore, the above mathematical state has an inherent ambiguity in predicting an experimental result and we need a fundamental assumption of the theory, \symmetriza- tion postulate"[2]. 2.1.2 Statement of the postulate A physical state which represents a real system consisting of identical quantum particles is either symmetric or anti-symmetric with respect to the permutation of any two particles. A particle which obeys the former is called a boson and a particle which obeys the latter is called a fermion. For instance, the quantum states of two identical bosons and fermions are 1 Boson : p [j1;R; 2;Li + j1;L; 2;Ri]; 2 (2.2) 1 Fermion : p [j1;R; 2;Li ¡ j1;L; 2;Ri]; 2 if the two states are orthogonal. To demonstrate the important conseque of this new postulate, next let us calculate the probability of ¯nding the two particles in the same output port (L). This probability consists of two possibilities. 2 2 1 1 2 + direct term exchange term Figure 2.2: Two possibilities for an output state j1;L; 2;Li. 3 à ! 1 1 The beam splitter scattering matrix is U^ = p1 in the basis of the two trans- 2 ¡1 1 " # jk i mission modes R!L . Therefore, the probability is jkL!Ri The probability for obtaining j1;R; 2;Ri state is identical to (2.3). We can calculate the probability of ¯nding one particle in each output port. The probability of obtaining j1;L; 2;Ri state consists of two possibilities: 2 1 1 2 direct term exchange term Figure 2.3: Two possibilities for an output state j1;L; 2;Ri. The probability for obtaining j1;R; 2;Li state is identical to (2.4). The scattering patterns of identical bosons and fermions are shown in Fig. 2.4. 4 boson fermion 1 1 ψ =[]1,;,,;,RRLL 2 − 1 2 ψ =[]1,;,,;,RLLR 2 − 1 2 out 2 out 2 Figure 2.4: The scattering behaviours of two identical bosons and fermions at 50-50% beam splitter. Comments : 1. The (2, 0) and (0, 2) output characteristics of bosons are called a \bunching" e®ect. The constructive interference enhances the probability of ¯nding two particles in the same state. This is the ultimate origin of ¯nal state stimulation in lasers, Bose Einstein condensation and superconductivity. 2. The deterministic (1, 1) output characteristics of fermions are called \anti-bunching" e®ect. The destructive interference between the direct and exchange terms sup- presses the probability of ¯nding more than two particles in the same state. This is the manifestation of Pauli's exclusion principle. 1 2.1.3 Symmetrization postulate for spin ¡ 2 particles 1 Next let us consider a following thought experiment. Two spin ¡ 2 particles are incident upon a 50%{50% beam splitter. The scattering matrix of a beam splitter is assumed to be spin-independent. If the two particles are in spin triplet states (symmetric spin states), the orbital states are symmetric for bosons and anti-symmetric for fermions to satisfy the symmetrization postulate for overall states : 8 > j "i j "i > 1 2 <> 1 j #i1j #i2 Boson : p [jRi1jLi2 + jLi1jRi2] ­ 2 > > 1 : p [j "i1j #i2 + j #i1j "i2] 2 8 > j "i j "i > 1 2 <> 1 j #i1j #i2 Fermion : p [jRi1jLi2 ¡ jLi1jRi2] ­ (2.5) 2 > > 1 : p [j "i1j #i2 + j #i1j "i2] 2 5 Their collision characteristics are the same as those for spin-less particles mentioned above. However, if the two particles are in a spin singlet state (anti-symmetric spin state), the orbital states are anti-symmetric for bosons and symmetric for fermions to satisfy the symmetrization postulate for overall states : 1 1 Boson : p [jRi1jLi2 ¡ jLi1jRi2] ­ p [j "i1j #i2 ¡ j #i1j "i2] 2 2 (2.6) 1 1 Fermion : p [jRi1jLi2 + jLi1jRi2] ­ p [j "i1j #i2 ¡ j #i1j "i2] 2 2 Now, the bosons feature a fermionic collision, i.e. (1, 1) output, and the fermions feature a bosonic collision, i.e. (2, 0) or (0, 2) output. This example illustrates a very important 1 nature of two spin ¡ 2 fermions which can occupy the same state. Conversely if two identical fermions occupy the same orbital state, their spin state is always a spin singlet state. 1 ¡jxi1jxi2 ­ p [j "i1j #i2 ¡ j #i1j "i2] (2.7) 2 AKA ¢¢¸ same orbital state 2.2 Thermodynamic Partition Functions 2.2.1 A small system in contact with a large heat reservoir Let us consider an ideal gas of non-interacting identical particles in a volume V and at a P temperature T . We assume the total number of particles, r nr = N, is constant, where r designates a microscopic state and nr is the number of particles in that state. The total energy of such an ideal gas system is X ER = "rnr ; (2.8) r where "r is the kinetic energy of a microscopic state r and R designates a macroscopic state represented by the occupation number of each microscopic state, fn1; n2; ¢ ¢ ¢ ; nr; ¢ ¢ ¢g. The thermodynamic partition function Z is de¯ned by X Z = e¡¯ER ; (2.9) R P where R stands for summation over all possible macroscopic states, ¯ = 1=kBT is a temperature parameter and kB is a Boltzmann constant. Next let us obtain the probability of ¯nding a particular macroscopic state R. For this purpose we assume that our system is a small system A in thermal contact to a large heat reservoir A0. We also assume the interaction between A and A0 is extremely small, so that their energies are additive. The energy of A is not ¯xed and depends on the speci¯c 6 macroscopic state R. It is the total energy of the combined system A and A0 which has a 0 constant value ET . This constant total energy is split into those of A and A : 0 ET = ER + E ; (2.10) where E0 is the energy of A0. According to the fundamental postulate of statistical mechanics, the probability of ¯nding A in a speci¯c macroscopic state R is proportioned to the number of states 0 0 0 ­ (ET ¡ ER) accessible to A when its energy lies in a range ±E near E = ET ¡ ER. Hence 0 0 PR = C ­ (ET ¡ ER) ; (2.11) where C0 is a constant independent of R and determined by the normalization condition P R pR = 1. 0 Since A is a much smaller system than A ;ER ¿ ET and (2.11) can be approximated 0 0 by expanding the logarithm of ­ (ET ¡ ER) about E = ET : · ¸ 0 0 @ 0 0 ln­ (ET ¡ ER) ' ln­ (ET ) ¡ 0 ln­ (E ) ¢ ER : (2.12) @E ET The derivative in (2.12) is the de¯nition of the temperature parameter of the heat reservoir according to the statistical mechanics [1], · ¸ @ 0 0 0 ln­ (E ) ´ ¯ = 1=kBT: (2.13) @E ET The above result means that the heat reservoir A0 is much larger than A so that its temperature T remains unchanged by such a small amount of energy it gives to A. Thus, we obtain 0 0 ¡¯ER ­ (ET ¡ ER) = ­ (ET ) e : (2.14) 0 since ­ (ET ) is a constant independent of R, (2.11) becomes ¡¯ER PR = Ce ; (2.15) where C = C0­0 (E ) can be determined by the above mentioned normalization condition P T R pR = 1. The probability of ¯nding the gas in a speci¯c macroscopic state R = fn1; n2; ¢ ¢ ¢ ; nr; ¢ ¢ ¢g is ¯nally given by e¡¯ER e¡¯ER PR = X = : (2.16) e¡¯ER0 Z R0 If the above is true, such a system is called a canonical ensemble[1; 3].