量子力学 Quantum mechanics School of Physics and Information Technology Shaanxi Normal University Chapter 5 IDENTICAL PARTICLES 5.1 Two-Particle Systems 201 5.2 Atoms 210 5.3 Solids 218 5.4 Quantum Statistical Mechanics 230 5.4 Quantum Statistical Mechanics If we have a large number of N particles, in thermal equilibrium at temperature T, what is the probability that a particle would be found to have the specific energy, Ej? The fundamental assumption of statistical mechanics is that in thermal equilibrium every distinct state with the same total energy, E, is equally probable. The temperature, T, is a measure of the total energy of a system in thermal equilibrium in classical mechanics. What is the new in quantum mechanics? How to count the distinct states! Why? Give a example to demonstrate! 5.4.1 An Example Suppose we have just have three noninteracting particles, A, B, and C, (all of mass m) in the one-dimensional infinite square well. The total energy is E = E A + E B + EC = π 2h 2 2ma 2 (n A2 + nB2 + nC2 ), where nA,nB, and nC are positive integers. Now suppose, for the sake of argument, that total energy E = 363 π 2h 2 2ma 2 , which is to say, n A2 + nB2 + nC2 = 363. nA = ? nB = ? nC = ? Thus (nA,nB,nC ) can be one of the following: (11, 11, 11) 1 (13, 13, 5) (13, 5, 13) (5, 13, 13) 3 (1, 1, 19) (1, 19, 1) (19, 1, 1) 3 6 (5, 7, 17) (5, 17, 7) (7, 5, 17) (7, 17, 5) (17, 5, 7) (17, 7, 5) 13 total For example, (nA,nB,nC )=(11,11,11) means nA=11, nB=11, nC=11, and A,B,C in the single states ψ A = ψ 11 ( x A ); ψ B = ψ 11 ( xB ); ψ C = ψ 11 ( xC ). If the particles are distinguishable, the three-particle state is ψ ( x) = ψ 11 ( x A )ψ 11 ( xB )ψ 11 ( xC ) The total number of probable (nA,nB,nC ) is 13. The most important quantity is the number of particles in each state, that is, the occupation number, Nn, for the single state ψ n (x ). Configuration(排布): The collection of all occupation numbers for a given (3-particle) state we will call the configuration. ψ1, ψ 2 , ψ 3 , ψ 4 , ψ 5 , ψ 6 , ψ 7 , L (N 1 , N 2 , N 3 , N 4 , N 5 , N 6 , N 7 , L). 1. Distinguishable condition: If the particles are distinguishable, each of these (nA,nB,nC ) represents a distinct quantum state, and the fundamental assumption of statistical mechanics says that in thermal equilibrium they are all equally likely. If all the three particles are in ψ 11 ( x) , the configuration is ψ1 ψ 2 ψ 3 ψ 4 L (0, ψ11 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, L ). N11 = 3 (11, 11, 11) one state ψ4 ψ3 ψ2 ψ1 If two are in ψ 13 and one is in ψ 5 , the configuration is ψ1 ψ2 (0, ψ5 ψ13 ψ13 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, L). N 5 = 1, N13 = 2 ψ5 (13, 13, 5) (13, 5, 13) (5, 13, 13) three different states If two are in ψ1 (2, ψ1 and one is inψ 19 , the configuration is ψ19 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0 L ). N1 = 1, N19 = 2 (1, 1, 19) (1, 19, 1) (19, 1, 1) three different states If one is in ψ 5 , one in ψ 7 , and one is inψ 17 , the configuration is ψ7 ψ17 (0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 L). ψ5 ψ 7 N 5 = 1, N 7 = 1, N17 = 1 ψ17 (0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 L). (5, 7, 17) C A B A B (5, 17, 7) C C B A (7, 5, 17) six different states C A B (7, 17, 5) C B A (17, 5, 7) C B A (17, 7, 5) ψ5 Of course, the last is the most probable configuration, because it can be achieved in six different ways, whereas the middle two occur three ways, and the first only one. Under the above condition, if we select one of these three particles at random, what is the probability (Pn)of getting a specific (allowed) energy En ? E1 : Only the third configuration In the third configuration, two particles are in E1 Probability 3/13 Probability 2/3 3 2 2 × = P1 13 3 13 E2 : 0 E3 : 0 E4 : 0 E5 : the second configuration In the second configuration, one particle is in E5 the fourth configuration In the fourth configuration, one particle is in E5 Probability 3/13 Probability 1/3 Probability 6/13 Probability 1/3 3 1 1 × = 13 3 13 6 1 2 × = 13 3 13 1 2 3 + = 13 13 13 P5 E7 : Only the fourth configuration In the fourth configuration, one particle is in E7 E11 : Only the first configuration In the first configuration, three particles are in E11 Probability 6/13 Probability 1/3 Probability 1/13 Probability 3/3 6 1 2 P7 × = 13 3 13 1 3 1 × = P11 13 3 13 Similarly P13 = 3 2 2 × = 13 3 13 P17 = 6 1 2 × = 13 3 13 P19 = 3 1 1 × = 13 3 13 We can check this by total probability 2 3 2 1 2 2 1 P = P1 + P5 + P7 + P11 + P13 + P17 + P19 = + + + + + + = 1 13 13 13 13 13 13 13 Above analysis is based on the assumption that the three particles are distinguishable! 2. Identical fermions: For fermions, no two particles are in the same state. This antisymmetrization requirement exclude the configurations where two particles are in the same state. Only the fourth configuration is available now! ψ7 ψ17 (0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 L). ψ5 E1 : 0 E5 : Only one configuration In the fourth configuration, one particle is in E5 Probability 1 1 1 1× = 3 3 Probability 1/3 E7 : In this configuration, one particle is in E7 E17 : In this configuration, one particle is in E17 Probability 1/3 P5 P7 Probability 1/3 P17 3. Identical bosons: For bosons, each configuration enables one state, so E1 : The third configurations In this configuration, two particles are in E1 E5 : the second configuration In the second configuration, one particle is in E5 the fourth configuration In the fourth configuration, one particle is in E5 Probability 1/4 Probability 2/3 Probability 1/4 Probability 1/3 Probability 1/4 Probability 1/3 1 1 1 × = 4 3 12 1 1 1 × = 4 3 12 1 1 1 × = 4 3 12 Similarly 1 1 1 P7 = × = 4 3 12 1 1 1 P17 = × = 4 3 12 1 3 1 1 2 1 P11 = × = P13 = × = 4 3 4 4 3 6 1 1 1 P19 = × = 4 3 12 P1 1 1 1 + = 12 12 6 P5 Conclusion: (1) This example shows that the nature of the particles determines the counting properties, or the statistical properties! The number of internal distinct states is different and the probability of getting specific energy is different too. (2) This example gives a system of three particles. If the number of particles in huge, we can conclude: The distribution of individual particle energies, at equilibrium, is simply their distribution in the most probable configuration. 5.4.2 The General Case Now consider an arbitrary potential, for which one particle energies are (E1 , E 2 , E 3 , L L , E n , L L ), with degeneracies (d1 , d 2 , d 3 , L L , d n , L L ). Suppose we put N particles (all with the same mass) into this potential; we are interested in the configuration (N 1 , N 2 , N 3 , L L , N n , L L ), for which there are N1 particles with energy E1, N2 particles with energy E2, and so on. Now we consider general question: how many distinct states correspond to this particular configuration? The answer: The number of the distinct states Q(N1,N2,N3,……) depends on whether the particles are distinguishable, identical fermions, or identical bosons. 1. Distinguishable particles: ( N 1 , N 2 , N 3 , L L , N n , L L ) (1) Choose N1 from N for energy bin: the binomial coefficient d2 N2 N N! ≡ N1 N1!( N − N1 )! N! N ( N − 1)( N − 2) L( N − N1 + 1) = ( N − N1 )! N1! N1 1 2 N! N1!( N − N1 )! d1 (2) Arrangement of the N1 particles within the bin on the degenerate d1 states: (d1 )N 1 (3) Thus the number of ways to put N1 particles, selected from a total population of N, into a bin containing d1 distinct option, is N ! (d1 ) 1 N1!( N − N1 )! N (4) The same goes for energy bin E2, of course, except that there are now only N-N1 particles left to work with: (N − N1 )! (d 2 )N 2 N 2 !( N − N1 − N 2 )! ; LLL (5) Finally, it follows that ( N ! (d1 ) 1 N − N1 )! (d 2 ) 2 Q ( N1 , N 2 , N 3 , L) = LLL N1!( N − N1 )! N 2 !( N − N1 − N 2 )! N N ∞ ( dn ) N ! (d1 ) 1 (d 2 ) 2 L = = N !∏ N1! N 2 !L N n! n =1 N N Nn 2. Identical fermions: ( N 1 , N 2 , N 3 , L L , N n , L L ) (1) The particles are identical. N1 (2) The antisymmetrization requires that only one particle can occupy any given state. d1 ≥ N1 Here we pick N1 draws from d1 draws to locate particles. C C N2 d2 N1 d1 d1 d1! = = N1 N1!(d1 − N1 )! d2 = N2 LLL C ∞ Nn dn dn = Nn d n! Q( N1 , N 2 , N 3 , L) = ∏ n =1 N n !( d n − N n )! 1 2 d1 N1 3. Identical bosons: (N 1 , N 2 , N 3 , LL , N n , LL) (1) The particles are identical. N1 1 2 d1 (2) Although the wave function of the N-particle state is symmetry, more than one particles can occupy the draws in certain bin. N1 N1 + (d1 − 1)! ( N1 + d1 − 1)! N1 + d1 − 1 = N1 N1!(d1 − 1)! N1! (d1 − 1)! ( N n + d n − 1)! N n + d n − 1 = Nn N n !( d n − 1)! ∞ ( N n + d n − 1)! Q( N1 , N 2 , N 3 ,L) = ∏ n =1 N n !( d n − 1)! N1 + (d1 − 1) 5.4.3 The Most Probable Configuration In thermal equilibrium, every state with a given total energy E and a given particle number N is equally likely. So the most probable configuration (N1, N2, N3, ……) is the one that can be achieved in the largest number of different ways—— it is that particular configuration for which Q(N1,N2,N3,……..) is a maximum, subject to the constraints ∞ ∑N n =1 ∞ n =N ∑E N n n = E. n =1 The problem of maximizing a function F(x1, x2, x3,……) of several variables, subject to the constraints f (x1, x2, x3,……)=0, f (x1, x2, x3,……)=0, etc., is most conveniently handled by the method of Lagrange multipliers. We introduce the new function G ( x1 , x2 , x3 , L) ≡ F + λ1 f1 + λ2 f 2 + L , and set all its derivatives equal to zero: ∂G = 0; ∂xn ∂G = 0. ∂λn In our case it’s a little easier to work with the logarithm of Q, instead of Q itself——this turns the products into sums. Since the logarithm is a monotonic function of its argument, the maxima of Q and ln(Q) occur at the same point. So we let ∞ ∞ G ≡ ln(Q ) + α N − ∑ N n + β E − ∑ N n En , n =1 n =1 where α and β are the Lagrange multipliers. ∂G =0⇒ ∂α ∂G =0⇒ ∂β ∂G = 0; ∂N n 1. Distinguishable particles: ∞ Q ( N1 , N 2 , N 3 ,L) = N !∏ (d n )Nn n =1 ∞ (d n ) N n G = ln( N !) + ln ∏ n =1 N n ! N n! ∞ ∞ + α N − ∑ N n + β E − ∑ N n En , n =1 n =1 ∞ G = ln( N !) + ∑ [N n ln(d n ) − ln( N n !)] n =1 ∞ ∞ + α N − ∑ N n + β E − ∑ N n En , n =1 n =1 Assuming the relevant occupation numbers (Nn) are large, we can invoke stirling’s approximation: ln (z!) ≈ z ln z − z for z >> 1 ∞ G ≈ ∑ [N n ln(d n ) − N n ln( N n ) + N n − αN n − β En N n ]+ ln( N !) + αN + β E n =1 It follows that ∂G = ln(d n ) − ln( N n ) − α − β En = 0 ∂N n The most probable occupation numbers, for distinguishable particles, are ln( N n ) = ln(d n ) − α − β En N n = d n e − (α + βEn ). 2. Identical fermions: ∞ dn! Q ( N1 , N 2 , N 3 ,L) = ∏ n =1 N n !( d n − N n )! ∞ G = ∑ [ln(d n !) − ln( N n !) − ln[(d n − N n )!]] n =1 ∞ ∞ + α N − ∑ N n + β E − ∑ N n En , n =1 n =1 Assume Nn>>1 and dn>>Nn, so the stirling’s approximation applies ∞ G ≈ ∑[ln(dn!) − Nn ln(Nn ) + Nn − (dn − Nn ) ln(dn − Nn ) + (dn − Nn ) − αNn − βEn Nn ] n=1 + αN + βE ∂G = − ln( N n ) + ln(d n − N n ) − α − β En = 0 ∂N n The most probable occupation numbers, for identical fermions, are ln( N n ) = ln(d n − N n ) − α − β En N n = ( d n − N n ) e − (α + β E n ) Nn = dn e (α + βEn ) +1 . 3. Identical bosons: ∞ ( N n + d n − 1)! Q( N1 , N 2 , N 3 ,L) = ∏ n =1 N n !( d n − 1)! ∞ G = ∑ [ln[( N n + d n − 1)!] − ln( N n !) − ln[(d n − 1)!]] n =1 ∞ ∞ + α N − ∑ N n + β E − ∑ N n En , n =1 n =1 Assuming Nn>>1 and using stirling’s approximation ∞ G ≈ ∑[( Nn + dn −1) ln(Nn + dn −1) − ( Nn + dn −1) − Nn ln(Nn ) + Nn − ln[(dn −1)!] − αNn − βEn Nn ] n=1 + αN + βE ∂G = ln( N n + d n − 1) − ln( N n ) − α − β En = 0 ∂N n ln( N n ) = ln( N n + d n − 1) − (α + β En ) N n = ( N n + d n − 1)e − (α + βEn ) dn −1 N n = (α + βEn ) −1 e dn>>1 Nn = dn e (α + βEn ) −1 5.4.4 Physical significance of α and β The parameters α and β came into the story as Lagrange multipliers, associated with the total number of particles and the total energy. Mathematically, they are determined by substituting the most probable occupation numbers Nn back into the constraints. N n = d n e − (α + βEn ) Nn = Nn = dn e (α + βEn ) e ∑N n =N n =1 +1 dn (α + βEn ) ∞ α and β ∞ ∑E N n n =E n =1 −1 To carry out the summation, En and dn should be know for particular potential. By using an example to do this: ideal gas Idea gas: a large number of noninteracting particles, all with the same mass, in the three dimensional infinite square well—— a box! We know that the allowed energies of the particle are 2 h 2 Ek = k where 2m r πn πn y πn k = x , , z l x l y lz l . z ly lx A shell of thickness dk contains a volume dτ 1 dτ = 4π k 2 dk , 8 ( ) so the “degeneracy” is (the number of electron states in the shell ) dτ V 2 dk = 3 = k dk . 2 π / V 2π ( ) ∫ 0 For distinguishable particles, the first constraint becomes N n = d ne ∑N n x e − ax ∞ ∞ − (α + β E n ) ∞ N = ∑ dke =N n n =1 n! dx = n +1 a − (α + βE k ) n =1 In the k-space, the sum will be converted into an integral, treating k as a continuous variable, then V 2 dk = k dk , 2 2π 2 h k Ek = , 2m V −α N = ∫ dN = e ∫ e 2 0 2π ∞ ∫ ∞ 0 2 n − ax 2 x e 2 h 2k 2 −β 2m V −α dN = e e 2 2π h2k 2 −β 2m m k dk = Ve 2 2πβh −α 2 1 ⋅ 3 ⋅ 5 ⋅ L ⋅ (2n − 1) π dx = 2 n +1 a n a ∫ ∞ 0 2 e − ax x 2 dx = k 2 dk . 3/ 2 ⋅ π 1 16 a 3/ 2 so e −α N 2πβh = V m 2 3/ 2 ⋅ ∞ The second constraint ∑E N n n = E becomes n =1 2 V −α h ∞ E = ∫ dE = e e 2 ∫ 2π 2m 0 Or, putting in e −α : −β h2k 2 2m 3N E= 2β 3V −α m k dk = e 2 2β 2πβh 4 3/ 2 ⋅ E 3 = N 2β The average kinetic energy of an atom at temperature T, in classical mechanic, is E 3 = k BT N 2 Boltzmann constant This suggests that β is related to the temperature: 1 β= k BT Different substances in thermal equilibrium with one another have the same value of β , and which can be adopted as s definition of T. Then N 2πh = V k BTm 2 e −α 3/ 2 It is customary to replace α by the so-called chemical potential, µ (T ) ≡ −α k BT α =− µ (T ) k BT . By using the chemical potential, we can rewrite the most probable number of particles in a particular (one-particle) state with energy ε : N n = d ne − (α + β E n ) Nn n(ε ) = dn = e − (ε − µ )/ k BT Maxwell-Boltzmann distribution Nn = Nn = dn e (α + βEn ) dn e (α + βEn ) +1 Nn n(ε ) = dn = 1 e (ε − µ ) / k BT +1 Fermi-Dirac distribution −1 Nn n(ε ) = dn = 1 e (ε − µ ) / k BT −1 Bose-Einstein distribution The Maxwell-Boltzmann distribution is the classical result, for distinguishable particles; the Fermi-Dirac distribution applies to identical fermions, and the Bose-Einstein distribution is for identical bosons. Maxwell-Boltzmann distribution n(ε ) = e − (ε − µ ) / k BT T →0 (ε −µ) / kBT e 0, if ε < µ (0), → ∞, if ε > µ (0). −(ε −µ) / kBT e 0, if ε > µ (0), → ∞, if ε < µ (0). 1 2 ε = mv 2 n(v ) = e ( ) − mv 2 − µ / 2 k BT The Fermi-Dirac distribution has a particularly simple behavior as T n(ε ) = 1 e (ε − µ ) / k BT +1 . kB As (ε −µ) / kBT e →0 : 0, if ε < µ (0), → ∞, if ε > µ (0). 1, if ε < µ (0), n(ε ) → 0, if ε > µ (0). µ ( 0) kB All states are filled, up to an energy µ(0), and none are occupied for energies above µ(0). Evidently the chemical potential at absolute zero is precisely the Fermi energy: µ ( 0) = E F Bose-Einstein distribution n(ε ) = n(ε ) > 0 1 e (ε − µ )/ k BT − 1 e(ε −µ )/ kBT −1 > 0 e (ε − µ )/ k BT > 1 ε > µ (T ) µ (T ) < ε hk 2 ε= 2m V →0 hk 2 ε= →0 2m ε <0 Returning to the special case of an ideal gas, for distinguishable particles we found that the total energy at temperature T is 3 E = Nk BT 2 and the chemical potential is N 2πh = V k BTm 2 e −α µ (T ) ≡ −α k BT 3/ 2 N 3 2πh 2 µ (T ) = k BT ln + ln V 2 mk BT 5.4.5 The Blackbody Spectrum Photons (quantum of the electromagnetic field) are identical bosons with spin 1, but they are very special, because they are massless particles, and hence intrinsically relativistic. There are four properties belong to nonrelativistic quantum mechanics: E = hν = hω (2) Wave number: k = 2π/λ = ω / c (1) Energy: (3) Spin: two spin states occur, m=1 or –1. (4) The number of the photons are not conserved:α Nn = dk e (α + βEk ) −1 Nn = =0 dk e hω / k B T − 1 For free photons in a box of volume V, dk is given by V 2 dk = k dk , 2 2π multiplied by 2 for two spin states, and expressed in terms of ω = kc V d k = 2 3 ω 2 dω. π c So the energy density, V 2 ω dω 2 3 π c E N n hω = = hω / k B T V V e That is 3 h ω hω dω = 2 3 hω / k B T − 1) − 1 V π c (e hω 3 E (ω ) = 2 3 hω / k BT dω π c (e − 1) We introduce energy density per unit frequency: hω 3 ρ (ω ) = 2 3 hω / k BT π c (e − 1) This is Plank’s famous formula for the blackbody spectrum, giving the energy per unit volume, per unit frequency, for an electromagnetic field in equilibrium at temperature T. ω = 2πν = kc = 2πc dω = − λ 2πc 2 λ dλ hω 3 E (ω ) = ρ (ω ) dω = 2 3 hω / k BT dω − 1) π c (e E (λ ) = 8πh λ3 (e 2πhc / k Tλ B 16π 2 hc dλ = 5 2πhc / k BTλ dλ = ρ ( λ ) dλ 2 − 1) − 1) λ λ (e 2πc 16π 2 hc ρ (λ ) = 5 2πhc / k BTλ λ (e − 1)