Introduction to the Second Quantization Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 27, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 1 / 35 Identical particles Examples of quantum many body problems: I Photons in a block-body cavity I Electrons in a solid I Nucleons in a nucleus I Neutrons in a neutron star I Helium atoms in a superfluid The concept of identical particles: I Intrinsic: Those do not depend on the dynamical state of the particle, such as the rest mass, the charge, or the total spin. I Extrinsic: Those depend on the state of the particle or on its preparation: the position, and velocity, spin orientation and other internal parameters. Here two particles are considered identical if they possess the same intrinsic properties. More mathematically speaking, we say n particles are identical if all O(x1; p1; :::; ; xn; pn) are invariant under the exchange of their coordinates (xi ; pi ), i = 1;:::; n. n 2 X jpi j X H(n) = H (n) + V (n) = + V (jx − x j): 0 2m i j i=1 i<j Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 2 / 35 One-particle system: states The collection of states of a single particle form a Hilbert space Z 2 3 2 H = L (R ) = f'(x); dxj'(x)j < 1g with the inner product Z h'j i = dx'∗(x) (x): If a particle posses a spin s, its wavefunction will have 2s + 1 components '(x; σ), σ = −s; −s + 1;:::; s − 1; s + 1. i.e. 0 '(x; −s) 1 B '(x; −s + 1) C B C B . C 2s+1 '(x) = B . C 2 H ⊗ C : B C @ '(x; s − 1) A '(x; s) Now, I will focus on the case s = 0. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 3 / 35 One-particle system: Observables Position q and momentum p are linear operators on H : (q')(x) = x'(x) @ @ @ (p')(x) = −i r'(x); r = ; ; ~ @x1 @x2 @x3 The canonical commutation relations α β [q ; p ] = i~δα,β; α; β = 1; 2; 3: In the Fourier representation 1 Z '(k) = dx e−ik·x'(x) e (2π)3=2 the position operator q and the momentum operator q becomes (q'e)(k) = −irk'e(k) (p'e)(k) = ~k'(k) Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 4 / 35 One-particle system: Evolution equations The evolution of the particle is governed by Schrodinger equation @ i '(x; t) = (H')(x; t) ~@t where H is the given Hamiltonian. The formal solution is given by '(x; t) = (U(t)')(x; 0); U(0) = I where U(t) is the evolution operator (semi-group on H). For example, if the external fields are independent of time, then itH U(t) = exp − ~ ∗ −1 U (t) = U (t); U(t1)U(t2) = U(t1 + t2): Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 5 / 35 n-particle system: States The collection of the states of n non-relativistic spinless particles form a Hilbert space Z n 2 3n 2 H = L (R ) = fΦ(x1;:::; xn); dx1 ::: dxnjΦ(x1;:::; xn)j < 1g: If the particles carry a spin s, the Hilbert space of states is n 2s+1 H ⊗ C . Now we are going to construct the n-particle states in Hn from the one-particle states. hx1;:::; xnj'1 ⊗ :::; ⊗'ni = '1(x1) :::'n(xn) Here j'i i, i = 1;::: n are the single-particle state in H. We interpret it as the joint probability of the state j'1 ⊗ :::; ⊗'ni to find particle 1 in x1, ::: , and particle n in xn. The inner product of j'1 ⊗ :::; ⊗'ni and j 1 ⊗ :::; ⊗ ni is h 1 ⊗ · · · ⊗ nj'1 ⊗ :::; ⊗'ni = h'1j 1ih'2j 2i ::: h'nj ni Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 6 / 35 n-particle system: Basis Suppose fj'i i; i = 1; 2;::: g form a basis in the Hilbert space of a single-particle system. We know fj'j1 ⊗ :::; ⊗'jn i; j1; j2; ··· = 1; 2;::: g form a basis of the space H⊗n = H ⊗ · · · ⊗ H. For all wavefunction Φ(x1;:::; xn), we can express as X Φ(x1;:::; xn) = cj1:::jn 'j1 (x1) :::'jn (xn): j1:::jn We could also write as X hx1;:::; xnjΦi = cj1:::jn hx1;:::; xnj'j1 ⊗ · · · ⊗ 'jn i: j1:::jn X jΦi = cj1:::jn j'j1 ⊗ · · · ⊗ 'jn i: j1:::jn This means we can identify n-particle space Hn with the tensor product H⊗n. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 7 / 35 n-particle system: One-body observables Observables of a n-identical-particle system can be constructed from observalbes in a single-particle system. If A is a single-particle observable in H, we define the action A on the jth factor of the tensor product H⊗n as (I ⊗· · ·⊗A⊗· · ·⊗I )j'1⊗· · ·⊗'j ⊗· · ·⊗'ni = j'1⊗· · ·⊗A'j ⊗· · ·⊗'ni and denote it as Aj . The corresponding extensive observable for the n-particle system is defined by n X A(n) = Aj : j=1 Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 8 / 35 n-particle system: Two-body observables Let V be an observable in two-particle space H ⊗ H, such as interaction potential. ⊗n ⊗n Define Vij on H as the operator acts on ith and jth factors of H and trivially on the others. The corresponding observable for a n-particle system is n X V (n) = Vij : i<j If V is acting on two identical particles, we have Vij = Vji . Therefore we have n 1 X V (n) = V : 2 ij i6=j Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 9 / 35 Symmetrization: Permutation operator ⊗n Permutation operator Pπ actin on H is Pπj'1 ⊗ · · · ⊗ 'ni = j'π(1) ⊗ · · · ⊗ 'π(n)i 1 2 ::: n where π = . π(1) π(2) : : : π(n) Some properties: ∗ −1 Pπ1π2 = Pπ1 Pπ2 ; Pπ = Pπ Let A be an observable on a single-particle space H. We have [A(n); Pπ] = 0 for all permutation π Pn where A(n) = j=1 Aj . Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 10 / 35 Symmetrization: Exchange degeneracy Suppose jΦi is an eigenstate of a n-identical-particle system. i.e. H(n)jΦi = EjΦi We will find H(n)PπjΦi = EPπjΦi for all permutation π In a n-identical-particle system, PπjΦi and their linear combinations describe the same physical state. Principle of symmetrization: the state of n identical particles are either completely symmetric or completely antisymmetic. Bosons are particles in the symmetric case. Fermions are particles in the antisymmetic case. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 11 / 35 Symmetrization: Symmetric and Antisymmetric States For any jΦi in H⊗n, we can construct corresponding one which is totally symmetric or totally antisymmetric: jΦi+ = S+jΦi jΦi− = S−jΦi where two operators S+ and S− defined by 1 X 1 X S = P ; S = (−1)πP : + n! π − n! π π2Pn π2Pn Properties: 2 2 ∗ S+ = S+; S− = S− = S−; S+S− = S−S+ = 0: π Pπ = S+Pπ = S+; PπS− = S−Pπ = (−1) S−: According to the principle of symmetrization, the space of state of a system of n identical particle particles is not entire H⊗n. ⊗n ⊗n I Space for bosons (symmetric): H+ ≡ S+H . ⊗n ⊗n I Space for fermions (anti-symmetric): H− ≡ S−H . Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 12 / 35 Hilbert spaces for bosons or fermions ⊗n ⊗n Denote the symmetric or antisymmetric state of H+ or H− by j'1;'2;:::;'niν ≡ Sνj'1 ⊗ '2 ⊗ · · · ⊗ 'ni where ν = 1 for bosons and ν = −1 for fermions. Exchanging the ith and jth components, we find j'1;:::;'i ;:::;'j ;:::;'ni+ = j'1;:::;'j ;:::;'i ;:::;'ni+ j'1;:::;'i ;:::;'j ;:::;'ni− = −|'1;:::;'j ;:::;'i ;:::;'ni− The inner product is 1 X h' ;:::;' j ; : : : ; i = (ν)πh' j i ::: h' j i: ν 1 n 1 n ν n! 1 π(1) n π(n) π2Pn In particular for the fermions case, 1 h' ;:::;' j ; : : : ; i = detfh' j ig: − 1 n 1 n − n! i j Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 13 / 35 Occupation Number Representation Let fj r ig be a basis of a single-particle space H. The occupation number normalized to 1 are defined by p n! jn1; n2;:::; nr ;::: i+ = p S+j r1 ⊗ · · · ⊗ rn i n1! ::: nr ! ::: p jn1; n2;:::; nr ;::: i− = n! S−j r1 ⊗ · · · ⊗ rn i where nr is the number of indices r1;:::; rn which are equal to r. For bosons, nr = 0; 1; 2; 3;::: . For fermions, nr = 0; 1. P The system has exactly n = r nr particles. Tien-Tsan Shieh (Institute of MathematicsAcademicIntroduction Sinica) to the Second Quantization July 27, 2011 14 / 35 Occupation Number Representation: Observables Let j r i are the eigenvector of a single-particle observable A such that Aj r i = αr j r i: Then jn ; n ;:::; n ;::: i are also eigenvectors of the one-body 1 2 t ν P observable A(n) with eigenvalue r nr αr . n ! X X A(n)j r1 ⊗ · · · ⊗ rn i = Aj j r1 ⊗ · · · ⊗ rn i = nr αr j r1 ⊗ · · · ⊗ rn i j=1 r Therefore if [Sν; A(n)] = 0, we have ! X A(n)jn1; n2;:::; nr ;::: iν = nr αr jn1; n2;:::; nr ;::: iν: r Orthogonality: νhm1; m2;:::; mr ;::: jn1; n2;:::; nr ;:::; iν = δm1;n1 δm2;n2 : : : δmr ;nr ::: ⊗n Basis: fjn1; n2;::: iνg form a basis of Hν .