2.4.8 Heisenberg Algebra, Fock Space and Harmonic Oscillator
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1 .4. SPECTRAL DECOMPOSITION 63 Let P +, P, P 1,..., P p be the corresponding orthogonal complimentary system of projections, that is, P + + P + p P i = I. i=1 Then there exists a corresponding system of operators N 1,..., N p satisfying the equations N i = P i, N i P i = P i N i = N i, N i P j = P j N i = 0, if i j and the angles θ 1,... θ k such that π < θ i < π and O = P + P + p R i (θ i ) i=1 where R i (θ i ) = cos θ i P i + sin θ i N i. are the two-dimensional rotation operators in the planes corresponding to P i. Theorem.4.9 Every invertible operator A on a real vector space can be written in a unique way as a product A = OR of an orthogonal operator O and a symmetric positive operator R..4.8 Heisenberg Algebra, Fock Space and Harmonic Oscillator Heisenberg Algebra. The Heisenberg algebra is a 3-dimensional Lie algebra with generators X, Y, Z satisfying the commutation relations [X, Y] = Z, [X, Z] = 0, [X, Z] = 0. A representation of the Lie algebra A is a homomorphism ρ : A L(V) from the Lie algebra to the space of operators on a vector space V such that ρ([s, T]) = [ρ(s ), ρ(t)]. mathphyshass1.tex; September 4, 013; 9:58; p. 63
2 64 CHAPTER. FINITE-DIMENSIONAL VECTOR SPACES The Heisenberg algebra can be represented by matrices X =, Y = 0 0 1, Z = or by the differential operators C (R 3 ) C (R 3 ) defined by X = x 1 y z, Y = y + 1 x z, Z = z Properties of the Heisenberg algebra [X, Y n ] = nzy n 1 [Y, X n ] = nzx n 1 [X, exp(by)] = bz exp(by) [Y, exp(ax)] = az exp(ax) exp( by)x exp(by) = X + bz exp(ax)y exp( ax) = Y + az exp(ax) exp(by) = exp(abz) exp(by) exp(ax) exp( by) exp(ax) exp(by) = exp(abz) exp(ax) exp(ax) exp(by) exp( ax) = exp(abz) exp(by) exp(ax) exp(by) = exp(abz) exp(by) exp(ax) Another useful formula is X exp ( 1 ) Y = exp ( 1 ) Y (X ZY) Campbell-Hausdorff formula exp(ax + by) = exp ( ab ) Z exp(ax) exp(by) ( ) ab = exp Z exp(by) exp(ax) mathphyshass1.tex; September 4, 013; 9:58; p. 64
3 .4. SPECTRAL DECOMPOSITION 65 Heisenberg group. The Heisenberg group is a 3-dimensional Lie group with the generators X, Y, Z. An arbitrary element of the Heisenberg group is parametrized by canonical coordinates (a, b, c) as g(a, b, c) = exp(ax + by + cz) Obviously, and the inverse is defined by g(0, 0, 0) = I [g(a, b, c)] 1 = g ( a, b, c) The group multiplication law in the Heisenberg group takes the form ( g(a, b, c)g(a, b, c ) = g a + a, b + b, c + c + 1 ) (ab a b) Notice that ( g(a, b, c) = g 0, 0, c + ab ) g(0, b, 0)g(a, 0, 0) A representation of a Lie group G is a homomorphism ρ : G Aut (V) from the group G to the space of invertible operators on a vector space V such that for any g, h G ρ(gh) = ρ(g)ρ(h) and ρ(g 1 ) = [ρ(g)] 1, ρ(e) = I Representations of the Heisenberg group. The elements of the Heisenberg group could be represented by the uppertriangular matrices. Notice that X = Y = Z = XZ = YZ = 0 mathphyshass1.tex; September 4, 013; 9:58; p. 65
4 66 CHAPTER. FINITE-DIMENSIONAL VECTOR SPACES and or Therefore, 0 a c 0 0 b = XY = YX = Z 0 0 ab g(a, b, c) =, 1 a c + ab 0 1 b a c 0 1 b 0 0 1, 3 = 0. Another representation is defined by the action on functions in R 3. Notice that exp(ax) f (x, y, z) = f (x + a, y, z a ) y exp(by) f (x, y, z) = f (a, y + b, z + a ) x exp(cz) f (x, y, z) = f (x, y, z + c) Therefore, g(a, b, c) f (x, y, z) = f (x + a, y + b, z + c + b x a ) y Fock space. Let us define the operator N = YX It is easy to see that [N, Y] = ZY, [N, X] = ZX. Suppose that there exists a unit vector v 0 called the vacuum state such that Let us define a sequence of vectors Xv 0 = 0 v n = 1 n! Y n v 0 mathphyshass1.tex; September 4, 013; 9:58; p. 66
5 .4. SPECTRAL DECOMPOSITION 67 By using the properties of the Heisenberg algebra it is easy to show that Yv n = n + 1 v n+1, n 0, Xv n = n Zv n 1, n 1 Therefore Nv n = n Zv n, n 0 Let us define vectors w(b) = exp(by)v 0 = n=0 b n n! v n called the coherent states. Then by using the properties of the Heisenberg algebra we get Xw(b) = bzw(b) Now, suppose that and Y = X Z = I Then, the vectors v n are orthonormal (v n, v m ) = δ nm and are the eigenvectors of the self-adjoint operator N = X X with integer eigenvalues n 0. Then the space span {v n n 0} is called the Fock space and the operators X and X are called the annihilation and creation operators and the operator N is called the operator of the number of particles. mathphyshass1.tex; September 4, 013; 9:58; p. 67
6 68 CHAPTER. FINITE-DIMENSIONAL VECTOR SPACES Note, also that the coherent states are not orthonormal (w(a), w(b)) = eāb Finally, we compute the trace of the heat semigroup operator Tr exp( tx X) = e tn = n=0 1 1 e t Harmonic oscillator. Let D be an anti-self-adjoint operator and Q be a self-adjoint operator satisfying the commutation relations [D, Q] = I The harmonic oscillator is a quantum system with the (self-adjoint positive) Hamiltonian Then the operators H = 1 D + 1 Q X = 1 (D + Q), X = 1 ( D + Q). are the creation and annihilation operators. The operator of the number of particles is and, therefore, the Hamiltonian is N = X X = 1 D + 1 Q 1 H = N + 1 The eigenvalues of the Hamiltonian are λ n = n + 1 with the eigenvectors v n mathphyshass1.tex; September 4, 013; 9:58; p. 68
7 .4. SPECTRAL DECOMPOSITION 69 It is clear that the vectors ψ n (t) = e itλ n v n = exp satisfy the equation [ ( it n + 1 )] (i t H)ψ n = 0 which is called the Schrödinger equation. The vacuum state is determined from the equation (D + Q)v 0 = 0 1 n/ n! ( D + Q)n v 0 and has the form with ψ 0 satisfying v 0 = exp ( 1 ) Q ψ 0 Dψ 0 = 0. The heat trace (also called the partition function) for the harmonic oscillator is 1 Tr exp( th) = sinh(t/).4.9 Exercises 1. Find the eigenvalues of a projection operator.. Prove that the span of all eigenvectors corresponding to the eigenvalue λ of an operator A is a vector space. 3. Let E(λ) = Ker (A λi). Show that: a) if λ is not an eigenvalue of A, then E(λ) =, and b) if λ is an eigenvalue of A, then E(λ) is the eigenspace corresponding to the eigenvalue λ. 4. Show that the operator A λi is invertible if and only if λ is not an eigenvalue of the operator A. mathphyshass1.tex; September 4, 013; 9:58; p. 69
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