By Dirac

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N=0 D. The realization of the Hilbert space establishes the isomorphism 2 H as the Hilbert space D of entire functions is called holomorphic representation. It follows from the matrix representations of creation-annihilation operators that in the holomorphic representation a= d , dz a∗ = z 64 2. FOUNDATIONS OF QUANTUM MECHANICS and d 1 . + dz 2 The holomorphic representation is characterized by the property that Hamiltonian H of the harmonic oscillator is diagonal. Moreover, since H has a simple spectrum, every bounded operator which commutes with H is a function of H.

Bounded positive operator A is of trace class if there is an orthonormal basis {en }∞ n=1 for H such that ∞ (Aen , en ) < ∞. 2. Basic axioms. A1. With every quantum system there is an associated separable complex Hilbert space H , in physics terminology called the space of states 4. A2. The set of observables A of a quantum system with the Hilbert space H consists of all self-adjoint operators on H . A3. Set of states S of a quantum system with a Hilbert space H consists of all positive (and hence self-adjoint) M ∈ S1 such that Tr M = 1.

A = ⎢ 0 √ a=⎢ 2 0 0 . ⎦ 0 . . . . . and ⎡ 0 ⎢0 ⎢ ∗ N =a a=⎢ ⎢0 ⎣0 . 0 1 0 0 . 0 0 2 0 . 0 0 0 3 . 2 ⎤ . ⎥ ⎥. ⎦ . Thus in this representation the Hamiltonian of the harmonic oscillator is represented by a diagonal matrix, H=ω N+ 1 2 = diag ω 3ω 5ω , , ,... 2 2 2 . Let D be the Hilbert space of entire functions, ⎫ ⎧ ⎬ ⎨ 1 2 2 −|z|2 2 |f (z)| e d z , D = f entire function : f = ⎭ ⎩ π C where d2 z = 2i dz ∧ d¯ z . The functions zn √ , n = 0, 1, 2, . . , n! form an orthonormal basis for D, and the assignment 2 c= {cn }∞ n=0 ∞ zn → f (z) = cn √ ∈ D n!