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--- hamburgminicourse2017:why_hilbert_space [2017/03/29 11:42] – markus
+++ hamburgminicourse2017:why_hilbert_space [2017/04/03 16:11] (current) – markus
@@ Line 1: / Line 1: @@
 ====== Why Hilbert space? ======
-**Pragmatic answer:** A Hilbert space is like an euclidean space with possibly infinite dimensions, i.e. it has a //lot// of structure, like that of a vector space, a scalar product, and completeness! Sometimes a Banach space (with just a norm and completeness) is enough.
+**Pragmatic answer.** A Hilbert space is like an euclidean space with possibly infinite dimensions, i.e. it has a //lot// of structure, like that of a vector space, a scalar product, and completeness! Sometimes a Banach space (with just a norm and completeness) is enough.
-**Historical answer:** Modern day QM started out as "Transformationstheorie" (Born-Jordan 1925, Dirac 1927) that unifies Heisenberg's matrix mechanics and Schrödinger's wave mechanics. The "Transformation" is actually the diagonalization of a matrix and refers to eigenvalue problems involving Heisenberg's infinite-dimensional matrices. Those matrices represent the usual PDOs of QM, but written as integral operators involving Dirac-deltas.
+**Historical answer.** Modern day QM started out as "Transformationstheorie" (Born--Jordan 1925, Dirac 1927) that unifies Heisenberg's matrix mechanics and Schrödinger's wave mechanics. The "Transformation" is actually the diagonalization of a matrix and refers to eigenvalue problems involving Heisenberg's infinite-dimensional matrices. Those matrices represent the usual PDOs of QM, but written as integral operators involving Dirac-deltas.
 \[ \partial_x^n \quad\longleftrightarrow\quad \delta^{(n)}(x-x') \]
@@ Line 11: / Line 11: @@
 \]
-(Here the $x_n$ in the sum term can be the coordinates of the eigenstates of $H$ as an infinite matrix realtive to some basis, represented as a sequence $\mathbb{N} \rightarrow \mathbb{C}$.)
+(Here the $x_n$ in the sum term can be the coordinates of the eigenstates of $H$ as an infinite matrix relative to some basis, represented as a sequence $\mathbb{N} \rightarrow \mathbb{C}$.)
-The **Riesz-Fischer theorem** (1907) then told von Neumann, that the spaces of all such (normalized) objects are //isomorphic// as Hilbert spaces.
+The **Riesz--Fischer theorem** (1907) then told von Neumann, that the spaces of all such (normalized) objects are //isomorphic// as Hilbert spaces.
 \[
 L^2(\Omega) \simeq \ell^2(\mathbb{N})
 \]
-e.g., $L^2(\mathbb{R}/2\pi) \simeq \ell^2(\mathbb{Z})$ with Fourier series or with any other orthonormal basis $(e_i)_{i \in I}$ of $L^2(\Omega)$:
+e.g., $L^2(\mathbb{R}/2\pi) \simeq \ell^2(\mathbb{Z})$ with Fourier series or with any other orthonormal basis $(e_i)_{i \in \mathbb{N}}$ of $L^2(\Omega)$:
 \[
-\psi \quad\longleftrightarrow\quad (\langle \psi,e_i \rangle)_{i \in I}.
+\psi \quad\longleftrightarrow\quad (\langle \psi,e_i \rangle)_{i \in \mathbb{N}}.
 \]
-The isomorphy holds on the level of //state spaces// but not on the level of //configuration spaces//, i.e. $\mathbb{R}^3$ and $\mathbb{N}$ (the numbering of orbitals). This explains the focus of QM on the Hilbert space (state space) instead of configuration space.
+The isomorphy holds on the level of //state spaces// but not on the level of //configuration spaces//, e.g. $\Omega=\mathbb{R}^3$ and $\mathbb{N}$ (the numbering of orbitals). This explains the focus of QM on the Hilbert space (state space) instead of configuration space.
  >> [[Why self-adjoint operators]]?