Hilbert Space

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A Hilbert Space $\mathcal{H}$ is an infinite dimensional Inner Product Space that is closed regarding the norm, produced by the inner product.


Once we have chosen an inner product (also called skalar product) it defines the norm and the metric of the space. The metric allows to calculate the distance between two vector elements. The vectors $x,y \in \mathcal{H}$ shall be elements of the Hilber Space. We define the norm \begin{align} \|x \| = \sqrt{ \langle x , x \rangle } \end{align} thus the metric is \begin{align} d(x,y)=\|x -y \| =\|y -x \| \end{align}

An Inner Product Space is closed regarding the norm, if there exists a Cauchy Sequence.

A sequence $\{ x_n \}$ in a normed Vector Space is called Cauchy Sequence, if for each $\varepsilon > 0$ there exists a Number $N \in \mathbb{N}$ so that \begin{align} \|x_n -x_m \| \le \varepsilon \quad \forall \quad n,m \geq N \end{align}

In specific \begin{align} \|x_n -x_m \| \rightarrow 0 \quad \text{for} \quad n,m \rightarrow \infty \end{align} Examples for closed Vector Spaces are the vector spaces over $\mathbb{R}$ and $\mathbb{C}$.


Further properties of the Hilbert space are the validity of

  • $\| \phi +\psi \| \leq \| \phi \| + \| \psi\|$ (triangle inequality)
  • $ \langle \phi | \psi \rangle ^2 \leq \langle \psi | \psi \rangle \langle \phi | \phi \rangle $ or equivalently
$\|\langle \phi | \psi \rangle \|^2 \leq \| \psi \|^2 \cdot \| \phi \|^2$ (Chauchy-Schwarz inequality)


Dirac Notation

The scalar product assignes a scalar quantity to each pair of two functions $\phi, \psi, \chi, \dots \in V$ \begin{align} \langle \phi , \psi \rangle = \delta_{\phi}[\psi] \end{align} and is a linear functional since $\delta_{\phi}[\psi + \chi]=\delta_{\phi}[\psi] + \delta_{\phi}[\chi]$. The set of all linear functionals $\{\delta_{\phi}[\psi],\delta_{\chi}[\psi], \dots \}$ in $V$ form the dual vector space $\tilde{V}$ of $V$.


Let us denote a vector in the vector space $V$ by $|\psi \rangle$ and call it a ket-vektor. Every ket-vector has a unique dual in the vector space $\tilde{V}$ we will denote $\langle \phi|$ and call it a bra-vector. Since in this space $|\psi \rangle$ is it's own dual $\langle \psi|$ the space is called self-dual.


For example in the vector space $\mathbb{C}^n$ with the complex inner product

\begin{align} |\psi \rangle=\begin{pmatrix}\alpha_1\\ \alpha_2\\ \vdots \\ \alpha_n\end{pmatrix} \in V \end{align} and \begin{align} \langle \psi|=(\alpha_1^*, \alpha_2^*, \dots ,\alpha^*_n) \in \tilde{V} \end{align} likewise we have for the continuous inner product \begin{align} |\psi \rangle= \psi(x) \in V \end{align} and \begin{align} \langle \psi|=\psi^*(x) \in \tilde{V} \end{align} in order for the product $\langle \psi | \psi \rangle$ called a bra-ket or bracket (that's where the notation bra and ket comes from) to be valid $\psi(x)$ must be square integrable $\psi(x) \in L^2$ \begin{align} \int_{-\infty}^{\infty} |\psi(x)|^2 dx < \infty \Rightarrow \psi(x) \in L^2 \end{align} The vectorspace of functions in $L^2$ is self-dual and there exists a $\tilde{\psi}(k) \in \tilde{V}$ for every $\psi(x) \in V$, since by Parceval's theorem \begin{align} \int_{-\infty}^{\infty} |\psi(x)|^2 dx =\int_{-\infty}^{\infty} |\tilde{\psi}(k)|^2 dk \end{align}


Completeness and Separability

A Vector Space is called complete (or a Banach Space) if every sequence $\{\psi_r \}_{r \in \mathbb{N}}$ is a Cauchy-Sequence, so that it holds for every sequence \begin{align} \|\psi_n -\psi_m \| \rightarrow 0 \quad \text{for} \quad n,m \rightarrow \infty \end{align} If this is the case, every limit \begin{align} \lim_{n \rightarrow \infty} \psi_n =\psi \end{align} exists and every sequence $\psi_n$ converges to $\psi$. In specific \begin{align} |\langle \phi_n | \psi \rangle -\langle \phi_m | \psi \rangle | = | c_n -b_m| = 0 \quad \text{for} \quad n,m \rightarrow \infty \end{align} in other words in the limit $\quad n,m \rightarrow \infty$ it's true that $c_n=b_m$. This is one requirement for the representation to be manifest and is used in the completeness relation. \begin{align} \sum_{n=1}^{\infty} | \phi_n \rangle \langle \phi_n |=\mathbb{1} \end{align}


An infinite dimensional Vector Space is seperable if and only if there is a countable orthonormal basis $\{\phi_n\}_{n \in \mathbb{N}}$ in $\mathcal{H}$. In a complete seperable Hilbert Spaces every element $\psi \in L^2$ can be uniquely represented as a convergent series \begin{align} \psi = \sum_{n=1}^{\infty} c_n |\phi_n \rangle \end{align} with coefficients \begin{align} \langle \phi_m | \psi \rangle &= \sum_{n=1}^{\infty} c_n \langle \phi_m |\phi_n \rangle \\ c_n &= \langle \phi_m | \psi \rangle \end{align} Once a basis $|\phi_n \rangle$ is given the coefficients are uniquely determined, thus eather $\psi$ or the coefficients $c_n$ are sufficient to specify a state vector completely. Furthermore, the property \begin{align} \langle \phi_m |\phi_n \rangle=\delta_{ij} \end{align} called orthogonality is valid for separable Hilbert Spaces.



Furhter Reading:

  • Eduard Prugovečki - Quantum Mechanics in Hilbert Space