9  Hilbert Spaces

9.1 Definition of a Hilbert Space

Finally we get to the most important definition of the course (you will need to learn this!!):

A Hilbert space is an inner product space that is also a complete metric space with respect to the norm induced by the inner product.

9.2 Examples of Hilbert Spaces

  • The completion of any inner product space \(\mathcal{I}\), using as the metric the norm induced by the inner product, is a Hilbert space: \(\mathcal{I}\hookrightarrow\mathcal{H}\).

  • All finite dimensional inner product spaces are Hilbert spaces. An example in quantum mechanics is the 2D complex inner product space describing spin-1/2.

  • However, there are also infinite-dimensional Hilbert spaces, such as the space spanned by energy eigenstates of the particle in a box (1D quantum well), or particle on a ring.

9.3 Key Example of a Hilbert Space: Quantum Mechanics

One presentation of the axioms of quantum mechanics is as follows.

  • [QM1] (states): States of a quantum system are represented by equivalence classes of unit vectors (kets) \([|\psi\rangle]\) in a Hilbert space \(\mathcal{H}\), \([|\psi\rangle]\in\mathcal{H}\), under \(|\psi\rangle\sim\exp(i\theta)|\psi\rangle\).

  • [QM2] (observables):

    • [QM2a]: Observables are represented by self-adjoint linear maps (operators) \(\hat{A}:\)\(\mathcal{H}\rightarrow\mathcal{H}\).

      • Corollary: by the spectral theorem, \(\hat{A}=\int_{\mathbb{R}}\lambda\textrm{d}\hat{E}_{A}(\lambda)\).
      • Corollary: if the spectrum is pure point, then \(\hat{A}=\sum_{n}^{\textrm{dim}\mathcal{H}}a_n|a_n\rangle\langle a_n|\).

    • [QM2b]: One such observable/operator pair is the energy/Hamiltonian \(\hat{H}\).

  • [QM3] (measurement):

    • [QM3a]: The possible results of a measurement of \(\hat{A}\) on state \(|\psi\rangle\) are in the spectrum of \(\hat{A}\).

      • Corollary: if the spectrum is pure point, then the outcomes are \(a_n\).

    • [QM3b] (Born rule): The probability of obtaining a result in the spectral set \(\Delta\) is \(\langle\psi|\hat{E}_A\left(\Delta\right)|\psi\rangle\).

      • Corollary: if the spectrum is pure point, then the probability is \(\left|\langle a_{n}|\psi\rangle\right|^{2}\).

    • [QM3c]: Immediately after obtaining result \(a_{n}\) in a discrete spectrum, the system is in the corresponding eigenstate \(|a_{n}\rangle\).

  • [QM4] (Dynamics): In the absence of measurement, states evolve unitarily: \(|\psi(t)\rangle=\exp\left(-i\hat{H}t/\hbar\right)|\psi(0)\rangle\).

9.4 Resolution of the Identity

9.4.1 Pre-amble: types of infinity

The number of elements in a set is called the cardinality of the set. We denote the cardinality of the integers to be \(\aleph_{0}\) (aleph-null). It is also called countable infinity. This is known to be the smallest infinite cardinality. Interestingly, the cardinality of the rationals is also \(\aleph_{0}\).

The cardinality of the reals, aka the cardinality of the continuum, is denoted \(\mathfrak{c}\). Cantor’s diagonal argument proves that \(\mathfrak{c}=2^{\aleph_{0}}>\aleph_{0}\), i.e. the cardinality of the continuum is strictly greater than the cardinality of the integers.

9.4.2 Separable Hilbert spaces

Definition: A Hilbert space is separable iff it admits an orthonormal basis of dimension at most \(\aleph_{0}\).

N.B. Non-separable Hilbert spaces exist. For example, a chain of \(N\) spins-1/2 (the 1D Ising model) has Hilbert space dimension \(2^{N}\), and so an infinitely long chain has dimension \(2^{\aleph_{0}}>\aleph_{0}\) . But often we are safe in assuming our Hilbert space is separable, and separable spaces will be the focus here.

The statement that a separable Hilbert space is complete is equivalent to the statement that one can write the

Resolution of the identity:

given an orthonormal basis \(\left\{ |e_{i}\rangle\right\}\) for a separable Hilbert space \(\mathcal{H}\),

\[\mathbb{I}=\sum^{\text{dim}\mathcal{H}}_{i=1}|e_{i}\rangle\langle e_{i}|\]

where \(\text{dim}\mathcal{H}\) is the dimension of the Hilbert space (the number of linearly independent basis vectors \(|e_{i}\rangle\)).

To motivate this, consider the simple case of a 2D real inner product space \(\mathcal{I}=\left(\mathbb{R}^{2},\langle\cdot|\cdot\rangle\right)\), which, with the norm induced by the inner product, is a Hilbert space. Here, the dimension of the Hilbert space is finite (2), so it is separable.

Let

\[|v\rangle=v_{1}|e_{1}\rangle+v_{2}|e_{2}\rangle \tag{9.1}\]

where \(\left\{ |e_{i}\rangle\right\}\), \(i\in\left[1,2\right]\) form a complete orthonormal basis for \(\mathcal{I}\). That is,

\[\langle e_{i}|e_{j}\rangle=\delta_{ij}\]

where we have defined The Kronecker delta:

\[\delta_{ij}\triangleq\begin{cases} \begin{array}{c} 1,\\ 0, \end{array} & \begin{array}{c} i=j\\ i\ne j. \end{array}\end{cases} \tag{9.2}\]

Acting \(\langle e_{i}|\) from the left on Equation 9.1 gives

\[v_{i}=\langle e_{i}|v\rangle\]

and so

\[|v\rangle=\left(\langle e_{1}|v\rangle\right)|e_{1}\rangle+\left(\langle e_{2}|v\rangle\right)|e_{2}\rangle\]

or

\[|v\rangle=|e_{1}\rangle\langle e_{1}|v\rangle+|e_{2}\rangle\langle e_{2}|v\rangle.\]

Factoring out the vector, we find

\[|v\rangle=\left(|e_{1}\rangle\langle e_{1}|+|e_{2}\rangle\langle e_{2}|\right)|v\rangle.\]

Since \(|v\rangle\) is any vector in the space, the term in parentheses must be the identity operator. That is,

\[\mathbb{I}=|e_{1}\rangle\langle e_{1}|+|e_{2}\rangle\langle e_{2}|.\]

The same reasoning follows for any separable Hilbert space, defined to have a dimension that is at most \(\aleph_{0}\).