Introduction: The Arena of Quantum Mechanics
A Hilbert Space is an inner product space that is complete—meaning limits of convergent sequences always exist within the space. This is the mathematical stage on which all of quantum mechanics is performed.
What is Completeness?
A space is complete if every Cauchy sequence converges to a point in the space. A Cauchy sequence is one where terms get arbitrarily close to each other:
\[\forall \epsilon > 0, \exists N : m, n > N \Rightarrow \|\vec{v}_m - \vec{v}_n\| < \epsilon\]Example of Incompleteness: The rational numbers \(\mathbb{Q}\) are not complete. The sequence \(1, 1.4, 1.41, 1.414, \ldots\) converges to \(\sqrt{2}\), which is not in \(\mathbb{Q}\).
Completing the Space: Adding all limits gives \(\mathbb{R}\), which is complete.
The Hilbert Space Definition
A Hilbert Space \(\mathcal{H}\) is:
- A vector space over \(\mathbb{C}\)
- Equipped with an inner product \(\langle \cdot, \cdot \rangle\)
- Complete with respect to the norm induced by the inner product
Key Examples
Example 1: \(\mathbb{C}^n\)
The space of \(n\)-dimensional complex vectors with the standard inner product is a finite-dimensional Hilbert space. It describes systems with finitely many states (like spin).
Example 2: \(L^2(\mathbb{R})\)
The space of square-integrable functions:
\[L^2(\mathbb{R}) = \left\{ f : \mathbb{R} \to \mathbb{C} \mid \int_{-\infty}^{\infty} |f(x)|^2 \, dx < \infty \right\}\]with inner product:
\[\langle f, g \rangle = \int_{-\infty}^{\infty} \overline{f(x)} g(x) \, dx\]This is the Hilbert space for a particle on a line—wavefunctions live here.
Example 3: \(\ell^2\)
The space of square-summable sequences:
\[\ell^2 = \left\{ (c_1, c_2, \ldots) \mid \sum_{n=1}^{\infty} |c_n|^2 < \infty \right\}\]This is the Hilbert space for systems with countably many states (like the harmonic oscillator energy levels).
Completeness of a Basis
A basis \(\{|n\rangle\}\) for a Hilbert space is complete if any state can be expanded:
\[|\psi\rangle = \sum_n c_n |n\rangle\]The completeness relation (resolution of identity) is:
\[\sum_n |n\rangle\langle n| = \hat{I}\]The Quantum Connection
The Hilbert space structure guarantees that quantum mechanics is mathematically consistent. The completeness of \(L^2\) ensures that wavefunctions can be expanded in any complete basis (position, momentum, energy). Without completeness, infinite sums might not converge, and superposition would be ill-defined.