From Finite to Infinite Dimensions: A First Look
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You should be comfortable with the definitions of inner products, linear operators, and the spectral theorem for finite-dimensional self-adjoint operators, as established in previous lectures.
Why this matters
Physical systems often possess infinitely many degrees of freedom, requiring a generalization of the finite-dimensional tools we have mastered.
The transition from finite to infinite dimensions is not merely a matter of changing the index range from \(1\) to \(n\) to \(1\) to \(\infty\). It involves subtle topological distinctions between Hilbert spaces and Banach spaces, the convergence of operator norms, and the emergence of continuous spectra. Understanding these differences is crucial for correctly applying quantum mechanical postulates to systems like the harmonic oscillator or the hydrogen atom, where the state space is infinite-dimensional.
The Infinite Dimensional Leap
In finite dimensions, every linear operator on a vector space is automatically continuous, and the space is complete. In infinite dimensions, these properties are no longer guaranteed. We must work within a Hilbert space, which is a complete inner-product space. This means that every Cauchy sequence of vectors converges to a limit within the space. This completeness is essential for defining limits of operators and ensuring that physical states remain well-defined. The key difference is that in infinite dimensions, an operator may not have a discrete set of eigenvectors that form a basis. Instead, we may encounter a continuous spectrum, where the "eigenvectors" are not normalizable and do not belong to the Hilbert space itself, but rather to a larger rigged Hilbert space. This distinction is fundamental to the mathematical rigor of quantum mechanics.
Convergence and Operator Norms
When dealing with infinite-dimensional spaces, we must specify the type of convergence we are considering. The most common is convergence in norm, defined by the operator norm \(\lVert \hat{A} \rVert = \sup_{\lVert \psi \rVert = 1} \lVert \hat{A} \psi \rVert\). An operator is bounded if this norm is finite. Bounded operators are continuous, meaning that small changes in the input vector result in small changes in the output vector. This is a critical property for physical observables, as it ensures that measurements are stable and well-behaved. In contrast, unbounded operators, such as the momentum operator \(\hat{p} = -i\hbar \frac{d}{dx}\), are not continuous in the norm topology. However, they are densely defined, meaning their domain is a dense subspace of the Hilbert space. This distinction is vital because many important physical operators are unbounded, and their spectral properties differ significantly from those of bounded operators.
Discrete vs. Continuous Spectra
In finite dimensions, the spectrum of a self-adjoint operator consists entirely of discrete eigenvalues. In infinite dimensions, the spectrum can be discrete, continuous, or a combination of both. The discrete spectrum corresponds to bound states, such as the energy levels of the hydrogen atom. The continuous spectrum corresponds to scattering states, such as the energy of a free particle. The spectral theorem for self-adjoint operators in infinite dimensions states that the operator can be decomposed into a sum (or integral) of projections onto its spectral subspaces. This is a powerful generalization of the finite-dimensional case, where the decomposition is a finite sum of projectors. The continuous part of the spectrum is handled using the spectral measure, which assigns a projection-valued measure to Borel sets in the spectrum. This allows us to define functions of operators, such as \(f(\hat{A})\), in a rigorous manner.
Orthonormal Bases in Infinite Dimensions
Even in infinite dimensions, we can often find orthonormal bases, provided the Hilbert space is separable. A Hilbert space is separable if it contains a countable dense subset. Most physical Hilbert spaces, such as \(L^2(\mathbb{R}^3)\), are separable. In such spaces, we can apply the Gram–Schmidt process to any countable linearly independent set to generate an orthonormal basis. This basis allows us to expand any vector in the space as an infinite series. The convergence of this series is in the norm topology, meaning that the partial sums converge to the vector in the Hilbert space norm. This is a crucial tool for practical calculations, as it allows us to approximate infinite-dimensional problems with finite-dimensional ones by truncating the series. The accuracy of the approximation depends on the rate of convergence, which is determined by the properties of the operator and the vector.
The derivations
Gram–Schmidt Orthonormalization
Let \((V,\langle\cdot,\cdot\rangle)\) be an inner-product space over \(F=\mathbb{R}\) or \(\mathbb{C}\), and let \(\{v_1,\dots,v_n\}\subset V\) be linearly independent. Then there exists an orthonormal set \(\{e_1,\dots,e_n\}\) — meaning \(\langle e_i,e_j\rangle=\delta_{ij}\) — such that for every \(k\le n\), \(\operatorname{span}\{e_1,\dots,e_k\}=\operatorname{span}\{v_1,\dots,v_k\}\). The set is produced constructively, and is unique up to the phase (sign, in the real case) of each \(e_k\).
Spectral Theorem for Self-Adjoint Operators
Let \( \hat{A} \) be a self-adjoint operator on a finite-dimensional complex inner-product space \( V \) with \( \dim V = n \), so that \( \hat{A}=\hat{A}^{\dagger} \). Then \( \hat{A} \) has real eigenvalues, its eigenvectors belonging to distinct eigenvalues are orthogonal, and \( V \) possesses an orthonormal basis \( \{|e_1\rangle,\dots,|e_n\rangle\} \) consisting entirely of eigenvectors of \( \hat{A} \). Equivalently, the unitary \( \hat{U} \) whose columns are those eigenvectors diagonalizes \( \hat{A} \): \( \hat{U}^{\dagger}\hat{A}\hat{U}=\hat{\Lambda}=\mathrm{diag}(\lambda_1,\dots,\lambda_n) \), and \( \hat{A}=\sum_{k}\lambda_k\,|e_k\rangle\langle e_k| \).
A common pitfall is assuming that the eigenvectors of an unbounded self-adjoint operator form a complete orthonormal basis in the same sense as in finite dimensions. In reality, for operators with continuous spectra, the "eigenvectors" are generalized eigenvectors that do not belong to the Hilbert space. They are elements of a larger space, and their "orthogonality" is expressed via delta functions rather than Kronecker deltas. This distinction is essential for correctly normalizing states and computing expectation values in scattering problems. C
Worked example
Problem. Consider the Hilbert space \(L^2([0,1])\) with the inner product \(\langle f, g \rangle = \int_0^1 f(x)^* g(x) \, dx\). Let \(\hat{A}\) be the multiplication operator defined by \((\hat{A}f)(x) = x f(x)\). Determine the spectrum of \(\hat{A}\) and explain why it does not have a discrete set of eigenvectors that form a basis.
- Identify the domain of \(\hat{A}\) as all \(f \in L^2([0,1])\) such that \(x f(x) \in L^2([0,1])\). Since \(x\) is bounded on \([0,1]\), \(\hat{A}\) is a bounded operator.
- Suppose \(\hat{A} f = \lambda f\). Then \(x f(x) = \lambda f(x)\) for almost every \(x \in [0,1]\). This implies \(f(x) = 0\) for almost every \(x \neq \lambda\). Since \(\lambda\) is a single point, \(f(x) = 0\) almost everywhere, so \(f = 0\). Thus, there are no non-trivial eigenvectors in \(L^2([0,1])\).
- The spectrum of \(\hat{A}\) is the essential range of the function \(x\) on \([0,1]\), which is the interval \([0,1]\). This is a continuous spectrum.
- Since there are no eigenvectors in the Hilbert space, we cannot form a basis of eigenvectors. Instead, we use the spectral theorem for bounded self-adjoint operators, which expresses \(\hat{A}\) as an integral over its spectrum.
Units check. The operator \(\hat{A}\) multiplies by \(x\), which has units of length if \(x\) is a position coordinate. The spectrum \([0,1]\) has the same units, consistent with the eigenvalue equation \(\hat{A}f = \lambda f\).
Three quick tests
- Is every linear operator on an infinite-dimensional Hilbert space bounded?
Answer
No, only those with finite operator norm are bounded. Unbounded operators, like the momentum operator, are densely defined but not continuous in the norm topology. - What is the difference between a discrete and a continuous spectrum?
Answer
A discrete spectrum consists of isolated eigenvalues with finite multiplicity, corresponding to bound states. A continuous spectrum is an interval of values where the operator has no eigenvectors in the Hilbert space, corresponding to scattering states. - Can an infinite-dimensional Hilbert space have an orthonormal basis?
Answer
Yes, if it is separable. Most physical Hilbert spaces are separable and admit a countable orthonormal basis, allowing vectors to be expanded as convergent infinite series.
Problem set for PU-104 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-075 · D-082 · Threads: symmetry, waves, matter