Spectral Decomposition of Hermitian Operators

Welcome back. In our previous lesson, we categorized the fundamental operators in quantum mechanics—Hermitian, unitary, and projection operators—and uncovered their core properties. We established that Hermitian operators, which represent physical observables, possess real eigenvalues and orthogonal eigenvectors. However, we left a crucial third property, completeness, to be explored.

This lesson is dedicated to that property, which is formally captured by the Spectral Theorem. Our goal is to apply this theorem to decompose any Hermitian operator in terms of its eigenvalues and eigenstates. This decomposition is not merely an abstract mathematical tool; it is the foundation for calculating functions of operators (like the time evolution operator) and provides a powerful geometric picture of how operators transform quantum states. We will develop this for both discrete and continuous spectra, adding layers of mathematical rigor that will be valuable for your advanced studies.

The Spectral Theorem: Decomposing Operators

The spectral theorem formalizes the idea that the orthonormal eigenvectors of a Hermitian operator are complete—that is, they form a basis for the Hilbert space. This allows any state vector to be expressed as a linear combination of these eigenvectors. For a finite-dimensional space, this is a standard result from linear algebra. Its real power in quantum mechanics comes from its extension to infinite-dimensional Hilbert spaces.

Let's begin with the case of a discrete spectrum, which applies to many bound state problems like the quantum harmonic oscillator.

Discrete Spectra

If the eigenvectors ${∣ a_{n} ⟩}$ of a Hermitian operator $A$ form a complete orthonormal basis, we can write the identity operator $I$ as a sum over the projectors onto these basis states. This is the completeness relation, or resolution of unity:

I = n \sum ∣ a_{n} ⟩ ⟨ a_{n} ∣

where $⟨ a_{m} ∣ a_{n} ⟩ = δ_{mn}$ .

With this tool, we can derive the spectral decomposition of $A$ . By applying $A$ to the identity operator, we get:

A = A I = A (n \sum ∣ a_{n} ⟩ ⟨ a_{n} ∣) = n \sum A ∣ a_{n} ⟩ ⟨ a_{n} ∣

Since $∣ a_{n} ⟩$ is an eigenvector of $A$ with eigenvalue $a_{n}$ , we have $A ∣ a_{n} ⟩ = a_{n} ∣ a_{n} ⟩$ . Substituting this in, we arrive at the spectral decomposition:

A = n \sum a_{n} ∣ a_{n} ⟩ ⟨ a_{n} ∣

This elegant equation states that any Hermitian operator can be expressed as a sum of its corresponding eigenvalues, each weighting the projection operator onto its associated eigenspace.

The following reading provides a concise derivation and introduces the concept of using this decomposition to define functions of an operator.

November 2009 The Dirac Approach to Quantum Theory ...

This text formalizes the spectral decomposition for a Hermitian operator with a discrete spectrum.

Please read Section 14, "Spectral Decomposition of a Hermitian Operator" (p. 17-18). Focus on the derivation of the spectral decomposition and its interpretation. Then, see how this naturally leads to the definition of a function of an operator, f(A).

The ability to define functions of an operator, $f (A) = \sum_{n} f (a_{n}) ∣ a_{n} ⟩ ⟨ a_{n} ∣$ , is immensely powerful. It allows us to compute expressions like $A$ , $e^{A}$ , or $sin (A)$ in a well-defined way. This is precisely the formalism needed to construct the time evolution operator $U (t) = e^{- i H t /ℏ}$ from the Hamiltonian $H$ .

Let's see this in action with a concrete example involving the Pauli spin matrices, which are central to describing spin-1/2 systems.

Chapter 3 Mathematical Formalism of Quantum Mechanics

This reading provides a detailed, step-by-step example of applying the spectral theorem.

In Section 3.4.2, "Spectral Theorem" (p. 64), follow the Example that works with the Pauli matrix \sigma_x. Pay attention to the entire calculation: Finding the eigenvalues. Determining the normalized eigenvectors, |+\rangle and |-\rangle. Constructing the projection operators P_+ and P_-. Writing the spectral decomposition of \sigma_x. Finally, see how this is used to calculate the exponential exp(-i\alpha\sigma_x/2), which represents a rotation.

A Geometric Interpretation

The spectral decomposition offers a powerful geometric picture of a linear transformation. Applying an operator $A$ to a vector $∣ ψ ⟩$ can be understood as a three-step process:

Change of Basis: Project the vector $∣ ψ ⟩$ onto the eigenbasis of $A$ . This is equivalent to rotating your coordinate system to align with the eigenvectors of $A$ .
Scaling: Stretch or shrink the components of the vector along each of these new coordinate axes by a factor equal to the corresponding eigenvalue.
Inverse Change of Basis: Rotate the coordinate system back to the original standard basis.

This is the essence of matrix diagonalization, where a matrix $A$ is written as $A = U D U^{†}$ , with $U$ being the unitary matrix whose columns are the eigenvectors (the change-of-basis matrix) and $D$ being the diagonal matrix of eigenvalues (the scaling factors).

The following image provides a beautiful visualization of this process for a symmetric matrix $S$ (a real Hermitian matrix).

This image visualizes the action of a matrix S on a set of vectors V. The direct transformation S.V is shown in the top right. The bottom row decomposes this transformation into three steps: a change of basis to the eigenvector coordinate system (U⁻¹V), scaling along the new axes by the eigenvalues (Λ.U⁻¹V), and a change back to the original basis (U.Λ.U⁻¹V), which reconstructs the final transformed shape.

Continuous Spectra

The framework of spectral decomposition extends naturally to operators with a continuous spectrum, such as the position ( $x^$ ) and momentum ( $p^$ ) operators. The key difference is that sums over discrete indices are replaced by integrals over continuous variables.

For the position operator $x^$ , the eigenvectors are denoted $∣ x ⟩$ , where $x$ is a continuous real number. These states represent a particle perfectly localized at position $x$ .
The completeness relation becomes an integral:

I = \int_{- \infty}^{\infty} ∣ x ⟩ ⟨ x ∣ d x

And the spectral decomposition of the position operator is:

x^= \int_{- \infty}^{\infty} x ∣ x ⟩ ⟨ x ∣ d x

Here, the operator $x^$ is expressed as a weighted integral over its projectors, where the weight at each point is simply the position eigenvalue $x$ itself.

A subtlety arises with these continuous eigenkets. States like $∣ x ⟩$ are not normalizable and thus are not, strictly speaking, members of the Hilbert space $L^{2} (R)$ . They are sometimes called "improper states." This technical issue is resolved formally by introducing a larger structure called a rigged Hilbert space (or Gelfand triple), a concept that is good to be aware of as you delve deeper into mathematical physics.

For our purposes, we can work with them using the generalized orthonormality condition:

⟨ x^{'} ∣ x ⟩ = δ (x^{'} - x)

where $δ (x^{'} - x)$ is the Dirac delta function. This reflects that the "inner product" of two different position eigenstates is zero, while the "norm squared" of any single eigenstate is infinite.

The following reading elaborates on this extension to continuous spectra.

November 2009 The Dirac Approach to Quantum Theory ...

This text extends the concepts of spectral decomposition and completeness to operators with continuous spectra.

Please read Section 16, "Operators With Continuous Spectra" (p. 20-21). Pay attention to: The statement of the completeness relation and spectral decomposition as integrals. The connection between the abstract bracket \langle x|\psi\rangle and the familiar wavefunction \psi(x). The introduction of the generalized orthonormality condition using the Dirac delta.

A Deeper Look: The Structure of the Spectrum

So far, we have used "spectrum" as a synonym for the set of eigenvalues. While this is true for finite-dimensional matrices, the situation in infinite-dimensional Hilbert spaces is more nuanced. For a bounded linear operator $T$ , the spectrum $σ (T)$ is defined as the set of complex numbers $λ$ for which the operator $T - λ I$ is not invertible.

Non-invertibility can happen for several reasons, leading to a decomposition of the spectrum into three disjoint parts. The following video provides a clear and concise overview.

Functional Analysis 28 | Spectrum of Bounded Operators

This video from The Bright Side of Mathematics gives a clear breakdown of the different parts of an operator's spectrum.

Please watch the segment from Point, Continuous, and Residual Spectrum. The speaker lucidly defines the three distinct ways an operator can fail to be invertible, which correspond to the three parts of the spectrum.

To summarize the video's points:

Point Spectrum ( $σ_{p} (T)$ ): The set of $λ$ for which $T - λ I$ is not injective. This corresponds to the familiar eigenvalues, where there exists a non-zero vector $∣ ψ ⟩$ such that $T ∣ ψ ⟩ = λ ∣ ψ ⟩$ .
Continuous Spectrum ( $σ_{c} (T)$ ): The set of $λ$ for which $T - λ I$ is injective but not surjective, and its range is dense in the Hilbert space. This is typical for scattering states in quantum mechanics.
Residual Spectrum ( $σ_{r} (T)$ ): The set of $λ$ for which $T - λ I$ is injective, but its range is not dense in the Hilbert space.

For the Hermitian operators of quantum mechanics, the situation simplifies considerably:

The spectrum is entirely real: $σ (A) \subset R$ .
The residual spectrum is empty: $σ_{r} (A) = \emptyset$ .
The spectrum of a Hermitian operator therefore consists only of a point part (eigenvalues) and a continuous part.

For an important subclass of operators known as compact self-adjoint operators, the structure is even simpler. These operators are crucial as they describe the Hamiltonians of many bound systems (e.g., a particle in a box). A key result from functional analysis states that for a non-trivial compact self-adjoint operator, the spectrum (excluding zero) consists entirely of a discrete set of eigenvalues which can only accumulate at zero. In this case, the continuous spectrum is absent (except possibly at $λ = 0$ ). This powerfully connects the abstract theory back to the discrete energy levels you are familiar with.

The following short clip from an MIT OpenCourseWare lecture on functional analysis makes this crucial point.

Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator

This clip highlights the special nature of the spectrum for compact self-adjoint operators.

Please watch the segment from Fredholm Alternative, which introduces the idea that for these operators, any non-zero point in the spectrum must be an eigenvalue. Then, jump to Spectrum vs. Eigenvalues, where the lecturer summarizes the conclusion: the non-zero spectrum is nothing but eigenvalues.

Conclusion

Today we have fully developed the spectral theorem, one of the most important mathematical results underpinning quantum mechanics.

Key Takeaways:

The spectral theorem allows any Hermitian operator $A$ to be decomposed in terms of its eigenvalues and eigenvectors (or eigenprojectors).
For a discrete spectrum, the decomposition is a sum: $A = \sum_{n} a_{n} ∣ a_{n} ⟩ ⟨ a_{n} ∣$ .
For a continuous spectrum, it is an integral: $A = \int a (k) ∣ k ⟩ ⟨ k ∣ d k$ .
This decomposition provides a practical method for defining and calculating functions of operators, like $e^{- i H t /ℏ}$ .
Geometrically, the theorem represents an operator's action as a change of basis to the eigenbasis, a scaling along the new axes, and a change back.
In infinite dimensions, the spectrum is formally defined by non-invertibility and can be split into point, continuous, and residual parts. For Hermitian operators, the spectrum is real and has no residual part.

The spectral theorem provides the framework for representing observables. In our next lesson, we will use it to address a practical and fundamental question: under what conditions can two different physical quantities be measured simultaneously with arbitrary precision? This will lead us to the concept of commuting operators and simultaneous diagonalization.

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