Hermitian
Introduction
In mathematics and physics, the term "Hermitian" refers to a class of operators or matrices that are fundamental in various branches of these fields. Hermitian operators, also known as self-adjoint operators, have properties that make them particularly important in quantum mechanics, linear algebra, and functional analysis. This article delves into the definition, properties, applications, and significance of Hermitian operators and matrices.
Definition
A Hermitian operator \( \hat{H} \) on a Hilbert space \( \mathcal{H} \) is defined by the property that it equals its own adjoint, i.e.,
\[ \hat{H} = \hat{H}^\dagger \]
where \( \hat{H}^\dagger \) denotes the adjoint of \( \hat{H} \). In the context of matrices, a Hermitian matrix \( H \) is a square matrix with complex entries that satisfies
\[ H = H^\dagger \]
where \( H^\dagger \) is the conjugate transpose of \( H \). This means that for any Hermitian matrix \( H \), the element in the \( i \)-th row and \( j \)-th column is the complex conjugate of the element in the \( j \)-th row and \( i \)-th column:
\[ H_{ij} = \overline{H_{ji}} \]
Properties
Real Eigenvalues
One of the most significant properties of Hermitian operators and matrices is that their eigenvalues are always real. This can be shown using the definition of a Hermitian operator and the properties of inner products in a Hilbert space. If \( \hat{H} \) is Hermitian and \( \psi \) is an eigenvector with eigenvalue \( \lambda \), then
\[ \hat{H} \psi = \lambda \psi \]
Taking the inner product with \( \psi \) and using the Hermitian property, we get
\[ \langle \psi | \hat{H} \psi \rangle = \lambda \langle \psi | \psi \rangle \]
Since \( \hat{H} \) is Hermitian,
\[ \langle \psi | \hat{H} \psi \rangle = \langle \hat{H} \psi | \psi \rangle = \overline{\lambda} \langle \psi | \psi \rangle \]
Thus,
\[ \lambda \langle \psi | \psi \rangle = \overline{\lambda} \langle \psi | \psi \rangle \]
Since \( \langle \psi | \psi \rangle \) is non-zero, it follows that \( \lambda = \overline{\lambda} \), meaning \( \lambda \) is real.
Orthogonal Eigenvectors
Another important property is that the eigenvectors corresponding to distinct eigenvalues of a Hermitian operator are orthogonal. If \( \hat{H} \) is Hermitian and \( \psi_i \) and \( \psi_j \) are eigenvectors with distinct eigenvalues \( \lambda_i \) and \( \lambda_j \), respectively, then
\[ \hat{H} \psi_i = \lambda_i \psi_i \] \[ \hat{H} \psi_j = \lambda_j \psi_j \]
Taking the inner product of the first equation with \( \psi_j \) and the second with \( \psi_i \), we get
\[ \langle \psi_j | \hat{H} \psi_i \rangle = \lambda_i \langle \psi_j | \psi_i \rangle \] \[ \langle \hat{H} \psi_j | \psi_i \rangle = \lambda_j \langle \psi_j | \psi_i \rangle \]
Using the Hermitian property,
\[ \langle \psi_j | \hat{H} \psi_i \rangle = \langle \hat{H} \psi_j | \psi_i \rangle \]
Thus,
\[ \lambda_i \langle \psi_j | \psi_i \rangle = \lambda_j \langle \psi_j | \psi_i \rangle \]
Since \( \lambda_i \neq \lambda_j \), it follows that \( \langle \psi_j | \psi_i \rangle = 0 \), meaning \( \psi_i \) and \( \psi_j \) are orthogonal.
Applications
Quantum Mechanics
In quantum mechanics, Hermitian operators represent observable quantities. The eigenvalues of these operators correspond to the possible measurement outcomes, which are always real numbers. For example, the Hamiltonian operator, which represents the total energy of a system, is Hermitian. The time evolution of a quantum state is governed by the Schrödinger equation, where the Hamiltonian plays a central role.
Linear Algebra
In linear algebra, Hermitian matrices are used extensively due to their nice spectral properties. They are diagonalizable, and their eigenvalues are real, which simplifies many problems. Hermitian matrices also appear in optimization problems, where they often represent quadratic forms.
Functional Analysis
In functional analysis, Hermitian operators are studied within the context of operator theory. They are essential in the spectral theorem, which provides a framework for understanding the structure of operators on Hilbert spaces. The spectral theorem states that any Hermitian operator can be decomposed into a direct integral of multiplication operators, which generalizes the concept of diagonalization.
Examples
Simple Hermitian Matrices
Consider the following 2x2 Hermitian matrix:
\[ H = \begin{pmatrix} a & b + ic \\ b - ic & d \end{pmatrix} \]
where \( a \), \( b \), \( c \), and \( d \) are real numbers. This matrix satisfies the Hermitian condition \( H = H^\dagger \).
Pauli Matrices
The Pauli matrices are a set of three 2x2 Hermitian matrices that are widely used in quantum mechanics and quantum information theory. They are defined as follows:
\[ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]
These matrices are Hermitian and also satisfy the commutation and anticommutation relations that define the algebra of the Pauli matrices.
Mathematical Formulation
Inner Product Spaces
In the context of inner product spaces, a Hermitian operator \( \hat{H} \) satisfies
\[ \langle \hat{H} \psi | \phi \rangle = \langle \psi | \hat{H} \phi \rangle \]
for all vectors \( \psi \) and \( \phi \) in the space. This property ensures that the operator is self-adjoint, which is crucial for the operator to have real eigenvalues and orthogonal eigenvectors.
Spectral Theorem
The spectral theorem for Hermitian operators states that any Hermitian operator \( \hat{H} \) on a Hilbert space can be expressed as
\[ \hat{H} = \int_{\sigma(\hat{H})} \lambda \, dE(\lambda) \]
where \( \sigma(\hat{H}) \) is the spectrum of \( \hat{H} \) and \( E(\lambda) \) is the projection-valued measure associated with \( \hat{H} \). This theorem generalizes the concept of diagonalization to infinite-dimensional spaces.
Advanced Topics
Unbounded Hermitian Operators
In many physical applications, Hermitian operators are unbounded. For example, the position and momentum operators in quantum mechanics are unbounded Hermitian operators. These operators require careful mathematical treatment to define their domains and ensure they are self-adjoint.
Hermitian Forms
A Hermitian form is a complex-valued function \( H(x, y) \) on a vector space that is linear in one argument and conjugate-linear in the other, and satisfies
\[ H(x, y) = \overline{H(y, x)} \]
Hermitian forms generalize the concept of inner products and are used in various areas of mathematics, including the study of complex manifolds and algebraic geometry.