Adjoint Operators in Quantum Mechanics: Hermitian Conjugates and Dirac Notation

Overview

In this MIT OpenCourseWare lecture, the instructor develops the concept of adjoint or Hermitian conjugate in quantum mechanics, starting from linear functionals and inner products and building toward the adjoint T† of a linear operator. He shows how any linear functional on a finite dimensional vector space can be represented by an inner product with a unique vector, and defines the dagger operation via the relation = for all vectors u and v. The talk covers essential properties, including (ST)† = T†S† and (T†)† = T, and provides a concrete 3D complex vector space example to connect abstract definitions with matrix representations. The lecture also uses Dirac bracket notation to relate operators, bras, and kets to their matrix elements.

Readers will see how the dagger operation connects to the conjugate transpose in an orthonormal basis and learn about basis dependence and the role of projectors in this formalism.

Intro and Context

The lecture, part of a Quantum Mechanics course, emphasizes the adjoint or Hermitian conjugate of an operator as a central concept in quantum theory. Beginning with linear functionals and inner products, the instructor motivates why every linear functional on a finite dimensional space can be represented as an inner product with a fixed vector. This leads to a basis independent definition of the adjoint operator and a systematic way to translate between different notations used by physicists and mathematicians.

Linear Functionals and Inner Products

A linear functional F on a vector space V maps a vector to a scalar, preserving linearity. The instructor recalls that every linear functional can be represented by with a suitable fixed vector u, given an inner product on V. The proof is sketched using an orthonormal basis {E1, ..., EN}: any V can be expanded as V = sum Vi Ei, and linearity of F together with the inner product structure yields F(V) as a sum of coefficients multiplied by inner products with the basis vectors. The uniqueness of the representing vector follows from an inner product property.

Definition of the Adjoint

For a linear operator T on V, the adjoint T† is defined by the requirement that = for all u, v in V. The speaker shows that this definition indeed yields a linear operator T†, proving linearity by considering complex scalars and using the conjugate in the inner product. He also proves basic composition properties, such as (ST)† = T†S†, and the involution property (T†)† = T.

Worked Example and Matrix Connection

To connect the abstract definition with concrete computations, the lecturer works through an explicit example in C^3. He derives the matrix representation of T† from the condition = by carefully collecting terms in the inner product. An orthonormal basis is used to simplify the calculation, and the resulting T† is shown to be the conjugate transpose of the matrix of T in that basis. The teacher emphasizes that this identification relies on the basis being orthonormal; with a non orthonormal basis, the relation involves a metric matrix G and is not simply a conjugate transpose.

Dirac Notation and Brackets

The talk then recaps how Dirac notation encodes operators via |v> kets and . The identity operator is written as sum_i |i>

Important Theorems and Physical Implications

The lecturer highlights a crucial theorem: in a complex inner product space, if = 0 for all v, then T must be the zero operator. The proof uses clever linear combinations with imaginary coefficients to exploit complex scalar structure, underscoring a key difference between real and complex spaces. He then shows a practical application: if is real for all v, then T is self-adjoint, because T† = T follows from the defining relation and the reality of the quadratic form. Unitary operators, which preserve norms, will be discussed further, with the same theorem yielding essential insights about their structure.

Summary and Takeaways

The lecture consolidates the idea that the adjoint is a basis independent concept defined by a fundamental compatibility with the inner product, yet its practical computation in a given basis reduces to the conjugate transpose when using an orthonormal basis. The Dirac bracket formalism offers a compact, elegant way to encode matrix elements and operator actions. Students are encouraged to move between notations and recognize when the dagger corresponds to the matrix conjugate transpose and when a more general approach is required because the basis is not orthonormal.