Linear Operators in Quantum Mechanics: Vector Spaces, Operators, and Eigenvalues | MIT OpenCourseWare

Overview

This MIT OpenCourseWare lecture explores linear operators as maps from a vector space to itself, showing how they generalize matrices and enable analysis in abstract and infinite-dimensional settings. The talk covers key ideas such as linearity, operator spaces, and the relationship between operators and matrices, setting the stage for eigenvalues, eigenvectors, and inner products in quantum mechanics.

Through concrete examples like polynomials, left and right shift operators, and simple differentiation and multiplication operators, the instructor demonstrates foundational concepts such as the null space, range, injectivity, surjectivity, and the basics of invertibility. The discussion also highlights the distinction between basis-dependent and basis-independent quantities and previews how eigenvalues and eigenvectors arise from invariant subspaces.

Introduction to Linear Operators

The lecture begins by defining linear maps T: V -> W and then specializes to operators on a vector space V where the domain and codomain coincide. Linear operators act like multiplication in the sense that they respect addition and scalar multiplication. The speaker emphasizes the utility of this abstract perspective, including the ability to handle infinite-dimensional spaces and to discuss basis-independent properties.

Examples of Vector Spaces and Operators

Several illustrative spaces are introduced: the real polynomials in one variable, with differentiation and multiplication by x as linear operators; and the space of infinite sequences, with left and right shift operators. These examples reveal how operators can resemble matrices in action, while also highlighting the peculiarities of infinite dimensions, such as information loss in shift operations.

• Zero operator and identity operator
• Left shift and right shift on sequences
• Differentiation and multiplication by x on polynomials

The Space of Operators L(V)

The set of all linear operators from V to itself, denoted L(V), is itself a vector space, with an additional multiplication defined by composition. The multiplication is associative but not commutative, and invertibility is not guaranteed for all operators. The talk gives intuitive checks of associativity and identity via action on arbitrary vectors.

Injectivity, Surjectivity, and Invertibility

The lecture defines injectivity (one-to-one) and surjectivity (onto) in terms of the null space and the range. In finite dimensions, injectivity is equivalent to surjectivity for linear maps, making invertibility equivalent to both properties. The instructor discusses how the dimension formula dim(V) = dim(Null(T)) + dim(Range(T)) underpins these relations and provides intuition with simple examples.

From Operators to Matrices

By choosing a basis, any operator can be represented as a matrix, and matrix-vector multiplication arises as a concrete realization of applying T to a vector. This connection shows that matrix algebra is a coordinate-based reflection of the underlying operator theory, and it clarifies why basis-independence is desirable for physical quantities.

Eigenvalues, Eigenvectors, and Invariant Subspaces

The discussion introduces invariant subspaces and the eigenvalue equation T U = λ U for nonzero U. Eigenvalues form the spectrum of T, and eigenvectors reveal directions in which T acts as simple scaling. The speaker distinguishes trivial zero vectors from genuine eigenvectors and introduces the spectrum as the collection of eigenvalues. The concept of invariant subspaces serves as the gateway to eigenproblems.

Finite vs Infinite Dimensions; Examples

Two important contrasts are highlighted: in finite dimensions, many properties become equivalent (injective, surjective, invertible), whereas in infinite dimensions, these equivalences can fail. The left shift operator on infinite sequences is surjective but not injective, while the right shift is injective but not surjective. The rotation operator in 3D space is used to illustrate how eigenvalues can be complex when real spaces are extended to complex numbers, foreshadowing why complex eigenvalues simplify factorization in linear algebra.

Closing and Outlook

The lecture closes by connecting the operator viewpoint to quantum observables, density operators, and symmetries, and it hints at the broader program of studying spectra, traces, determinants, and basis-independent quantities in future sessions.