MIT OCW Lecture: Linear Algebra Foundations in Quantum Mechanics

Overview

In this MIT OpenCourseWare lecture, the instructor revisits spin operators and Pauli matrices, develops basic linear algebra concepts, and introduces the language of vector spaces, subspaces, and linear maps. The discussion highlights eigenvalues, eigenvectors, and traces, and demonstrates how Pauli matrices encode angular momentum properties. The lecturer also shows that the spin along any direction has eigenvalues ±ħ/2 and introduces a compact A·σ notation to express products of Pauli matrices. The session ties these linear-algebra ideas to quantum mechanics while beginning to build the matrix-representation viewpoint without requiring an inner product.

The talk also surveys a variety of vector spaces drawn from physics and mathematics, and lays the groundwork for understanding subspaces, direct sums, and dimensionality as essential concepts for describing physical states.

Overview

This MIT OpenCourseWare lecture surveys the linear algebra toolkit used in quantum mechanics, focusing on the Pauli matrices, eigenvalues, and the algebra of angular momentum. The instructor begins from a matrix equation M^2+αM+βI=0 and derives how eigenvalues must satisfy λ^2+αλ+β=0, illustrating the method by which spectrum information is extracted from operator identities. The Pauli matrices σ1, σ2, and σ3 satisfy σi^2 = I and have zero trace, leading to eigenvalues ±1, and the trace condition forces the eigenvalues to balance as +1 and -1 for σ1. The lecture then introduces the anticommutator {σi, σj} = 2δij I and develops a powerful decomposition of operator products using the anti-commutator and commutator, culminating in the identity σi σj = δij I + i εijk σk.

To streamline notation, the instructor defines the dot product A·σ = A1σ1 + A2σ2 + A3σ3 for a normal vector A and a Pauli triplet, and shows how this compact notation yields geometric interpretations of operator products. A·σ A·B expands to (A·B)I + i(A×B)·σ, bridging linear algebra with vector operations. This framework clarifies why the spin along an arbitrary direction can be described by S_N = (ħ/2) N·σ, and why its eigenvalues are ±ħ/2 regardless of direction. The discussion also briefly introduces the cross product for operator triplets and hints at a calculation for S×S. The section ends with an informal segue into a linear-algebra module that will cover vector spaces, subspaces, bases, and dimension.

Vector Spaces, Fields, and Subspaces

The lecture then pivots to a formal linear-algebra foundation. A vector space consists of vectors and scalars from a field F, with operations of vector addition and scalar multiplication satisfying familiar axioms. The instructor illustrates with examples that span from real and complex vector spaces to spaces of matrices, polynomials, infinite sequences, and complex-valued functions. The distinction between real and complex vector spaces is emphasized as a structural difference, not a property of the vectors themselves. Subspaces are subsets that are themselves vector spaces, always containing the zero vector, and direct sums allow decomposing a space into a unique sum of subspaces. The idea of writing any vector as a sum of components from each subspace is illustrated with R^2 as R^1 ⊕ R^1.

Dimensionality and Bases

Dimensionality is defined via bases: a finite-dimensional vector space has a basis, a linearly independent spanning set, whose elements determine the dimension as the number of basis vectors. The lecturer stresses that a basis does not depend on inner products or norms; a basis exists for any finite-dimensional vector space, and its size is invariant across all bases. Infinite-dimensional spaces are discussed conceptually, with examples like the space of polynomials or the space of all complex-valued functions on an interval, which require infinite lists to span. The space of Hermitian 2×2 matrices is shown to be 4-dimensional, with the basis consisting of the identity and the three Pauli matrices, validating a finite dimensional structure for a key physical space.

Linear Operators and Matrix Representations

A linear operator maps a vector space to itself (an endomorphism) and preserves addition and scalar multiplication. The matrix representation of a linear operator arises from its action on a chosen basis, a viewpoint that allows discussing observables and transformations without invoking bra-ket language or inner products at the outset. The instructor teases future topics in which one will introduce trace, eigenvalues, and eigenvectors in a matrix formalism, and explains how a carefully chosen basis makes the matrix representation straightforward to work with.

Examples and Takeaways

The lecture surveys several examples: finite-dimensional spaces like R^n with real entries, the space of complex matrices, and the four-dimensional Hermitian space; infinite-dimensional spaces like polynomials and complex-valued functions on an interval. Through these, the speaker emphasizes the central role of linear algebra in describing quantum states as vectors in a complex space, and observables as linear operators acting on those spaces. The talk closes by reemphasizing the independence of inner products for developing the matrix formalism and primes students to engage with the Axler-inspired approach to linear algebra, laying the groundwork for future depth in structure and representation theory.