Hermitian and Unitary Operators in Quantum Mechanics: Eigenvalues, Eigenvectors, and Dirac Notation

In this MIT OpenCourseWare lecture, the professor introduces foundational quantum mechanics concepts including Hermitian operators, real eigenvalues, orthogonality of eigenvectors, unitary operators, and Dirac notation. The talk demonstrates how Hermitian operators yield real eigenvalues, how eigenvectors with distinct eigenvalues are orthogonal, and how unitary operators preserve norm and inner product. It also covers non-enumerable bases, the resolution of identity, and the bridge between position and momentum representations via wavefunctions and Fourier transforms. The session blends concise proofs with intuition, and points to homework exercises that deepen understanding of these essential quantum tools.

Overview

The lecture surveys core quantum mechanics ideas centered on Hermitian and unitary operators, the Dirac bra-ket notation, and the link between different quantum bases. It emphasizes that Hermitian operators have real eigenvalues, proves orthogonality of eigenvectors corresponding to different eigenvalues, and introduces unitary operators as norm and inner-product preserving maps. The talk also discusses how these properties underpin the Dirac formalism and the usage of non-enumerable bases in quantum theory, setting the stage for more advanced topics like diagonalization and simultaneous eigenbases in later lectures.

Real Eigenvalues of Hermitian Operators

The instructor demonstrates a standard proof: if V is an eigenvector of a Hermitian operator H with eigenvalue λ, then by evaluating 〈V|H|V〉 in two ways one finds that λ must equal its complex conjugate, hence λ is real. This result ensures expectation values of Hermitian operators are real, aligning with physical observables.

Orthogonality of Eigenvectors

The second theorem shows that eigenvectors belonging to distinct eigenvalues of a Hermitian operator are orthogonal. By considering inner products like 〈V2|H|V1〉 and using H† = H, the difference λ1 − λ2 multiplies 〈V2|V1〉 to yield zero, forcing the inner product to vanish when λ1 ≠ λ2. The talk also discusses degeneracy, where a given eigenvalue can have a multi-dimensional eigenspace, and notes that you can choose an orthonormal basis within that subspace.

Unitary Operators and Isometries

The lecture then introduces unitary operators as isometries, preserving norms: ∥SUx∥ = ∥x∥ for all x. It shows that a unitary operator S satisfies S†S = I, implying S is invertible with inverse S†. This leads to the important property that unitary transformations preserve inner products and thus the structure of the vector space, including orthonormal bases under a unitary change of basis.

Dirac Notation and Non-Ennumerable Bases

The notes emphasize the practicality and clarity of Dirac notation when working with non-enumerable bases, such as position states |x⟩. Here the label x is not a vector in the usual sense but a basis label, while the state |ψ⟩ remains a vector in the Hilbert space. The bra-ket language helps distinguish these objects and keeps track of operations like resolutions of identity without needing explicit coordinate listings.

Position and Momentum Bases

The discussion then turns to position states |x⟩ with eigenvalue x for the position operator X and momentum states |p⟩ with eigenvalue p for the momentum operator P. The inner products satisfy ⟨x|x'⟩ = δ(x − x') and ⟨p|p'⟩ = δ(p − p'). The resolution of the identity is expressed as ∫ dx |x⟩⟨x| = ∫ dp |p⟩⟨p| = I, enabling wavefunctions ψ(x) = ⟨x|ψ⟩ and φ(p) = ⟨p|ψ⟩ to be viewed as coordinates of the state in the respective bases.

Wavefunctions and Transformations

The wavefunction in coordinate space is ψ(x) = ⟨x|ψ⟩, and the state |ψ⟩ can be written as ∫ dx |x⟩ψ(x). Momentum space wavefunctions φ(p) emerge as the Fourier transform of ψ(x), illustrating the deep connection between bases. The text also shows how operators act in different representations, for example deriving P in the x-representation as −iħ∂/∂x and illustrating how the X operator acts in the p-representation via differentiation with respect to p.

Uncertainty and the Geometric Picture

The video closes the section by introducing uncertainty in a state, defining ΔA = ∥(A − ⟨A⟩)ψ∥, which vanishes if and only if ψ is an eigenstate of A. The squared uncertainty expands to ΔA² = ⟨A²⟩ − ⟨A⟩², a form familiar from statistics. A geometric interpretation is offered: projecting Aψ onto the ℓ direction spanned by ψ yields ⟨A⟩ψ, while the orthogonal component ψ⊥ has length ΔA, providing an intuition for why uncertainties arise and how eigenstates minimize them.

Looking Ahead

The lecturer foreshadows the uncertainty principle, hinting at the use of Cauchy–Schwarz inequality to relate the uncertainties of two non-commuting observables, A and B. The material connects the algebraic manipulations to the geometry of Hilbert space and prepares students for a deeper dive into quantum measurement and commutation relations in future lectures.

Notes and Further Reading

Students are encouraged to consult the course notes for additional explanations and to complete related homework exercises that reinforce the proofs and transform the abstract formalism into practical computation.