**Matrix Mechanics: The Foundation of Quantum Mechanics** - **Origins:** Matrix Mechanics was born in 1925, developed by [[Werner Heisenberg]], Max Born, and Pascual Jordan. It provided one of the earliest, complete, and self-consistent formulations of [[quantum mechanics]]. - **Departure from Classical Physics:** This framework represented a radical shift from the classical picture of electrons orbiting the nucleus like planets. Instead, Matrix Mechanics described physical observables (like position, momentum, and energy) as matrices that evolve over time. - **Quantum Jumps:** Matrix Mechanics elegantly explained the phenomenon of quantum jumps, where an electron suddenly transitions between discrete energy levels. These transitions were represented by sudden changes in the matrices representing observables. **Linear Algebra: The Language of Quantum Mechanics** The very heart of Matrix Mechanics is built upon the concepts of [[linear algebra]]. Here's how it weaves into quantum mechanics: - **Vectors: States of Quantum Systems** - A quantum system's state is represented as a vector in a complex vector space (often called a [[Hilbert space]]). - Example: The spin state of an electron can be represented as a vector in a two-dimensional space. - **Matrices: Operators** - Observables like position, momentum, energy, etc., are represented by matrices (more precisely, [[Hermitian|Hermitian matrices]]). - Operations on these matrices correspond to making measurements or performing transformations on the quantum system. - **Matrix Multiplication: Dynamics and Measurements** - The time evolution of a quantum system is governed by matrix multiplication involving the Hamiltonian matrix (represents the system's energy). - Measurement outcomes are determined by the eigenvalues of the matrix representing the observable. - **Eigenvalues and Eigenvectors: The Core of the Theory** - Eigenvalues of an observable's matrix represent the possible values that can be obtained if a measurement is made. - Eigenvectors represent the quantum states which yield those specific values upon measurement. **Key Points** - **Equivalence to Wave Mechanics:** Matrix Mechanics, while initially very different in appearance, was later proven to be mathematically equivalent to Schrödinger's wave mechanics. They are simply different ways to view the same quantum phenomena. - **Computational Strength:** Matrix Mechanics is especially well-suited for problems in quantum mechanics that involve a finite number of discrete energy states. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Matrix Mechanics") or contains(subject, "Quantum Mechanics") sort title, authors, modified ```