Quantum states represented by vectors in Hilbert space from "summary" of The Principles of Quantum Mechanics by P. A. M. Dirac
In quantum mechanics, the state of a physical system is represented by a vector in a complex linear vector space called Hilbert space. This mathematical formalism provides a precise way to describe the properties of a quantum system and predict the outcomes of measurements. Hilbert space is a generalization of ordinary three-dimensional Euclidean space, allowing for an infinite number of dimensions. Each quantum state corresponds to a unique vector in Hilbert space, which contains all the information about the system at a given time. The vectors in Hilbert space are called state vectors or ket vectors, denoted by the symbol |ψ⟩. These vectors are normalized to unity, meaning that the inner product of a state vector with itself is equal to one. This normalization condition ensures that the probability of finding the system in any possible state is always equal to one. Quantum measurements are represented by linear operators acting on the state vectors. These operators correspond to physical observables such as position, momentum, energy, and spin. When a measurement is made, the state vector collapses to one of the eigenstates of the corresponding operator, with the probability of each outcome given by the squared magnitude of the inner product between the state vector and the eigenstate. The evolution of a quantum system over time is governed by the Schrödinger equation, a linear differential equation that describes how the state vector changes with time. The solution to this equation gives the time evolution of the system and allows us to predict the future behavior of the quantum system.- Representing quantum states by vectors in Hilbert space is a powerful and elegant mathematical framework that underpins the formalism of quantum mechanics. It provides a rigorous foundation for understanding the behavior of microscopic particles and has been instrumental in the development of modern physics.
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