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orthovector
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can somebody explain eigenvalues inside hilbert spaces??
orthovector said:do you think the strang linear algebra book is the right intoductory linear algebra book?
also, when should I begin my studies on thermal and statistical physics? BEFORE OR AFTER quantum mechanics and linear algebra??
A Hilbert space is a mathematical concept that represents an infinite-dimensional vector space, where vectors can have any number of components. It is named after the German mathematician David Hilbert and is used in various areas of mathematics, physics, and engineering.
In quantum mechanics, Hilbert spaces are used to represent the state of a quantum system, which can have an infinite number of possible states. The eigenstates of a quantum system, which represent the possible values of an observable, are also elements of a Hilbert space.
Eigenvalues are numerical values that are associated with eigenstates in a Hilbert space. They represent the possible values that an observable can take in a quantum system. The eigenvalues of an observable correspond to the different energy levels of the system.
Eigenvalues and eigenvectors are closely related in Hilbert spaces. An eigenvector is a vector in a Hilbert space that remains unchanged, up to a scaling factor, when multiplied by a linear operator. The corresponding eigenvalue is the factor by which the eigenvector is scaled.
Unlike finite-dimensional vector spaces, Hilbert spaces cannot be visualized in the traditional sense since they have an infinite number of dimensions. However, they can be represented and studied using mathematical tools and techniques, such as basis vectors and inner product spaces.