Difference between revisions of "Hilbert Space"
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| − | * In | + | * In [[Quantum Mechanics]] the state of a physical system is represented by a vector in a Hilbert space: a complex vector space with an inner product. |
* The term “Hilbert space” is often reserved for an infinite-dimensional inner product space having the property that it is complete or closed. However, the term is often used nowadays, as in these notes, in a way that includes finite-dimensional spaces, which automatically satisfy the condition of completeness. | * The term “Hilbert space” is often reserved for an infinite-dimensional inner product space having the property that it is complete or closed. However, the term is often used nowadays, as in these notes, in a way that includes finite-dimensional spaces, which automatically satisfy the condition of completeness. | ||
* We will use Dirac notation in which the vectors in the space are denoted by |v>, called a ket, where v is some symbol which identifies the vector. | * We will use Dirac notation in which the vectors in the space are denoted by |v>, called a ket, where v is some symbol which identifies the vector. | ||
Revision as of 07:48, 27 April 2023
Full Title
Context
- In Quantum Mechanics the state of a physical system is represented by a vector in a Hilbert space: a complex vector space with an inner product.
- The term “Hilbert space” is often reserved for an infinite-dimensional inner product space having the property that it is complete or closed. However, the term is often used nowadays, as in these notes, in a way that includes finite-dimensional spaces, which automatically satisfy the condition of completeness.
- We will use Dirac notation in which the vectors in the space are denoted by |v>, called a ket, where v is some symbol which identifies the vector.
- One could equally well use something like v. A multiple of a vector by a complex number c is written as c|v> -think of it as analogous to cv.
- In Dirac notation the inner product of the vectors |v> with |w> is written <v|w>. This resembles the ordinary dot product ~v · ~w except that one takes a complex conjugate of the vector on the left, thus think of ~v∗· ~w.
