Difference between revisions of "Hilbert Space"
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==Context== | ==Context== | ||
| − | * In quantum mechanics the state of a physical system is represented by a vector in a Hilbert space: a | + | * 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. |
| − | 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 | + | * 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. |
| − | property that it is complete or closed. However, the term is often used nowadays, as in these notes, in a way | + | * 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 c'''v'''. |
| − | that includes finite-dimensional spaces, which automatically satisfy the condition of completeness. | + | * 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. |
| − | * 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 c'''v'''. | ||
| − | * 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. | ||
==References== | ==References== | ||
[[Category: Physics]] | [[Category: Physics]] | ||
Revision as of 07:47, 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.
