Difference between revisions of "Hilbert Space"
From MgmtWiki
(Created page with "==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 produc...") |
(→Context) |
||
| Line 2: | Line 2: | ||
==Context== | ==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. | 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 | 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. | 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. | is some symbol which identifies the vector. | ||
| − | One could equally well use something like | + | * One could equally well use something like '''v'''. A multiple of a vector by a complex number c is written |
| − | as c| | + | 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 | 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 07:46, 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.
