Hilbert Space

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. 1 ⋆ We will use Dirac notation in which the vectors in the space are denoted by |vi, called a ket, where v is some symbol which identifies the vector. One could equally well use something like ~v or v. A multiple of a vector by a complex number c is written as c|vi—think of it as analogous to c~v of cv. ⋆ In Dirac notation the inner product of the vectors |vi with |wi is written hv|wi. 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.