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.

• 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.