Creation and Annihilation Operators - Mathematical Details

Mathematical Details

The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract form of the operators satisfy the properties below.

Let H be the one-particle Hilbert space. To get the bosonic CCR algebra, look at the algebra generated by a(f) for any f in H. The operator a(f) is called an annihilation operator and the map a(.) is antilinear. Its adjoint is a†(f) which is linear in H.

For a boson,

,

where we are using bra-ket notation.

For a fermion, the anticommutators are

.

A CAR algebra.

Physically speaking, a(f) removes (i.e. annihilates) a particle in the state | f  whereas a†(f) creates a particle in the state | f .

The free field vacuum state is the state with no particles. In other words,

where | 0  is the vacuum state.

If | f  is normalized so that  f | f  = 1, then a†(f) a(f) gives the number of particles in the state | f .

Read more about this topic:  Creation And Annihilation Operators

Famous quotes containing the words mathematical and/or details:

    All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.
    Roger Bacon (c. 1214–c. 1294)

    Different persons growing up in the same language are like different bushes trimmed and trained to take the shape of identical elephants. The anatomical details of twigs and branches will fulfill the elephantine form differently from bush to bush, but the overall outward results are alike.
    Willard Van Orman Quine (b. 1908)