Bra–ket notation

The Dirac notation is a notation of state vectors, which is used in quantum mechanics. The advantage of this notation is that it is coordinate- free. The equations can be write down in general and one can later choose the coordinates that are most appropriate for solving the problem.

Paul Dirac himself invented both the spelling and the wording, ( engl. angle bracket ) alludes to the angle bracket, with the one often called the scalar product of two vectors. The notation is therefore also called Bra- Ket notation.

In the Bra- Ket notation to write the vectors of a vector space outside of an inner product with an angle bracket as Ket. Each chain corresponds to a Bra belonging to the dual space, ie a linear mapping represented in the underlying body, and vice versa. The result of the operation of a packet is written Bras, whereby the connection with the conventional notation of the scalar product is formed.

In physics, the notation is used, whether this is by vectors of a vector space or to functions in a Hilbert space. The mathematical justification for the Bra- Ket notation results proved independently from the set of Fréchet - Riesz, F. Riesz and the M. Fréchet in 1907. He says, among other things, that a Hilbert space and its topological dual space isometrically isomorphic to each other.


Be a vector of complex - dimensional vector space (). The chain expression can be represented as a vertical vector with complex elements ()

The Bra- expression can therefore be represented as a horizontal vector with the conjugated values


Using the notation

Can an electron in the state 1s with spin up of the hydrogen atom are called.

The polarization state of a photon can be used as the superposition of two base states, for example, ( vertical polarization ), and ( horizontal polarization ) to be interpreted:

In which


Scalar product

The scalar product of a bra with a ket is written in Bra- Ket notation as

= Application of Bras on the chain.

For arbitrary complex numbers, and the following applies:

Due to the duality relation is still


The tensor product of a chain with a Bra is written as

In the case of normal vectors corresponding to the tensor product of a matrix.

For a complete orthonormal basis performs the operation

A projection from the base state. This defines the projector onto the subspace of the state:

A particularly important application of the multiplication of chain with Bra is the unit operator, which is the sum of the projection operators as

(In infinite-dimensional Hilbert spaces is to be regarded with discrete basis of the Limes. )

This " representation of the unit operator " therefore is particularly so outstanding importance, since you can use it to develop each state in an arbitrary basis.

An example of a basic development by inserting the One:

This is the representation of the state - kets the base by the insertion of the so-called one.

The fact that this always works, is a direct consequence of the completeness of the Hilbert space in which the states, so the Kets, ' live '.

For a continuous basis is to be formed instead of the sum of an integral. Thus, for example, for the local area, the sum of the spatial continuum, and thus the device operator as the integral over the whole:

Of course, even with such a continuous basis based development is possible, which usually leads to a Fourier integral. Technically it is not, this is a development by the basis vectors of the Hilbert space, as there may be a continuum of pairwise orthogonal vectors in the considered separable spaces: Vectors of the type rather form a mathematically non-trivial extension of the considered Hilbert space, and they are therefore called also sometimes " improper vectors " because they are not like the delta function or as monochromatic plane waves square integrable. ( The concept of orthogonality must here be generalized by using instead of the usual Kronecker symbols delta functions. )

One considered in these calculations details that only the "Recipes " and run out basically, the base development remains a useful analogy.

Representations in quantum mechanics

In quantum mechanics, one often works with projections of state vectors to a particular base rather than with the state vectors themselves

The projection to a particular base is called representation. An advantage of this is that the wave functions thus obtained are complex numbers, of which the quantum mechanics formalism, the can be written as a partial differential equation.

  • Representation in the local space - based (local representation):

Be an eigenstate of the position operator with the property.

The wave function is obtained by projection as

The scalar product is

  • Representation in the momentum space basis ( momentum representation ):

Be an eigenstate of the momentum operator with the property.

The wave function is obtained by projection as

The scalar product is now the same as before

In general, scalar products are invariant under any change of basis. Examples are the transitions ( "Appearance Change" ) of a complete set of eigenvectors and / or improper eigenvectors of self-adjoint operators of the system to another, eg, the transition from a matrix system to another or the transition from a matrix representation to local or momentum representation.

  • Matrix elements of an invariant defined " measure ", with associated, dependent on the used base operator are equal in all bases, though the operators themselves ia have different representations. How do you calculate approximately in the position representation