Representations in state space

Choosing a representation means choosing an orthonormal basis in the state space E. Assume the set {|u_i>} is a discretely labeled basis and the set {|w_a>} is a continuously labeled basis for E.

<u_i|u_i>=d_ij , <w_a|w_a>=d(a-a’).

Any ket |y> may be written as

is the projector into the subspace spanned by {|u_i>}. If {|u_i>} is a basis then =I is the identity operator. This is called the closure relation.

Similarly, any ket |y> may be written as

is the projector into the subspace spanned by {|w_a>}. If {|w_a>} is a basis then = I is the identity operator. This is called the closure relation for a continuously labeled basis.

In the basis {|u_i>} the ket |y> is completely specified by its components c_i=<u_i|y>. The corresponding bra is completely specified by its components c_i*=<y|u_i> in the basis {<u_i|}.

By convention, we represent a ket as a one column matrix of its components in a given basis.

We represent a bra as one row matrix of its components in a given basis.

Matrix elements of a linear operator

Let W be a linear operator and let {|u_i>} be a basis. Let W|y>=|f>.

Then

We find

We can write W|y>=|f> as a matrix equation.

The column matrices are representations of the kets |f> and |y> and the square matrix is a representation of the operator W in the {|u_i>} basis. The W_ij=<u_i|W| u_j> are the matrix elements of the operator W. The matrix elements are scalars.

By knowing the components of |y> in the basis {|u_i>} and the actions of the operator W on the basis vectors, we can calculate the components of |f>=W|y> in the same basis.

The matrix elements of W are W_ij=<u_i|W|u_j>.

The matrix elements of W^T are W^T_ij=<u_i|W^T|u_j>=<u_j|W|u_i>^*=W_ji^*.

If W is Hermitian, W=W^T, then W_ij=W_ji^*. In particular, W_ii=W_ii^*. The diagonal matrix elements of a matrix representing a Hermitian operator are real.

Example:

The set

forms a basis for a two-dimensional vector space. Let

The vector A may be written as

,

with

and .

Image1775.gif (1256 bytes)

In the {|x_i>} basis |A> is represented by the matrix Let W be the projector onto the x axis. We know the action of W on the basis vectors. W|x₁>=|x₁>, W|x₂>=0. We therefore know what W doe to |A>.

W|A>=|B>. W|A>=A₁W|x₁>+A₂W|x₂>=A₁|x₁>=B₁|x₁>+B₂|x₂>. B₁=A₁, B₂=0.

. is the matrix of W in the {|x_i>} basis.

W₁₁=<x₁|W|x₁>, W₁₂=<x₁|W|x₂>, W₂₁=<x₂|W|x₁>, W₂₂=<x₂|W|x₂>.

W is a Hermitian operator, W_ij = W_ji^*.