next up previous
Next: Unitary operators Up: Bra-ket notation Previous: Projection operators

Matrix representation of operators

Above we suggested that operators in Hilbert space can be written as a sum of outer products. However, it is much easier to work with operators that are expressed as matrices. Given operator Y and a basis set of kets $ \{ \vert a_{1} > \, \vert a_{2}> ... \vert a_{n}> \}$, we can construct an $ n \times n$ table of complex numbers whose $ n \, m$th element is given by <an| Y | am>. The set of n2 complex numbers can most conveniently be written as a square matrix, where m is a column index and n a row index. We use the notation $ {\underbar Y}$ to represent this matrix, and the $ n \, m$th element of that matrix is $ {\underbar Y}_{nm}$. The matrix $ {\underbar Y}$ is called the matrix representation of operator Y. Show that we can express Y in the form

 \begin{displaymath}Y=\sum_{nm} \vert a_{n}><a_{m}\vert {\underbar Y}_{nm}
\end{displaymath} (1.39)

Consider the following equation in ket space,

 \begin{displaymath}Y \vert \Psi>= \vert \Psi'>
\end{displaymath} (1.40)

where Y is an operator and $ \vert\Psi> $ an arbitrary ket. We take the inner product of this equation with state | am>(Remember, taking an inner product of two kets involves multiplying on the left by a bra), or

 \begin{displaymath}<a_{m} \vert Y \vert \Psi>= <a_{m} \vert \Psi'>.
\end{displaymath} (1.41)

Evaluating this expression we obtain,

 \begin{displaymath}\sum_{n} Y_{m \, n} <a_{n}\vert\Psi> = <a_{m}\vert\Psi'>
\end{displaymath} (1.42)

which can be written as a matrix equation

 \begin{displaymath}{\underline Y} \,\, \underline{\Psi} = \underline{\Psi'}
\end{displaymath} (1.43)

where we defined a column matrix $ \underline{\Psi} $ whose n'th row has the entry $ <a_{n}\vert \Psi>$. The utility of a matrix representation for operator Y and ket $ \vert\Psi> $ is now apparent. We can replace any abstract operator equation, such as Eq. (17), with a more familiar matrix equation (20).

We can do the same in bra space. For example, consider the adjoint of Eq. (17)

 \begin{displaymath}< \Psi\vert = <\Psi'\vert Y^{\dag }.
\end{displaymath} (1.44)

Show that the matrix representation of this equation is given by

 \begin{displaymath}{\underline \Psi}^{\dag } = {\underline \Psi}^{\dag } {\underline Y}^{\dag }
\end{displaymath} (1.45)

where $ {\underline \Psi}^{\dag } $ is the adjoint matrix of the column matrix $ \underline{\Psi} $ (note: you can consider the adjoint of a column matrix to be a row matrix). Note that the n m 'th element of matrix ${\underline A}^{\dag } $ is the complex conjugate of the m n'th element of matrix $ \underline{A}$, i.e. ${\underline A}_{nm}^{\dag } = \underline{A}^{*}_{mn} $


next up previous
Next: Unitary operators Up: Bra-ket notation Previous: Projection operators
Bernard Zygelman
1999-09-21