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 <a_n| Y | a_m>. 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

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

Consider the following equation in ket space,

$$Y \vert \Psi>= \vert \Psi'>$$ (1.40)

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

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

Evaluating this expression we obtain,

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

which can be written as a matrix equation

$${\underline Y} \,\, \underline{\Psi} = \underline{\Psi'}$$ (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)

$$< \Psi\vert = <\Psi'\vert Y^{\dag }.$$ (1.44)

Show that the matrix representation of this equation is given by

$${\underline \Psi}^{\dag } = {\underline \Psi}^{\dag } {\underline Y}^{\dag }$$ (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. ${\underline A}_{nm}^{\dag } = \underline{A}^{*}_{mn}$