The Matrix Representation of Operators and Wavefunctions

We will define our vectors and matrices using a complete set of, orthonormal basis states \bgroup\color{black}$u_i$\egroup, usually the set of eigenfunctions of a Hermitian operator. These basis states are analogous to the orthonormal unit vectors in Euclidean space \bgroup\color{black}$\hat{x}_i$\egroup.

\begin{displaymath}\bgroup\color{black}\langle u_i\vert u_j\rangle = \delta_{ij}\egroup\end{displaymath}

Define the components of a state vector \bgroup\color{black}$\psi$\egroup (analogous to \bgroup\color{black}$x_i$\egroup).

\begin{displaymath}\bgroup\color{black}\psi_i\equiv\langle u_i\vert\psi\rangle\q...
...uad \vert\psi\rangle=\sum\limits_i\psi_i\vert u_i\rangle\egroup\end{displaymath}

The wavefunctions are therefore represented as vectors.Define the matrix element

\begin{displaymath}\bgroup\color{black}O_{ij}\equiv \langle u_i\vert O\vert u_j\rangle.\egroup\end{displaymath}

We know that an operator acting on a wavefunction gives a wavefunction.

\begin{displaymath}\bgroup\color{black}\vert O\psi\rangle=O\vert\psi\rangle=O\su...
... \vert u_j\rangle=\sum\limits_j \psi_j O\vert u_j\rangle\egroup\end{displaymath}

If we dot \bgroup\color{black}$\langle u_i\vert$\egroup into this equation from the left, we get

\begin{displaymath}\bgroup\color{black}(O\psi)_i=\langle u_i\vert O\psi\rangle=\...
u_i\vert O\vert u_j\rangle=\sum\limits_j O_{ij} \psi_j \egroup\end{displaymath}

This is exactly the formula for a state vector equals a matrix operator times a state vector.

\begin{displaymath}\bgroup\color{black} \left(\matrix{(O\psi)_1\cr (O\psi)_2\cr ...
...matrix{\psi_1\cr \psi_2\cr ...\cr \psi_j\cr ...}\right) \egroup\end{displaymath}

Similarly, we can look at the product of two operators (using the identity \bgroup\color{black}$\sum\limits_k \vert u_k\rangle\langle u_k\vert=1$\egroup).

\begin{displaymath}\bgroup\color{black}(OP)_{ij}=\langle u_i\vert OP\vert u_j\ra...
...le u_k\vert P\vert u_j\rangle=\sum\limits_k O_{ik}P_{kj}\egroup\end{displaymath}

This is exactly the formula for the product of two matrices.

& \left(\matrix{(OP)_{11}&(OP)_{12}&...&(OP)_{1j}&...\cr
... &... &...&... &... }\right)

So, wave functions are represented by vectors and operators by matrices, all in the space of orthonormal functions.

* Example: The Harmonic Oscillator Hamiltonian Matrix.*
* Example: The harmonic oscillator raising operator.*
* Example: The harmonic oscillator lowering operator.*

Now compute the matrix for the Hermitian Conjugate of an operator.

\begin{displaymath}\bgroup\color{black}(O^\dag )_{ij}=\langle u_i\vert O^\dag \v...
...vert u_j\rangle=\langle u_j\vert O u_i\rangle^*=O^*_{ji}\egroup\end{displaymath}

The Hermitian Conjugate matrix is the (complex) conjugate transpose.

Check that this is true for $A$ and $A^\dag$.

We know that there is a difference between a bra vector and a ket vector. This becomes explicit in the matrix representation. If \bgroup\color{black}$\psi=\sum\limits_j \psi_j u_j$\egroup and \bgroup\color{black}$\phi=\sum\limits_k \phi_k u_k$\egroup then, the dot product is

\begin{displaymath}\bgroup\color{black}\langle \psi\vert\phi\rangle=\sum\limits_...

We can write this in dot product in matrix notation as

\begin{displaymath}\bgroup\color{black}\langle \psi\vert\phi\rangle=
\phi_2\cr \phi_3\cr ...}\right)\egroup\end{displaymath}

The bra vector is the conjugate transpose of the ket vector. The both represent the same state but are different mathematical objects.

Jim Branson 2013-04-22