The Matrix Representation of Operators and Wavefunctions

We will define our vectors as matrices using an orthonormal basis $u_i$, like the eigenfunctions of a Hermitian operator. These basis states are analogous to the orthonormal unit vectors in Euclidean space $\hat{x}_i$.

$$\bgroup\color{black}\langle u_i\vert u_j\rangle = \delta_{ij}\egroup$$

Define the components of a state vector $\psi$ (analogous to $x_i$).

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

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

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

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

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

If we dot $\langle u_i\vert$ into this equation from the left, we get

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

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

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

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

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

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

$$\begin{eqnarray*} \normalsize & \left(\matrix{(OP)_{11}&(OP)_{12}&...&(OP)_{1j}&... ...P_{i1}&P_{i2}&...&P_{ij}&...\cr ... &... &...&... &... }\right) \end{eqnarray*}$$

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.

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

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 $\psi=\sum\limits_j \psi_j u_j$ and $\phi=\sum\limits_k \phi_k u_k$ then, the dot product is

$$\bgroup\color{black}\langle \psi\vert\phi\rangle=\sum\limits_... ...}\psi^*_j \phi_k\delta_{jk}=\sum\limits_k\psi^*_k\phi_k.\egroup$$

We can write this in dot product in matrix notation as

$$\bgroup\color{black}\langle \psi\vert\phi\rangle= \left(\matr... ...)\left(\matrix{\phi_1\cr \phi_2\cr \phi_3\cr ...}\right)\egroup$$

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