Matrix Representation of Operators and Wavefunctions
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Vector Representation of an eigenstate
For a set of vectors \(\{|1 \rangle, |2 \rangle, ... | \infty \rangle \}\) that spans the space we are interested in, the arbitrary eigenstate can be decomposed
\[ | \psi \rangle = \sum_i^n c_i | i \rangle = \begin{pmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{pmatrix} \label{1A} \]
The \(\{|1 \rangle, |2 \rangle, ... | \infty \rangle \}\) constitutes a basis (one of many possible) for the space.
Most of the operators we are discussed are linear so
\[ \hat{A} | \psi \rangle = \hat{A} \left( \sum_i^n c_i | i \rangle \right) = \sum_i^n c_i \hat{A} | i \rangle \label{2A}\]
Matrix Representation of an Operator
Operators can be expressed as matrices that "operator" on the eigenvector discussed above
\[ \hat{A} | i \rangle = \sum_i^n A_{ij} | i \rangle \label{3A} \]
The number \(A_{ij}\) is the \(ij^{th}\) matrix element of \(A\) in the basis select.
Hermitian operators are operators that satisfy the general formula
\[ \langle \phi_i | \hat{A} | \phi_j \rangle = \langle \phi_j | \hat{A} | \phi_i \rangle \label{Herm1}\]
If that condition is met, then \(\hat{A}\) is a Hermitian operator. For any operator that generates a real eigenvalue (e.g., observables), then that operator is Hermitian. The Hamiltonian \(\hat{H}\) meets the condition and a Hermitian operator. Equation \ref{Herm1} can be rewriten as
\[A_{ij} =A_{ji}\]
where
\[A_{ij} = \langle \phi_i | \hat{A} | \phi_j \rangle\]
and
\[A_{ji} = \langle \phi_j | \hat{A} | \phi_i \rangle\]
Therefore, when applied to the Hamiltonian operator
\[\boxed{H_{ij}^* =H_{ji}.}\]
Multiplication
We can define an inner product (dot product) of two eigenstates \(| \phi_1 \rangle\) and \(| \phi_2\ \rangle\)
\[ \langle \phi_1 | \phi_2 \rangle = \sum_{i=1}^n c_i^* c_j \label{4A}\]
which looks like this in vector representations
\[ \langle \phi_1 | \phi_2 \rangle = ( c_1^* \; c_2^* \; ... c_n^* ) \begin{pmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{pmatrix} \label{5A} \]
This form emphasizes the dot product nature of the inner product multiplication.
From equation \(\ref{5A}\), we can express the bra form of the eigenstate as
\[ \langle \phi | = \sum_i^n c_i^* \langle i | \label{6A}\]
Completeness Relation
For vectors \(|i \rangle\) forming an orthonormal basis \(\langle i | j \rangle = \delta_{ij}\) for all space then
\[ \sum_i^n | i \rangle \langle j| = 1 \label{7A}\]
Those terms in this sum are outer products and are matrices (remember inner products are scalars).
Diagonal Representation of an operator
\[\hat{A} = \sum_i ^n \lambda_i | i \rangle \langle i| \label{8A}\]
This is a matrix that has non-zero element everywhere except the diagonal. E.g.,
\[ \begin{pmatrix} 4 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -4 \end{pmatrix}\]