Heisenberg Representation

Consider the eigenvalue problem,R4(124) R3(240)

$$\hat{H} \vert \psi> = E\vert \psi>,$$ (20)

for an arbitrary system (e.g., an atom or molecule) described by a state $\vert\psi >$, expanded in a basis set $\{\phi_j\}$ as follows,

$$\vert \psi> = \sum_j C_j \vert \phi>,$$ (21)

where $C_j = <\phi_j\vert\psi>$, and $<\phi_j\vert\phi_k>= \delta_{jk}$.

Substituting Eq. (21) into Eq. (20) we obtain:

$$\sum_j \hat{H} \vert\phi_j > C_j =\sum_j E C_j \vert\phi_j >.$$

Applying functional $<\phi_k \mid$ to both sides of this equation we obtain,

$$\sum_j <\phi_k\vert \hat{H} \vert\phi_j > C_j =\sum_j E <\phi_k \vert\phi_j > C_j,$$ (22)

where $<\phi_k\vert\phi_j>=\delta_{kj}$ and $k$ = 1, 2, ..., $n$.

Introducing the notation $H_{kj} =<\phi_k \vert\hat{H}\vert \phi_j>$ we obtain,

$$\begin{matrix}(k=1) & \rightarrow \\ (k=2) & \rightarrow \\ & ...... ...n1}C_1 +H_{n2}C_2 +H_{n3}C_3+...+H_{nn}C_n =0C_1 +0C_2 + ... +EC_n, \end{cases}$$ (23)

that can be conveniently written in terms of matrices and vectors as follows,

$$\left [ \begin{matrix}H_{11} & H_{12} & ... & H_{1n} \\ H_{21} & ... ...right ] \left [ \begin{matrix}C_1\\ C_2 \\ ... \\ C_n \\ \end{matrix} \right ].$$ (24)

This is the Heisenberg representation of the eigenvalue problem introduced by Eq. (20). According to the Heisenberg representation, also called matrix representation, the ket $\vert\psi >$ is represented by the vector ${\bf C}$, with components $C_j = <\phi_j\vert\psi>$, with j=1, ..., n, and the operator $\hat{H}$ is represented by the matrix H with elements $H_{jk}=<\phi_j\vert\hat{H}\vert\phi_k>$.

The expectation value of the Hamiltonian,

$$<\psi\vert H\vert\psi> = \sum_j \sum_k C_k^* <\phi_k\vert\hat{H}\vert\phi_j> C_j,$$

can be written in the matrix representation as follows,

$$<\psi\vert H\vert\psi> = {\bf C}^{\dagger} {\bf H} {\bf C} =\left... ... ] \left [ \begin{matrix}C_1\\ C_2 \\ ... \\ C_n \\ \end{matrix} \right ].$$

Note:

(1) It is important to note that according to the matrix representation the ket-vector $\vert\psi >$ is represented by a column vector with components $C_j = <\phi_j\vert\psi>$, and the bra-vector $<\psi\vert$ is represented by a row vector with components $C_j^*$.

(2) If an operator is hermitian (e.g., $\hat{H}$) it is represented by a hermitian matrix (i.e., a matrix where any two elements which are symmetric with respect to the principal diagonal are complex conjugates of each other). The diagonal elements of a hermitian matrix are real numbers, therefore, its eigenvalues are real.

(3) The eigenvalue problem has a non-trivial solution only when the determinant det$[{\bf H} - \hat{{\bf 1}} E]$ vanishes:

det$$[{\bf H} - \hat{{\bf 1}} E] = 0,$$   where$$\hspace{.20cm} \hat{{\bf 1}}$$   is the unity matrix$$.$$

This equation has $n$ roots, which are the eigenvalues of H.