Matrix Representation of Quantum Mechanics
- Quantum Mechanics 1.5
from 「Modern Quantum Mechanics」: Sakurai, J. J.
Let’s take a look for Heisenberg’s matrix mechanics.
Matrix representation of state vectors
Any state vector can be represented by the sum of eigenvectors for a Hermitian operator $L$.
- $ \dps \ket{\psi}=\sum_n c_n\ket{\psi_n} $
Then, we can use those eigenvectors as a basis, with coordinates $c_n$. The state vector can be represented as a column vector:
\[ \ket{\psi} \longrightarrow \begin{bmatrix} c_1 \nl c_2 \nl c_3 \nl \vdots \end{bmatrix},\; \bra{\psi} \longrightarrow \begin{bmatrix} c_1^\ast & c_2^\ast & c_3^\ast & \cdots \end{bmatrix} \]
And if the eigenvectors are orthonormal; $\brkt{\psi_n}{\psi_m}=\delta_{nm}$; it can be represented with inner products:
\[ \ket{\psi} \longrightarrow \begin{bmatrix} \brkt{\psi_1}{\psi} \nl \brkt{\psi_2}{\psi} \nl \brkt{\psi_3}{\psi} \nl \vdots \end{bmatrix} \]
If we use $[\cdot]$ to denote matrix representation, an inner product is expressed as follows.
- $ \brkt{f}{g} = [f]^\dag [g]$
Matrix representation of operators
An operator can be represented as a matrix similarly:
\[ T \longrightarrow \begin{bmatrix} \brktop{\psi_1}{\hat{T}}{\psi_1} & \brktop{\psi_1}{\hat{T}}{\psi_2} & \brktop{\psi_1}{\hat{T}}{\psi_3} & \cdots \nl \brktop{\psi_2}{\hat{T}}{\psi_1} & \brktop{\psi_2}{\hat{T}}{\psi_2} & \brktop{\psi_2}{\hat{T}}{\psi_3} & \cdots \nl \brktop{\psi_3}{\hat{T}}{\psi_1} & \brktop{\psi_3}{\hat{T}}{\psi_2} & \brktop{\psi_3}{\hat{T}}{\psi_3} & \cdots \nl \vdots & \vdots & \vdots & \ddots \end{bmatrix} \]
Then, an operation at a state vector is identical to general matrix multiplication.
- $ \hat{T}\ket{f} \longrightarrow [T][f] $
- $ \brktop{f}{\hat{T}}{g} \longrightarrow [f]^\dag[T][g] $
Also, the following identity is true for any linear operator $T$.
- $ [T^\dag]=[T]^\dag $
