- #1
Javier2808
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- TL;DR Summary
- Misunderstanding the matrix representation of the [H,x] commutator
Hello!
I have checked commutator matrix form of $$\vec{p}=im/\hbar [H,\vec{x}]$$ but I realized i don't undertand something
I have $$[H,\vec{x}]=H\vec{x}-\vec{x}H$$ and
$$(H\vec{x})_{i}=H_{ij}x_j$$ & $$\ ( \vec{x}H)_{i}=x_jH_{ji}$$
but what is the second term matrix representation
$$\vec{x} H=
\left(
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}
\right)
\left(
\begin{array}{ccc}
H_{11} & H_{12}& H_{13}\\
H_{21} & H_{22}& H_{23}\\
H_{31} & H_{32}& H_{33}\\
\end{array}
\right) = ?$$
Should I introduce the transpose Vector?
$$\vec{u}=\vec{x}^T H=
\left(
\begin{array}{ccc}
x_{1}&x_{2} &x_{3}
\end{array}
\right)
\left(
\begin{array}{ccc}
H_{11} & H_{12}& H_{13}\\
H_{21} & H_{22}& H_{23}\\
H_{31} & H_{32}& H_{33}\\
\end{array}
\right) = \left(
\begin{array}{ccc}
u_{1}&u_{2} &u_{3}
\end{array}
\right)
$$
it is a row vector but
$$\vec{w}=H \vec{x} =
\left(
\begin{array}{ccc}
H_{11} & H_{12}& H_{13}\\
H_{21} & H_{22}& H_{23}\\
H_{31} & H_{32}& H_{33}\\
\end{array}
\right) \left(
\begin{array}{c}
x{1} \\
x_{2} \\
x_{3}
\end{array}
\right)= \left(
\begin{array}{c}
w_{1} \\
w_{2} \\
w_{3}
\end{array}
\right)$$
is a column vector, I could not sum them. What is wrong?
Thanks for your help
I have checked commutator matrix form of $$\vec{p}=im/\hbar [H,\vec{x}]$$ but I realized i don't undertand something
I have $$[H,\vec{x}]=H\vec{x}-\vec{x}H$$ and
$$(H\vec{x})_{i}=H_{ij}x_j$$ & $$\ ( \vec{x}H)_{i}=x_jH_{ji}$$
but what is the second term matrix representation
$$\vec{x} H=
\left(
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}
\right)
\left(
\begin{array}{ccc}
H_{11} & H_{12}& H_{13}\\
H_{21} & H_{22}& H_{23}\\
H_{31} & H_{32}& H_{33}\\
\end{array}
\right) = ?$$
Should I introduce the transpose Vector?
$$\vec{u}=\vec{x}^T H=
\left(
\begin{array}{ccc}
x_{1}&x_{2} &x_{3}
\end{array}
\right)
\left(
\begin{array}{ccc}
H_{11} & H_{12}& H_{13}\\
H_{21} & H_{22}& H_{23}\\
H_{31} & H_{32}& H_{33}\\
\end{array}
\right) = \left(
\begin{array}{ccc}
u_{1}&u_{2} &u_{3}
\end{array}
\right)
$$
it is a row vector but
$$\vec{w}=H \vec{x} =
\left(
\begin{array}{ccc}
H_{11} & H_{12}& H_{13}\\
H_{21} & H_{22}& H_{23}\\
H_{31} & H_{32}& H_{33}\\
\end{array}
\right) \left(
\begin{array}{c}
x{1} \\
x_{2} \\
x_{3}
\end{array}
\right)= \left(
\begin{array}{c}
w_{1} \\
w_{2} \\
w_{3}
\end{array}
\right)$$
is a column vector, I could not sum them. What is wrong?
Thanks for your help