- #1

chimay

- 81

- 7

Hi guys.

Recentely I'm approaching Quantum Mechanics starting from the mathematical basics.

In order to understand the benefit of representing a certain matrix in its eigenvectors basis my book makes the example I attached ( Principles of Quantum Mechanics by Shankar ).

Using matrix form it can be easily shown we can write this:

[tex] \begin{bmatrix} x_{1}'' \\ x_{2}'' \end{bmatrix}= \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} [/tex]

a, b, c, d being proper coefficient.

The author says that both the matrix and the vectors refers to the canonic basis; how can we be so confident about this?

I mean that analysing the physical problem does not require any reference to the basis we will refer when we use the matrix form; yet representing an operator in matrix form requires to have specified what is the basis...

Recentely I'm approaching Quantum Mechanics starting from the mathematical basics.

In order to understand the benefit of representing a certain matrix in its eigenvectors basis my book makes the example I attached ( Principles of Quantum Mechanics by Shankar ).

Using matrix form it can be easily shown we can write this:

[tex] \begin{bmatrix} x_{1}'' \\ x_{2}'' \end{bmatrix}= \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} [/tex]

a, b, c, d being proper coefficient.

The author says that both the matrix and the vectors refers to the canonic basis; how can we be so confident about this?

I mean that analysing the physical problem does not require any reference to the basis we will refer when we use the matrix form; yet representing an operator in matrix form requires to have specified what is the basis...