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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...

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# Classic Oscillator

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