Classic Oscillator: Understanding Matrix Form with Shankar

[chimay]

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:
\begin{bmatrix} x_{1}'' \\ x_{2}'' \end{bmatrix}= \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}
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...

 

[UltrafastPED]

Shankar is positing that condition. You must always use the same basis for all objects (vectors, matrices), except when you are converting from one basis to another.

So in this case you would write the Hamiltonian for the blocks and springs problem. If they are the coordinates for a valid Hamiltonian, then they are canonical.

If you instead write down any old coordinates and write the Lagrangian - well, your dynamics will be correct, but the coordinates will not be canonical until you carry out the transform to go from (p,q) of the Lagrangian to the (P,Q) of the Hamiltonian.

 

[MisterX]

If you instead write down any old coordinates and write the Lagrangian - well, your dynamics will be correct, but the coordinates will not be canonical until you carry out the transform to go from (p,q) of the Lagrangian ...

Aren't the coordinates in Lagrangian mechanics q_i, \dot{q}_i ?

Anyway I think maybe what was meant by "canonic basis" w.r.t. the OP is that they are the positions and accelerations of the individual masses - as opposed to normal coordinates.

 

[chimay]

Thank you for your answers.
I need to study the second chapter ( Review of classical mechanics ) before reading your answers; I will post again then.

Thank you