# Classic Oscillator: Understanding Matrix Form with Shankar

• chimay
In summary, the conversation discusses the use of matrix form and basis in representing a physical problem in quantum mechanics. The author explains that both the matrix and vectors must refer to the same basis, and that using any other basis would require a transformation. The conversation also mentions the use of canonical basis in Lagrangian mechanics. The participants agree to revisit the topic after studying the second chapter on classical mechanics.
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...

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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.

UltrafastPED said:
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.

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

Thank you

## 1. What is a Classic Oscillator?

A Classic Oscillator is a physical system that exhibits periodic motion. It consists of a mass attached to a spring and is often used to model simple harmonic motion.

## 2. How is the Classic Oscillator represented in matrix form?

In matrix form, the Classic Oscillator is represented by a set of coupled linear differential equations that describe the motion of the mass-spring system.

## 3. What is the significance of Shankar's understanding of the Classic Oscillator in matrix form?

Shankar's understanding of the Classic Oscillator in matrix form allows for a more elegant and concise representation of the system. It also provides a deeper understanding of the underlying mathematics and physics involved.

## 4. What are some applications of the Classic Oscillator in real-world situations?

The Classic Oscillator has many applications, including in mechanical engineering (e.g. in shock absorbers), electrical engineering (e.g. in circuits), and even in biological systems (e.g. in the beating of a heart).

## 5. How can understanding the Classic Oscillator in matrix form be useful for scientists?

Understanding the Classic Oscillator in matrix form can be useful for scientists in a variety of fields, as it provides a powerful mathematical tool for analyzing and modeling oscillatory systems. It can also be applied to more complex systems by using techniques such as perturbation theory.

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