Coupled 2D harmonic oscillators

[Apashanka das]

1. The problem statementhttps://www.physicsforums.com/attachments/225935

Homework Equations

3. I have rescaled coordinates which are X=(x₁+x₂)/√2 and Y=√3(x₁-x₂)/√2 for which the potential term becomes for a 2D harmonic oscillator of coordinates X and Y. But how to express Kinetic terms in terms of these new coordinates X and Y?

 

[Dr Transport]

solve for $x_1$ and $x_2$ in terms of $X, Y$ then find the kinetic energy.

 

[Apashanka das]

solve for $x_1$ and $x_2$ in terms of $X, Y$ then find the kinetic energy.

sorry sir I didn't get you here QM kinetic term is nedded

 

[Dr Transport]

$\hat{p}$ can be related to the $\dot{X}$, you just have to find the momentum operators in terms of your new coordinates.

 

[Apashanka das]

$\hat{p}$ can be related to the $\dot{X}$, you just have to find the momentum operators in terms of your new coordinates.

sir but in calculating p² there will be again interaction term between X And Y
Sir I am actually trying to find d²/dx₁² in terms of d²/dX² and d²/dY²

 

[Apashanka das]

$\hat{p}$ can be related to the $\dot{X}$, you just have to find the momentum operators in terms of your new coordinates.

O thank you sir I have got it
The ans is option B