Hamiltonian for a dimer approaching a surface

1 reply · 2K views · last reply
AllenFaust
Messages
3
Reaction score
0
Thread moved from the technical forums, so no Homework Template is shown
Hi! it's been a day since I have started this problem. I was wondering how I could arrive to this Hamiltonian?
And I'm a bit at a lost on how exactly to derive this? I hope anyone can help me with this, even a suggestion of good starting point would be much appreciated.

Hamiltonian written as H = -(ħ²/2M) ∂²/∂Z² - (ħ²/2μ) ∂²/∂r² + V(Z,r), with kinetic terms for centre-of-mass coordinate Z and relative coordinate r.


Basically the problem is to derive a hamiltonian for the system here.
Diagram of two spheres labelled mA and mB separated horizontally by distance r, with a dashed vertical line of length Z from the midpoint down to a horizontal surface.


Regarding this, I'm currently thinking of having a set of polar and azimuthal terms cancel out and thus leading to this maybe?

Thank you very much!
 
Attachments
  • Formula for reduced mass: mu equals m_A times m_B divided by the sum m_A plus m_B.
    upload_2017-6-9_18-0-52.png
    544 bytes · Views: 474

Discussion

The model appears to assume that the angle between the surface and the dimer is fixed, and neglects angular momentum. The kinetic energy term can be derived starting from two masses parallel to the surface, then separating out center of mass and relative coordinates.
 

Similar threads

Derive Formula for Velocity of Object Approaching Earth
  • directory_NA
  • · Replies 17 ·
  • Introductory Physics Homework Help
Replies
17