• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Quantum harmonical oscillator with electric field

  • Thread starter Chen
  • Start date
972
1
Hi,

I have a particle of mass m and charge q, which is located in the potential of an harmonic oscillator and also subject to a constant electric field. The Hamiltonian is given as:

[tex]H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x^2 - q E' x[/tex]

And I need to find a change of variables from x to u, so that the eigenvalue equation:

[tex]H \phi (x) = E \phi (x)[/tex]

Becomes:

[tex][-\frac{h^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 E'^2}{2m \omega ^2}) \phi (u)[/tex]

(It's an h-bar there, of course.) I don't even know where to start. I tried plugging u(x) into the original eigenvalue equation and find some constraint on u from there, to no avail.

Thanks
 
Last edited:

George Jones

Staff Emeritus
Science Advisor
Gold Member
7,223
772
Complete the square on the last 2 terms in the Hamiltonian, and the transformation might become a bit more obvious.

Regards,
George
 
972
1
Doh... thanks! :smile:
 

Related Threads for: Quantum harmonical oscillator with electric field

Replies
6
Views
2K
Replies
4
Views
2K
Replies
2
Views
4K
Replies
1
Views
924
Replies
1
Views
7K
Replies
3
Views
1K
  • Posted
Replies
3
Views
1K
  • Posted
Replies
9
Views
4K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top