Coupled harmonic oscillators QM

[valtorEN]

Homework Statement

Consider two coupled oscillators. The Hamiltonian is given as
H=p1^2/2m + p2^2/2m +1/2m*omega^2*[x1^2+x2^2+2*lambda*(x1-x2)^2]
Separate the center of mass and relative motion and find the eigenfunctions and eigenvalues.

Homework Equations

relative coordinate :: x=x1-x2
CM position :: X

The Attempt at a Solution

i assume that i have to reduce the 2 body to equivalent one body, using the fact that the potential only depends of the separation of the 2 particles x1-x2, that is, if V(x1,x2)=V(x1-x2)
now introduce 2 new variables,
x=x1-x2,
X=m1x1+m2x2/(m1+m2)
now introduce REDUCED MASS 1/mu:=1/m1+1/m2
not sure how to proceed

 

[valtorEN]

i guess i put X=x1-x2 into eq above, and t hen try to get into parts for X and reduced mass 1/mu=1/m1+1/m2, so the last term i get is 1/2*m*omega^2[x1^2+x2^2+2*lambda*X^2]
what should i do next?

 

[dextercioby]

I'm surprised no one came to help, since the problem's relatively easy.

Just like for the H-atom the separation of the radial and CM motion is done by expressing the particle position and canonical momentum wrt the coordinates and momenta of the CM and the virtual particle.

That is you have to find

p_{1}=p_{1}(X,P,x,p)
p_{2}=p_{2}(X,P,x,p)
x_{1}=x_{1}(X,P,x,p)
x_{2}=x_{2}(X,P,x,p)

and then plug back into the Hamiltonian. You then have to do simple algebraic manipulations which hopefully will lead to the desired separation.

Daniel.