3D isotropic harmonic oscillator vs. diatomic molecule

[Heirot]

The Hamiltonian of the diatomic molecule is given by H = p1^2 / 2m + p2^2 / 2m + 1/2 k R^2, where R equals the distance between atoms. Using this result, given in standard texbooks, I keep geting C = 9/2 kT instead of 7/2 kT for heat capacity. I've traced down my problem to the potential energy term. I seem to be calculating as if I have a 3D isotropic oscillator instead of two point particles connected by a spring. It appears as these two systems have the same Hamiltonian, but that surely can't be so. My question is, what's the right Hamiltonian for a given system and how to see that these two systems have different degrees of freedom?

Thanks!

 

[DrDu]

We had this very discussion already two or three weeks ago:

https://www.physicsforums.com/showthread.php?t=400150

 

[DrDu]

Basically, your error consists in writing R instead of R-R_0 with R_0 being the equilibrium distance of the molecule. Hence there is no longer a single minimum at R=0 but a sphere of degenerate minima.

 

[Heirot]

Thanks for the link! So, you're saying that the textbooks give the wrong Hamiltonian? It should be R = |r1 - r2| - r0, where r1 and r2 are vector positions of both particles? I don't see how this reduces the number od quadratic contributions to the Hamiltonian from 3 to 1 as is necessary for the correct heat capacity.

 

[DrDu]

"The textbooks"? How many did you check?

 

[DrDu]

You get 3/2kT for the translational degrees of freedom of the center of mass, kT for the approximately harmonic motion around r_0 in the radial co-ordinate (distance) and kT for the rotation trough the degenerate minima giving 7/2 kT in total.

 

[Heirot]

This is a problem from e.g. Huang. Doesn't the r0 term couple rotational and vibrational degrees of freedom? I.e. r0 is R dependent because of the centrifugal effect?

 

[DrDu]

Indeed it is, but you can usually treat this dependence as a small perturbation. Obviously, the dependence on R-R_0 is not exactly quadratic, etc. However, that Hamiltonian and wavefunction can be justified for a diatomic molecule as the zeroth order term in a development in the quotient of electron to nuclear mass. That was the content of the original paper by Born M, Oppenheimer R. 1927. Ann. Physik 84:457–84

 

[Heirot]

Thank you very much for the clarification! Is there, perhaps, an exact solution for the rotating oscillator?

 

[DrDu]

You are wellcome. I fear there is no exact solution for the rotator oscillator.