When is a hamiltonian separable and when isnt it?

[benbenny]

So if you have a 3D Shrodinger Equation problem, what allows you so assume that the wave function solution is going to be a product of 3 wave functions where each wave function is for a different independent variable?

And also is it true that in general in these cases the eigen-energies are going to be the additive combination of the eigen-energies for of the separate SE problems for each of the variables?

Thanks,
Ben

 

[Dr Transport]

Yes on both counts. The foundation of this is a course in PDE's and Fourier Analysis.

 

[benbenny]

thanks for the reply but I am afraid you havnt answered my first question - are you saying that the Hamiltonian is always separable? I was under the impression that it is separable only in a minority of cases.

 

[Meir Achuz]

H is separable in most physical coordinate systems, like Cartesian, Spherical, Cylindrical, Elliptic, Parabollic, and about eight others. This is a majority of cases. Arfken discussed this in earlier editions.

 

[dextercioby]

There's also a small number of classical potentials which lead to completely integrable solutions both in the classical mechanics and in the quantum one.

 

[Dr Transport]

Unless the potential is not separable, (I can't remember one that wasn't) you should be able to use separation of variables to decouple the differential equations. clem stated correctly that there are a bunch of coordinate systems where separation is possible, Morse and Feshback give them all and do the separating for you as does Stratton and Smythe (both classic E&M texts). Unfortunately, and this is a short coming in Jackson, he does not.

 

[benbenny]

thanks to you all.

 

[alxm]

The Schrödinger equation is not separable for any many-body problems (such as the electronic Schrödinger equation). The same situation applies to the classical equations of motion for a many-body ssytem.