I Perturbative versus nonperturbative quantum mechanics

[Riotto]

What is the nonperturbative approach to quantum mechanics as opposed to perturbative one? When does the latter method fail and one has to apply nonperturbative approach? Please keep your discussion confined within non-relativistic quantum mechanics.

 

[DrClaude]

What is the nonperturbative approach to quantum mechanics as opposed to perturbative one? When does the latter method fail and one has to apply nonperturbative approach? Please keep your discussion confined within non-relativistic quantum mechanics.

Is this a homework question?

And what is your level of QM? You're set the thread level to A, meaning graduate level, but at that level the answer should be obvious.

 

[Riotto]

Is this a homework question?

And what is your level of QM? You're set the thread level to A, meaning graduate level, but at that level the answer should be obvious.

No. It's not a homework. It's for my personal understanding. My exposure to quantum mechanics is that of the undergraduate level. Sorry for the advanced tag. Not a frequent user. Title changed now.

 

[DrClaude]

The basic idea of perturbation theories is to start from a problem that one can solve, and then modify the solution to take into account some additional effect that is relatively small.

One example would be to find the energy levels of the hydrogen atom inside a weak magnetic field. You start by solving the hydrogen atom, then calculate how the energy levels are shifted by the presence of the field.

There are also different orders to which a solution can be achieved. In the time-idenpendent version, usually, to first order one has shifted eigenvalues but unmodified eigenfunctions. But the theory can be taken to a higher level, modifying the eigenfunctions and refining the correction to the eigenenergies, and so on.

 

[Riotto]

The basic idea of perturbation theories is to start from a problem that one can solve, and then modify the solution to take into account some additional effect that is relatively small.

One example would be to find the energy levels of the hydrogen atom inside a weak magnetic field. You start by solving the hydrogen atom, then calculate how the energy levels are shifted by the presence of the field.

There are also different orders to which a solution can be achieved. In the time-idenpendent version, usually, to first order one has shifted eigenvalues but unmodified eigenfunctions. But the theory can be taken to a higher level, modifying the eigenfunctions and refining the correction to the eigenenergies, and so on.

Dear DrClaude, I appreciate your help but unfortunately, your answer didn't address my actual question. The question asks about an explanation for the nonperturbative approach of quantum mechanics and occasions when it becomes indispensable perhaps with an illustration. I'm aware of what is the basic picture of perturbation theory that you laid out. But again, thanks for the response.

 

[DrClaude]

Ok, then let me go back to the original question.

What is the nonperturbative approach to quantum mechanics as opposed to perturbative one?

The non-perturbative approach is simply trying to solve the problem at hand without resorting to perturbation theory.

When does the latter method fail and one has to apply nonperturbative approach?

The perturbative approach is always an approximation. It can "fail" mostly when the additional terms aren't small enough to allow perturbation theory to work. One example is the use of perturbation theory as a first approach to the helium atom, where the electron-electron interaction is treated as a perturbation, which it isn't because this interaction is of the same order of magnitude as the electron-nucleus interaction.

 

[Riotto]

One example is the use of perturbation theory as a first approach to the helium atom, where the electron-electron interaction is treated as a perturbation, which it isn't because this interaction is of the same order of magnitude as the electron-nucleus interaction.

Which method other than the perturbation theory should one apply in this example of yours? Do you have some other approximation method such as the variational principle in mind? Thanks!

 

[DrClaude]

Which method other than the perturbation theory should one apply in this example of yours? Do you have some other approximation method such as the variational principle in mind? Thanks!

As only a small number of problems have analytical solutions, if perturbation theory doesn't work, then one has to use numerical methods.

 

[Riotto]

As only a small number of problems have analytical solutions, if perturbation theory doesn't work, then one has to use numerical methods.

Even perturbation theory beyond a certain order will have to be dealt with numerically. Even when you don't use perturbation theory, your numerical algorithm must be based on some framework, some approximation method if not perturbation theory. The question is which framework(s) do people have in mind when they talk about nonperturbative methods. Thanks!

 

[A. Neumaier]

The question is which framework(s) do people have in mind when they talk about nonperturbative methods.

It depends on whose mind you are trying to read!

Everything apart from perturbation theory, such as the Rayleigh-Ritz variational principle, semiclassical (WKB) methods, Hartree-Fock calculations, coupled cluster theory, renormalization group techniques, etc. - and any of these refined by combination with perturbation theory, such as variational perturbation theory.

 

[Riotto]

It depends on whose mind you are trying to read!

Everything apart from perturbation theory, such as the Rayleigh-Ritz variational principle, semiclassical (WKB) methods, Hartree-Fock calculations, coupled cluster theory, renormalization group techniques, etc. - and any of these refined by combination with perturbation theory, such as variational perturbation theory.

So nonperturbative methods are also approximation methods?

 

[A. Neumaier]

So nonperturbative methods are also approximation methods?

Typically, yes.

But there are also families of exactly solvable problems where you can write down an explicit nonperturbative solution. These problems (harmonic oscillator, hydrogen atom, free particle on a sphere, etc.) are rare but instructive, hence often covered in textbooks.

 

[Riotto]

Thanks A. Neumaier. So for problems which are not exactly solvable, and in which the 'unsolvable part' of the Hamiltonian is not "small", other nonperturbative approximation methods become useful for calculating energy eigenvalues and eigenfunctions. Do I get it right?

 

[A. Neumaier]

So for problems which are not exactly solvable, and in which the 'unsolvable part' of the Hamiltonian is not "small", other nonperturbative approximation methods become useful for calculating energy eigenvalues and eigenfunctions. Do I get it right?

Yes.

 

[Riotto]

Thanks for the help. :-)