Is this a homework question?Riotto said: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.
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 said: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.
DrClaude said: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.
The non-perturbative approach is simply trying to solve the problem at hand without resorting to perturbation theory.Riotto said:What is the nonperturbative approach to quantum mechanics as opposed to perturbative one?
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 said:When does the latter method fail and one has to apply nonperturbative approach?
DrClaude said: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.
As only a small number of problems have analytical solutions, if perturbation theory doesn't work, then one has to use numerical methods.Riotto said: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 said:As only a small number of problems have analytical solutions, if perturbation theory doesn't work, then one has to use numerical methods.
It depends on whose mind you are trying to read!Riotto said:The question is which framework(s) do people have in mind when they talk about nonperturbative methods.
A. Neumaier said: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.
Typically, yes.Riotto said:So nonperturbative methods are also approximation methods?
Yes.Riotto said: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?
Perturbative quantum mechanics is an approach to solving quantum mechanical problems by expanding the solution in a power series, with the assumption that the perturbation (or disturbance) to the system is small. Nonperturbative quantum mechanics, on the other hand, involves solving the full equations of motion without making any assumptions about the size of the perturbation.
Perturbative methods are typically used when the perturbation is small enough to be considered a small disturbance to the system. Nonperturbative methods are necessary for larger perturbations or for systems that cannot be approximated as a small disturbance.
No, perturbative methods are only applicable to certain types of problems where the perturbation is small enough. In cases where the perturbation is large, nonperturbative methods must be used instead.
Perturbative methods are typically less accurate than nonperturbative methods because they involve making approximations and assumptions about the system. Nonperturbative methods, on the other hand, provide a more accurate solution by solving the full equations of motion without making any assumptions.
Some examples include systems with strong interactions, such as the interactions between quarks in a hadron, or systems with large energy scales, such as the early universe. These systems cannot be accurately described using perturbative methods and require nonperturbative approaches.