WKB and perturbation theory.

[tpm]

for a Hamiltonian [tex] H=H_0 + \epsilon V(x) [/tex]

my question is (for small epsilon) can WKB and perturbative approach give very different solutions ?? to energies eigenvalues and so on the index '0' means that is the Hamiltonian of a free particle.

problem arises perhaps in calculation of:

[tex] \int \mathcal D [x] exp(iS[x]/\hbar)[/tex]

with the action [tex] S[x]=S_0 [x] +\epsilon V[x] [/tex]

here the main problem is that in perturbation theory the functional integral may be divergent (due to IR and UV divergences) but in WKB (semiclassical approach) the integral can be 'calculated' (given finite meaning),

hence i'd like to know if at least for perturbative case you can use WKB approach (with some re-scaled constant) to deal with perturbation theory..note that for the 'free particle' no interaction WKB gives exact methods..thankx.

 

[StatMechGuy]

Technically, WKB and "perturbation theory" are both perturbative expansions in what are presumed to be "small quantities". In the case of your "perturbation theory", the small parameter is the [tex]\epsilon[/tex], whereas in the WKB approximation, you have expanded the propagator to first order in [tex]\hbar[/tex], so evidently the total action of the trajectories is large compared to Planck's constant, which sets the scale for action in quantum mechanics.

Thus, you can get two very different answers depending on what exactly is "small".

 

[denian]

I would say that both WKB and perturbation theory can give different solutions for the Hamiltonian H=H_0 + \epsilon V(x) when \epsilon is small. However, the results may not be drastically different, as both methods aim to provide an approximate solution to the problem.

In perturbation theory, we assume that \epsilon is a small parameter and expand the solution in a power series. This approach is useful when the perturbation term is small compared to the unperturbed Hamiltonian (H_0). However, as you mentioned, there can be issues with divergences in the functional integral that may arise from IR and UV divergences.

On the other hand, WKB (or semiclassical) approach involves treating the classical limit of the quantum system, where \hbar is small. Here, we use the stationary phase approximation to evaluate the functional integral, which can give a finite result in some cases. However, this method may not be accurate for highly excited states or when the potential is not well-behaved.

In general, both WKB and perturbation theory can be used to solve problems involving small \epsilon, but it is important to consider the limitations and potential issues of each method. It may be possible to combine both approaches, as you suggested, but this would require careful consideration and may not always be applicable. Ultimately, the choice of method would depend on the specific problem at hand and the accuracy required for the solution.