# SE equation with a strong potential

## Main Question or Discussion Point

let be the SE with two potentials V and V_0 with N>>>>1 a big number..

$$i\hbar\frac{d\psi}{dt}=-\frac{\hbar^{2}}{2m}D^{2}\psi+(V+NV_{0})\psi$$

then my question is how could we solve it approximately..thanks...

## Answers and Replies

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If you know a solution for V_0, then change t as tN and x² as Nx² and V as V/N. Then perform a first order development with V/N as pertubation.

Note that V0 probably depends on x, and therefore you need to manage the change of variables x² -> Nx² in the potential term too.
If V0 as a dependence like VO(x/xref), then xref² has simply to be replaced by N xref². Should be simple.

thanks..i manage in a very similar way described by you:
first i divide all equation by N e=1/N tehn we would have:

$$ie\hbar\frac{d\psi}{dt}=-\frac{e\hbar^{2}}{2m}D^{2}\psi+(eV+V_{0})\psi$$

after that i define the Hamiltonian $$H_{0}=-\frac{e\hbar^{2}}{2m}D^{2}\psi+V_{0}\psi$$ to solve this i use the WKB approach as e<<<1

then after that i treat V as a perturbation and solve it to first and second order....

But what is this good for?..let,s suppose we have a Lagrangian of the form L0+V with the potential then we could add a term NV0 in the Feynmann Path-integral, to obtain the K0 propagator we use the development of Taylor of S near its classical solution in the form:

$$S[\phi]=S[\phi_{c}]+(1/2)\delta^{2}S[\phi_{c}]\phi^{2}+.....$$

then we evaluate this functional integral to calculte K0,for the rest we use perturbation theory to calculate the corrections to first and second order...

Last edited:
Dr Transport
Gold Member
Solve for the $NV_0$ and then use perturbation theory for the other potential.....

reilly
Seems to me that more info is required. WKB is great, but not always valid. Can one solve with either potential; maybe one could solve exacly with both -- two square wells. Do you have a specific problem in mind? Are you talking bound states or scattering, or perhaps both? Given the magnitudes involved, will first order perturbation theory work? (One solvable case is a 1/r potential, with a very large angular momentum, with n=L*(L+1) so the effective potential is (-)q*q/r + n/(r*r), a good test case.

Regards,
Reilly Atkinson

Regards,
Reilly Atkinson