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SE equation with a strong potential

  1. Sep 19, 2005 #1
    let be the SE with two potentials V and V_0 with N>>>>1 a big number..

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

    then my question is how could we solve it approximately..thanks...
     
  2. jcsd
  3. Sep 19, 2005 #2
    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.
     
  4. Sep 19, 2005 #3
    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:

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

    after that i define the Hamiltonian [tex]H_{0}=-\frac{e\hbar^{2}}{2m}D^{2}\psi+V_{0}\psi [/tex] 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:

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

    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: Sep 19, 2005
  5. Sep 19, 2005 #4

    Dr Transport

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    Solve for the [itex] NV_0 [/itex] and then use perturbation theory for the other potential.....
     
  6. Sep 19, 2005 #5

    reilly

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    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
     
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