- #1

eljose

- 492

- 0

[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...

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter eljose
- Start date

- #1

eljose

- 492

- 0

[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

lalbatros

- 1,256

- 2

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.

- #3

eljose

- 492

- 0

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 Feynman 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...

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 Feynman 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:

- #4

- 2,589

- 747

Solve for the [itex] NV_0 [/itex] and then use perturbation theory for the other potential...

- #5

reilly

Science Advisor

- 1,077

- 1

Regards,

Reilly Atkinson

Regards,

Reilly Atkinson

Share:

- Replies
- 18

- Views
- 673

- Last Post

- Replies
- 2

- Views
- 448

- Replies
- 6

- Views
- 519

- Last Post

- Replies
- 15

- Views
- 706

- Last Post

- Replies
- 21

- Views
- 640

- Last Post

- Replies
- 5

- Views
- 353

- Replies
- 1

- Views
- 322

- Replies
- 2

- Views
- 321

- Replies
- 4

- Views
- 357

- Replies
- 7

- Views
- 209