Time dependent perturbations and method of succesive approximations

In summary: Griffiths discusses the method of successive approximations as a method for solving a two level system in time dependent perturbation theory. In summary, he says that ca = ca_0 + ca_1.
  • #1
mattlorig
24
0
In chapter nine of Griffiths' Quatum Mechanics text, he talks about the method of succesive approximations as a method for solving a two level system in time dependent perturbation theory.

d(ca)/dt = f(t) cb --> ca_n = int[ f(t') * cb_n-1, dt', 0 t]
d(cb)/dt = g(t) ca --> cb_n = int[ g(t') * ca_n-1, dt', 0 t]

So, for the case were ca(0) = 1 and cb(0) = 0 one would get:

ca1 = int [f(t') * 0, dt', 0, t] = 0
cb1 = int [g(t') * 1, dt', 0, t]

but griffiths says ca1 = 1.

So, my question is the following. Is ca = ca_0 + ca_1 + ca_2 + ...
or, is ca ~ ca_n (with larger n being more precise)?

I hope my question was clear. I should really learn LATEX.
 
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  • #2
Maybe I should clarify my question. Regarding the method of successive approxiamtions griffiths talks about in time dependent perturbation theory, is ca
= SUM (ca_n)
= LIM {ca_n}
?
Hopefully that's easier to understand than my first post.
 
  • #3
Nope,i don't really follow.Why would [itex] c_{a,1} [/itex] be 0...?

Daniel.
 
  • #4
because the integral of zero is zero.
 
  • #5
Okay,alright.What page of Griffiths ?

Daniel.
 
  • #6
p 302 of the (my book is black...I think it's the second newest edition). Griffiths does say "ca_2 includes the zeroth order term; the 2nd order correction would be the integral term alone". But, I think griffiths use of ca_n is ambiguous here. In any case, I'm pretty sure that if I just substitute cb_n-1 into the integral for ca_n, and let ca = SUM( ca_n), that would be correct.
 
  • #7
[tex] \left\{\begin{array}{c} \dot{c}_{a}(t)=\frac{1}{i\hbar}H'_{ab}(t)e^{-i\omega_{0}t}c_{b}(t)\\ \dot{c}_{b}(t)=\frac{1}{i\hbar}H'_{ba}(t)e^{i\omega_{0}t}c_{a}(t) \end{array}\right [/tex](1)

Initial conditions

[tex] \left\{\begin{array}{c}c_{a}(0)=1\\c_{b}(0)=0 \end{array}\right [/tex] (2)

Zero-th approximation

[tex] c_{a}^{(0)}(t)=1 [/tex] (3)

[tex] c_{b}^{(0)}(t)=0 [/tex] (4)

First order.Plug (3) & (4) in the equations (1) and integrate

[tex] \frac{dc_{a}^{(1)}(t)}{dt}=0\Rightarrow c_{a}^{(1)}(t)=\mbox{const}=c_{a}(0)=1 [/tex] (5)

[tex] \frac{dc_{b}^{(1)}(t)}{dt}=\frac{1}{i\hbar}H'_{ba}(t)e^{i\omega_{0}t} \Rightarrow c_{b}^{(1)}(t)=\frac{1}{i\hbar}\int_{0}^{t} H'_{ba}(t')e^{i\omega t'} \ dt [/tex] (6)

Second order.Plug the first order approx given by (5) & (6) into the system (1).

[tex] \frac{dc_{a}^{(2)}(t)}{dt}=\frac{1}{i\hbar}H'_{ab}(t)e^{-i\omega_{0}t}\frac{1}{i\hbar} \int_{0}^{t} H'_{ba}(t')e^{i\omega t'} \ dt \Rightarrow c_{a}^{(2)}(t)=\mbox{const}-\frac{1}{\hbar^{2}}\int_{0}^{t} H'_{ab}(t')e^{-i\omega_{0}t'} \left[\int_{0}^{t'} H'_{ba}(t'')e^{i\omega t''} \ dt'' \right] \ dt' [/tex] (7)

Imposing the condition [itex] c^{(2)}_{a}(0)=1 [/itex] (8),you get the formula (9.18) from Griffiths.

You try now for [itex] c_{b}^{(2)}(t) [/itex].See if you get what Griffiths says:it stays unchanged.

Go for the 3-rd order.:wink: Make sure you got it all clear.

Daniel.
 
Last edited:
  • #8
Thanks Daniel for the help. I will find the 3rd order approximations to ca and cb today. I think my main mistake was just not realizing why ca_1 = 1. Now that that's clear, the rest should be fairly simple.

Also, after doing a bit more research, I found this method in my old Diff Eq book. Apparently it is known as Picard's Iteration Method.

Lastly, I wanted to thank you for always being kind and helping me through the (many) problems I've asked about. You've been extremeley kind and helpful on a number of occasions, and I appreciate it very much.
 
  • #9
Thank you.Well,it resembles in a way the method of iterations when solving the integral equation which gives birth to the Born series.

But it's different.In that case,u get the solution as an infinite sum of perturbative approximations.In this case,it's not a sum anymore.

[tex]c_{a}(t)\neq c_{a}^{(0)}(t)+c_{a}^{(1)}(t)+c_{a}^{(2)}(t)+... [/tex]

,but [tex] c_{a}(t)\simeq c_{a}^{(0)}(t) \ \mbox{in the zero-th order} [/tex]

[tex] c_{a}(t)\simeq c_{a}^{(1)}(t) \ \mbox{in the first order} [/tex]

and so on.I hope u see the difference.U'll have to compare this case with the Born series (as i said before),with the Dyson series and with the series which appear in the stationary perturbative theory for the nondegerate energy levels and see where they look alike and where they are different.

Daniel.
 

Related to Time dependent perturbations and method of succesive approximations

1. What is the method of successive approximations in time dependent perturbations?

The method of successive approximations is a technique used in physics and mathematics to solve problems involving time dependent perturbations. It involves breaking down a complex problem into smaller parts and solving them one at a time, gradually improving the solution with each iteration.

2. How does the method of successive approximations work?

The method of successive approximations works by using an initial guess for the solution and then making small corrections to this guess in each iteration. These corrections are based on the previous iteration's solution and are calculated using a series of mathematical equations.

3. What types of problems can be solved using the method of successive approximations?

The method of successive approximations is commonly used to solve problems in quantum mechanics, such as the time-dependent Schrödinger equation. It can also be applied to problems in other areas of physics, such as classical mechanics and electromagnetism.

4. What are the limitations of the method of successive approximations?

The method of successive approximations is not always guaranteed to converge to the exact solution, especially for highly nonlinear problems. Additionally, it can be computationally expensive and time-consuming, especially for problems with many iterations.

5. Are there any alternatives to the method of successive approximations for solving time dependent perturbation problems?

Yes, there are other techniques that can be used to solve time dependent perturbation problems, such as the variational method and the perturbation theory. These methods may be more suitable for certain types of problems, and it is important to choose the most appropriate method for the specific problem at hand.

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