A Process of Successive Approximations

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In summary, the conversation discusses the process of successive approximations in Time-Dependent Perturbation Theory for a two-level system. This involves starting with an initial guess of the solution and assuming that solutions follow a certain pattern. The solution is then plugged into the equation, taking into account that the perturbation is small. The conversation ends with a clarification on how to equate terms of different orders in the solution.
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Viona
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Homework Statement
What is the Process of Successive Approximations, how I can use it?
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I need to simple mathematical example to understand it, where I can find that?
I was reading in the Book: Introduction to Quantum Mechanics by David J. Griffiths. In chapter Time-Dependent Perturbation Theory, Section: Two-level system. Every thing was fine till He said He will solve this equation:

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by a process of successive approximations. I have no idea what this process is and I did not find it in some books for Mathematical Methods. Please help me to understand it.
 
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Griffiths must have explained it. Basically, you start with an initial guess of the solution and assume that solutions look like:

$$c_a=\epsilon^0c^0_a+\epsilon^1 c^1_a+\epsilon^2c^2_a+...$$
$$c_b=\epsilon^0c^0_b+\epsilon^1 c^1_b+\epsilon^3c^2_b+...$$

Here the superscript denotes the order of that element. The ##\epsilon^n## too denotes the order of elements (This is just a book-keeping device. You will set it to 1 at the end). Now, you plug these into the expression remembering that ##H'## is small, so, it is itself of the order ##\epsilon##. Once you plug it, you equate terms of same order. Start by equating zeroth order terms first, then plug this solution in the equation you get after equating first order elements and so on...
 
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It worked now, but I equated the the first order of ##\epsilon## (in the left side) with the zeroth order of ##\epsilon ## (in the right side) because the right side is multiplied by ##H^{'}## which is small (of the same order of ##\epsilon##). For example for the first order of ## \epsilon## i wrote:
$$ dc^{(1)}_{a} /dt =-(i /\hbar) H^{'} c^{(0)}_{b}$$
I hope this is correct.
 
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Related to A Process of Successive Approximations

What is "A Process of Successive Approximations"?

"A Process of Successive Approximations" is a problem-solving method that involves breaking down a complex problem into smaller, more manageable parts and finding a solution for each part. The solutions are then combined to solve the original problem.

How does "A Process of Successive Approximations" work?

The process begins by identifying the main problem and breaking it down into smaller, more specific problems. Each problem is then solved using available information and resources. The solutions are then combined and refined until a satisfactory solution for the original problem is achieved.

What are the benefits of using "A Process of Successive Approximations"?

There are several benefits to using this problem-solving method. It allows for a systematic and organized approach to solving complex problems, breaks down a large problem into smaller, more manageable parts, and allows for continuous improvement and refinement of solutions.

When is "A Process of Successive Approximations" most useful?

This method is most useful when dealing with complex problems that cannot be solved using a single solution. It is also helpful when there is limited information or resources available, as it allows for finding solutions with the available resources and continuously improving upon them.

What are some examples of using "A Process of Successive Approximations"?

Some examples of using this problem-solving method include developing new technologies, designing complex systems, and solving mathematical equations. It can also be used in everyday situations, such as planning a project or organizing a large event.

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