What is "to the first order in H"?

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In summary, the expression means that for a system in which only terms of first order in ##H'## are kept, the second order in ##H'## is zero.
  • #1
Haorong Wu
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I'm learning Griffiths' QM (3rd edn).

In Chapter 11 (Quantum Dynamics), there is an expression I'm not familiar with:

## \left| C_b \right|^2 = \left[ - \frac i \hbar \int_0^t H'_{ba} \left( t' \right) e^{i \omega_0 t'} \, dt' \right] \left[ \frac i \hbar \int_0^t H'_{ba} \left( t' \right) e^{-i \omega_0 t'} \, dt' \right] =0 ## (to first order in ##H'##),
where ##H'## is a time-dependent perturbation of a two-level system.

There are some other places where the expression of "to the first order in H" appears. I can't remember anywhere I have learnd the expressions in calculus or other mathematics courses.
 
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  • #2
It means that in deriving the expression, only terms of first order in ##H'## where kept, and that higher order terms (##H'^2##, ##H'^3##, etc.) have been dropped.
 
  • #3
DrClaude said:
It means that in deriving the expression, only terms of first order in ##H'## where kept, and that higher order terms (##H'^2##, ##H'^3##, etc.) have been dropped.
Hi, DrClaude. I'm still confused. Shouldn't ##\left[ \int H'_{ba} \left( t' \right) \cdot \int H'_{ba} \left( t' \right) \right]## mean second order in ##H'##?
 
  • #4
I only have the 2nd ed. of Griffiths, so I can't check, but this looks like perturbation theory, so it would be ##c_b## that is expressed to 1st order in ##H'##.
 
  • #5
DrClaude said:
I only have the 2nd ed. of Griffiths, so I can't check, but this looks like perturbation theory, so it would be ##c_b## that is expressed to 1st order in ##H'##.

Yes, it is the perturbation theory. Well, the expression is from the solutions of the 2nd edn. The problem is 9.4.

Here is the solution.
9.4.jpg


Would you mind look at it in your freetime?

Thank you so much!
 
  • #6
I get it now. It is ##|c_b|^2 = 0## to first order in ##H'##.
 
  • #7
DrClaude said:
I get it now. It is ##|c_b|^2 = 0## to first order in ##H'##.
I'm sorry I still can not see how can it be first order in ##H'##. How does ##\left[ \int H'_{ba} \left( t' \right) \cdot \int H'_{ba} \left( t' \right) \right] ## relate to the first order in ## H'##?

Wait, is it because ##\left| H' \right| =H_{aa} H_{bb} -H_{ab} H_{ba}## and here ## H_{aa}=H_{bb}=0 ##?
 
  • #8
Haorong Wu said:
I'm sorry I still can not see how can it be first order in ##H'##. How does ##\left[ \int H'_{ba} \left( t' \right) \cdot \int H'_{ba} \left( t' \right) \right] ## relate to the first order in ## H'##?

Wait, is it because ##\left| H' \right| =H_{aa} H_{bb} -H_{ab} H_{ba}## and here ## H_{aa}=H_{bb}=0 ##?
No. As you noticed yourself
Haorong Wu said:
Shouldn't ##\left[ \int H'_{ba} \left( t' \right) \cdot \int H'_{ba} \left( t' \right) \right]## mean second order in ##H'##?
This expression is 2nd order in ##H'##, therefore, to first order, ##|c_b|^2 = 0##.
 
  • #9
DrClaude said:
No. As you noticed yourself

This expression is 2nd order in ##H'##, therefore, to first order, ##|c_b|^2 = 0##.

:woot:Ah...I understand it now. Thanks!
 
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1. What does it mean to be "to the first order in H"?

Being "to the first order in H" means that a mathematical expression or equation is simplified by considering only the first term in a series expansion involving the variable H. This is often used in scientific calculations to make the problem more manageable and to provide a good approximation of the solution.

2. How is "to the first order in H" different from "to the second order in H"?

"To the first order in H" considers only the first term in a series expansion, while "to the second order in H" considers the first two terms. This means that "to the second order in H" is a more accurate approximation than "to the first order in H".

3. Why is "to the first order in H" commonly used in scientific calculations?

Using "to the first order in H" simplifies a problem by considering only the most significant term in a series expansion. This makes the problem more manageable and often provides a good approximation of the solution, which is often sufficient for scientific calculations.

4. Can "to the first order in H" be used for any variable?

Yes, "to the first order in H" can be used for any variable. It is a general mathematical concept that can be applied to any variable in a series expansion. However, the accuracy of the approximation will depend on the specific problem and the value of the variable being considered.

5. What are the limitations of using "to the first order in H" in scientific calculations?

The main limitation of using "to the first order in H" is that it provides a simplified approximation of the solution, which may not be accurate enough for some scientific applications. In these cases, a higher order approximation, such as "to the second order in H", may be necessary for more precise results.

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