Time-dependent Perturbation Theory

  • #1
Viona
49
12
Homework Statement
Why the first-order correction equals 1 instead of arbitrary constant?
Relevant Equations
Why the first-order correction equals 1 instead of arbitrary constant?
I was reading in the Book: Introduction to Quantum Mechanics by David J. Griffiths. In chapter Time-Dependent Perturbation Theory, Section 9.12. I could not understand that why he put the first order correction ca(1)(t)=1 while it equals a constant.
tdd.png
 
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  • #2
How are the 's defined? I ask because (1) I'm too lazy to go get my copy of Griffiths and (2) it may provide the answer to your question.
 
  • #3
vela said:
How are the 's defined? I ask because (1) I'm too lazy to go get my copy of Griffiths and (2) it may provide the answer to your question.
He used a process of successive approximations, so for this two particle system the particle starts at state , then at time t=0: ca(0)=1 and cb(0)=0. If there were no perturbation the system will stay there forever, so we can say the zeroth- order terms are: ca(t)=1 and cb(t)=0. To calculate the first order terms we plug the zeroth order terms in the equations [9.13] below:
sddd.png
 
  • #4
Viona said:
I could not understand that why he put the first order correction ca(1)(t)=1 while it equals a constant.
Recall equation [9.15], which says .
 
  • #5
TSny said:
Recall equation [9.15], which says .
In equation [9.15] this is before the perturbation (at time ) no transition happened yet and the particle still in the upper state . But equation [9.17] after perturbation at time t and ca(1)(t) = 1 is the first-order correction. I understand that the zeroth-order but why the first-order correction also =1?
 
  • #6
Viona said:
In equation [9.15] this is before the perturbation (at time ) no transition happened yet and the particle still in the upper state . But equation [9.17] after perturbation at time t and ##c_a(1)(0) = 1c_a(t) = 1## but why the first-order correction also =1?
How many constant functions do you know that fulfill ?
 
  • #7
Viona said:
In equation [9.15] this is before the perturbation (at time ) no transition happened yet and the particle still in the upper state . But equation [9.17] after perturbation at time t and ca(1)(t) = 1 is the first-order correction. I understand that the zeroth-order but why the first-order correction also =1?

Initially I agreed with you, but the passage in your original post defines not as the "first order correction" but the "first order approximation".

To my mind, if we speak of perturbing a system subjec to then the perturbed system is for some small parameter . We would then pose an expansion of the form , and we would then call the first order correction. And if we don't perturb the initial condition as well then indeed and .

But here the text (at least the parts you've posted) doesn't mention a small parameter, and you have described the author's method as "successive approximation" rather than "asymptotic expansion". This would be like obtaining approximate solutions to our perturbed system by the iterative process for which always holds.
 
  • #8
Viona said:
He used a process of successive approximations, so for this two particle system the particle starts at state , then at time t=0: ca(0)=1 and cb(0)=0. If there were no perturbation the system will stay there forever, so we can say the zeroth- order terms are: ca(t)=1 and cb(t)=0. T
The initial conditions at are assumed to hold for all orders of approximation. So, when you solve the first-order equation , you can use the initial condition to show that for all for the first-order approximation.
 
  • #9
pasmith said:
Initially I agreed with you, but the passage in your original post defines not as the "first order correction" but the "first order approximation".

To my mind, if we speak of perturbing a system subjec to then the perturbed system is for some small parameter . We would then pose an expansion of the form , and we would then call the first order correction. And if we don't perturb the initial condition as well then indeed and .

But here the text (at least the parts you've posted) doesn't mention a small parameter, and you have described the author's method as "successive approximation" rather than "asymptotic expansion". This would be like obtaining approximate solutions to our perturbed system by the iterative process for which always holds.
He said he is going to use a process of successive approximations, I am not familiar with the "asymptotic expansion" so I can not tell if this what he did
as.png
 

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