ND Pertubation theory: Second order correction

In summary, the conversation was about calculating the second-order correction to the ground-state energy of a system using a perturbed Hamiltonian. The question was about identifying a mistake in the calculation and obtaining the correct result. The solution involved using the ladder operators and specific values for "m" in each term to determine the matrix elements.
  • #1
Schwarzschild90
113
1

Homework Statement


Calculate the second-order correction to the ground-state energy of the stationary states of the system.

The perturbed Hamiltonian is:

H' =- (/gamma /hbar m /omega)/2 (a+ - a-) ^2

2 & 3. Relevant equatio and the attempt at a solution
QM d.PNG

This is not right. I follow the same procedure as letting the ladder operators act on eigenvalues, i.e. that a+|n> becomes (√(n+1))|n+1>
 
Last edited:
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  • #2
Did you have a question?
 
  • #3
I didn't notice at first that my post is lacking some details.

My question is this: Where in my calculations did I make a mistake in calculating a solution to the problem?
 
  • #4
Schwarzschild90 said:

Homework Statement


Calculate the second-order correction to the ground-state energy of the stationary states of the system.

The perturbed Hamiltonian is:

H' =- (/gamma /hbar m /omega)/2 (a+ - a-) ^2

2 & 3. Relevant equatio and the attempt at a solution
View attachment 101566
This is not right. I follow the same procedure as letting the ladder operators act on eigenvalues, i.e. that a+|n> becomes (√(n+1))|n+1>
Why do you have factors of "n" all over the place? There should be only an "m" and in each term is has a specific value (like m=2!) so you should get numbers for all the matrix elements (and most of them are zero)
 
  • #5
How do I figure out which value of m each matrix element should have?
 
  • #6
I figured it out.

I got the following result: sqrt(1)sqrt(2).

(Which is correct)
 

Related to ND Pertubation theory: Second order correction

1. What is perturbation theory?

Perturbation theory is a mathematical tool used to approximate solutions to problems that cannot be solved exactly. It involves breaking down a complex problem into smaller, more manageable parts and solving them one at a time. In the context of quantum mechanics, perturbation theory is used to find corrections to the energy levels of a system that is slightly perturbed from its known or ideal state.

2. What is second order correction in perturbation theory?

Second order correction in perturbation theory is the second term in the series expansion of the perturbation solution. It takes into account the effects of the perturbation to the first order correction and calculates a more accurate approximation of the solution. Second order correction is usually more accurate than first order correction, but it may not be enough for highly perturbed systems.

3. What is the difference between first and second order correction?

The main difference between first and second order correction is the level of accuracy in the approximation of the solution. First order correction only considers the effects of the perturbation to the first order, while second order correction takes into account the effects of the perturbation to the second order. Therefore, second order correction is more accurate than first order correction.

4. When is second order correction necessary?

Second order correction is necessary when the perturbation is large and first order correction is not accurate enough. In some cases, second order correction may also be necessary for small perturbations if the system is highly sensitive to perturbations.

5. What are the limitations of second order correction in perturbation theory?

Second order correction is limited in its accuracy, especially for highly perturbed systems. In some cases, higher order corrections may be needed to obtain a more accurate solution. Additionally, perturbation theory is only applicable for small perturbations and may not work for large perturbations or strong interactions.

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