Perturbation Theory: Solving a/r+br Potential

In summary, perturbation theory is a mathematical method used to approximate solutions to difficult problems by breaking them down into simpler parts and adding the solutions together. In the case of the a/r+br potential, perturbation theory allows for the approximation of energy levels and wavefunctions by treating the perturbation as a small correction to the original Hamiltonian. This potential has significance in physics, particularly in quantum mechanics and atomic/molecular physics. However, there are limitations to perturbation theory when applied to this potential, as it may fail for highly excited or degenerate systems. To improve accuracy, higher-order terms can be included in the perturbation expansion, but this also increases computational complexity.
  • #1
Korova
2
0
A quick question from a high school student,

In perturbation theory, what is to be done with the found energy correction? I'm working out the solution to an a/r+br potential and using br as the perturbation. I set up the integral and normalized, but what do I do with the expression that I'm getting?

Thanks in advance,
Korova
 
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  • #2
bleh... nevermind... I figured it out.
 

FAQ: Perturbation Theory: Solving a/r+br Potential

What is perturbation theory?

Perturbation theory is a mathematical method used to approximate solutions to problems that are too difficult to solve exactly. It involves breaking down a complex problem into simpler, solvable parts and then adding these solutions together to get an overall solution.

How does perturbation theory help solve the a/r+br potential?

In the case of the a/r+br potential, perturbation theory allows us to approximate the energy levels and wavefunctions of the system by treating the perturbation (in this case, the br term) as a small correction to the original Hamiltonian (a/r). This allows us to solve for the energy levels and wavefunctions using techniques from linear algebra and calculus.

What is the significance of the a/r+br potential in physics?

The a/r+br potential is a simplified model of the potential energy of an electron in the presence of both a Coulomb potential (a/r) and a linear potential (br). This potential is commonly used in quantum mechanics to study the behavior of charged particles in electric fields, and it has applications in areas such as atomic and molecular physics.

Are there any limitations to perturbation theory in solving the a/r+br potential?

Yes, there are limitations to perturbation theory when applied to the a/r+br potential. This method is only accurate when the perturbation is small compared to the original Hamiltonian, and it may fail to give accurate results for highly excited states or highly degenerate systems. In these cases, more advanced techniques may be needed.

How can perturbation theory be extended to higher orders in the a/r+br potential?

In perturbation theory, we can improve our approximations by including higher-order terms in the perturbation expansion. This involves solving for corrections to the energy levels and wavefunctions at each order, and then adding these corrections to the previous approximations. This can lead to more accurate results, but it also becomes more computationally intensive as the order increases.

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