# Is the Correction $$c_{nn}^{(1)}$$ Zero in Quantum Perturbation Theory?

• T-7
In summary, the poster successfully showed that the correction c_{nn}^{(1)} is zero by setting the real part to zero and stating that the imaginary part can be set to zero without loss of generality. This result holds for any value of n since the summation is over all possible values of m.
T-7

## Homework Statement

Substituting

$$|n> = |n^{0}> + \sum_{m}c_{nm}|m^{0}>$$

into the normalisation condition

$$<n|n> = 1$$

show that the correction $$c_{nn}^{(1)}$$ is zero.

## The Attempt at a Solution

$$<n|n> = 1 => \left( <n^{0}| + \sum_{m}c_{nm}^{*}<m^{0}| \right) \left( |n^{0}> + \sum_{m}c_{nm}|m^{0}> \right) = 1$$

$$<n^{0}|n^{0}> + \sum_{m}c_{nm}<n^{0}|m^{0}> + \sum_{m}c_{nm}^{*}<m^{0}|n^{0}> + \sum_{m}c_{nm}^{*}\sum_{m'}c_{nm'}<m^{0}|m^{0}> = 1$$

$$1 + c_{nn} +c_{nn}^{*} + \sum_{m}c_{nm}^{*}\sum_{m'}c_{nm'} = 1$$

$$c_{nn} +c_{nn}^{*} + \sum_{m}c_{nm}^{*}\sum_{m'}c_{nm'} = 0$$

Now I believe I can say:

$$c_{nn} = \lambda c_{nn}^{(1)} + \lambda^{2} c_{nn}^{(2)} + ...$$

and so

$$c_{nn}^{*} = \lambda c_{nn}^{*(1)} + \lambda^{2} c_{nn}^{*(2)} + ...$$

And thus:

$$(\lambda c_{nn}^{(1)} + \lambda^{2} c_{nn}^{(2)} + ...) + (\lambda c_{nn}^{*(1)} + \lambda^{2} c_{nn}^{*(2)} + ...) + \sum_{m}(\lambda c_{nm}^{*(1)} + \lambda^{2} c_{nm}^{*(2)} + ...) \sum_{m'}(\lambda c_{nm'}^{(1)} + \lambda^{2} c_{nm'}^{(2)} + ...)= 0$$

Comparing coefficients of $$\lambda$$

$$\lambda c_{nn}^{(1)} + \lambda c_{nn}^{*(1)} = 0$$

Hence

$$Re(c_{nn}^{(1)}) = 0$$

Set imaginary part to zero; it is arbitrary.

So

$$c_{nn}^{(1)} = 0$$

Could someone tell me if this is along the right lines. I can't seem to do any more than prove that the real part is zero (if indeed I have proved this correctly), and it seems to me that the imaginary part, being arbitrary, could be set to zero. But perhaps this is too much hand waving, and something more should be said? Or perhaps my proof is wrong fullstop?

Cheers.

Your approach is definitely on the right track. You correctly showed that the real part of c_{nn}^{(1)} must be zero, and it is also true that the imaginary part can be set to zero since it is arbitrary. However, to make your proof more rigorous, you could explicitly state that the imaginary part is arbitrary and that it can be set to zero without loss of generality. Additionally, you could also mention that this result holds for any value of n since the summation is over all possible values of m. Overall, your proof is correct and well-structured. Good job!

## What is a quantum perturbation correction?

A quantum perturbation correction is a mathematical technique used in quantum mechanics to account for the effects of small perturbations or disturbances on a quantum system. These perturbations can include external forces or interactions between particles.

## Why are quantum perturbation corrections necessary?

Quantum perturbation corrections are necessary because they allow us to more accurately describe and predict the behavior of quantum systems. Without these corrections, our understanding of quantum mechanics would be incomplete and we would not be able to make precise predictions about the behavior of quantum systems.

## How do quantum perturbation corrections work?

Quantum perturbation corrections work by treating the perturbation as a small additional term in the quantum Hamiltonian, which is the mathematical description of a quantum system. This term is then used in a series of calculations to determine the corrected energy levels and other properties of the system.

## What is the difference between first and second order quantum perturbation corrections?

The difference between first and second order quantum perturbation corrections lies in the level of accuracy. First order corrections take into account the first order effect of the perturbation, while second order corrections take into account both the first and second order effects. Second order corrections are therefore more accurate, but also more computationally intensive.

## What are some applications of quantum perturbation corrections?

Quantum perturbation corrections have a wide range of applications in various fields, including atomic and molecular physics, solid state physics, and quantum chemistry. They are used to study and understand the behavior of quantum systems, as well as to develop new technologies such as quantum computers and quantum communication devices.

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