# First-Order Perturbation Theory Derivation in Griffiths

• BucketOfFish
In summary, the conversation discusses a claim made in Griffiths's quantum book regarding the relation between the unperturbed Hamiltonian and the first-order wavefunction correction. The claim states that <\psi^0|H^0\psi^1> = <H^0\psi^0|\psi^1>, and the conversation raises concerns about this claim in relation to Hermitian operators and eigenfunctions. The final conclusion is that the relation must hold for all states, not just eigenstates, if the operator is Hermitian.
BucketOfFish

## Homework Statement

On page 251 of Griffiths's quantum book, when deriving a result in first-order perturbation theory, the author makes the claim that $$<\psi^0|H^0\psi^1> = <H^0\psi^0|\psi^1>$$ where $H^0$ is the unperturbed Hamiltonian and $\psi^0$ and $\psi^1$ are the unperturbed wavefunction and its first-order correction.

## Homework Equations

$$H^0\psi^0=E^0\psi^0$$

## The Attempt at a Solution

From the derivations I've seen of Hermitian operators, I seem to understand that both sides of the expression in the braket have to be eigenfunctions of the operator for the operator to be symmetric. If this is the case, then I feel that Griffiths's explanation may be lacking something, since there is no guarantee that the first-order wavefunction correction is an eigenfunction of the unperturbed Hamiltonian. In this case, you could not switch the Hamiltonian operator around like this.

I must be misunderstanding something! Can someone help me clear this up?

Last edited:
That relation must hold for all states, not just eigenstates, if the operator is Hermitian.

Ah yes, I forgot that observables are not just for stationary states. Thanks for your help, vela!

## 1. What is First-Order Perturbation Theory?

First-Order Perturbation Theory is a technique used in quantum mechanics to approximate the energy and wavefunction of a perturbed system. It is based on the assumption that the perturbation is small enough to be treated as a first-order correction to the unperturbed system.

## 2. How is First-Order Perturbation Theory derived?

The derivation of First-Order Perturbation Theory involves using the time-independent Schrödinger equation and perturbation theory to obtain an expression for the first-order correction to the energy and wavefunction of a perturbed system. This derivation can be found in many quantum mechanics textbooks, including Griffiths.

## 3. What are the limitations of First-Order Perturbation Theory?

First-Order Perturbation Theory is only accurate for small perturbations and can break down if the perturbation is too large. It also assumes that the perturbation is time-independent and that there is no degeneracy in the unperturbed system. Additionally, it is only applicable to non-degenerate systems.

## 4. How is First-Order Perturbation Theory used in practical applications?

First-Order Perturbation Theory is used in many areas of physics, including quantum chemistry, solid-state physics, and nuclear physics. It is particularly useful in calculating the electronic structure of atoms and molecules, as well as understanding the behavior of quantum systems under external perturbations.

## 5. Are there higher-order perturbation theories?

Yes, there are higher-order perturbation theories, such as Second-Order and Third-Order Perturbation Theory, which take into account higher-order corrections to the energy and wavefunction of perturbed systems. However, these theories are more complex and require more computational resources, so First-Order Perturbation Theory is often the most practical choice in many situations.

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