# QFT for Gifted Amateur - Problem 2.2

• Daniel_C
In summary, the conversation discusses using first-order time-independent non-degenerate perturbation theory to solve for the energy eigenvalues of a Hamiltonian with a perturbation term. The formula for the energy eigenvalues is provided and the perturbation Hamiltonian is identified. The speaker expresses confusion about how to proceed with the calculation, specifically with the use of creation and annihilation operators. They ask for help to calculate the first order perturbation energies.
Daniel_C
Homework Statement
For the Hamiltonian $$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 \hat{x}^2 + \lambda \hat{x}^4$$ where lambda is small, use the creation annihilation operators with perturbation theory to find the energy eigenvalues of all the levels.
Relevant Equations
$$\hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{mw} \hat{p} \right)$$
$$\hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} - \frac{i}{mw} \hat{p} \right)$$
I'm getting confused by the perturbation theory aspect of problem 2.2 in this book. We have to show that the energy eigenvalues are given by

$$E_n = \left(n + \frac{1}{2}\right) \hbar \omega + \frac{3\lambda}{4} \left(\frac{\hbar}{m\omega}\right)^2 (2n^2 + 2n + 1)$$

For the Hamiltonian

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 \hat{x}^2 + \lambda \hat{x}^4$$

I'm going to try and apply first-order time-indepedent non-degenerate perturbation theory to this problem and use the equation
$$E^'_n = < n^0 | H' | n^0>$$
where H' is the perturbation of the Hamiltonian.

I've identified the perturbation of the Hamiltonian and have arrived at

$$E^'_n = < n^0 | \lambda \hat{x}^4 | n^0>$$

I'm getting confused as to start actually calculating from this point I've set myself up at with all the new creation and annihilation operators they've introduced. Could someone help put me on the right track?

My best guess would just be to explicitly evluate the integral by plugging in the quantum harmonic oscillator wave functions.

What is the perturbation Hamiltonian in terms of ##a## and ##a^{\dagger}##? Use that to calculate the first order perturbation energies. If you run into more problems, please post your work, not just the bottom line.

## 1. What is QFT for Gifted Amateur - Problem 2.2?

QFT for Gifted Amateur - Problem 2.2 is a problem from the book "Quantum Field Theory for the Gifted Amateur" by Tom Lancaster and Stephen J. Blundell. It is a theoretical exercise that involves using quantum field theory to solve a problem related to electromagnetism.

## 2. What is the purpose of QFT for Gifted Amateur - Problem 2.2?

The purpose of QFT for Gifted Amateur - Problem 2.2 is to provide a practical application of quantum field theory in the field of electromagnetism. It allows readers to practice and improve their understanding of the theory while solving a real-world problem.

## 3. What background knowledge is needed to solve QFT for Gifted Amateur - Problem 2.2?

To solve QFT for Gifted Amateur - Problem 2.2, one should have a basic understanding of quantum field theory, electromagnetism, and mathematical concepts such as vector calculus and complex numbers. It is recommended to have prior knowledge of the material covered in the first few chapters of the book.

## 4. Is QFT for Gifted Amateur - Problem 2.2 suitable for beginners?

No, QFT for Gifted Amateur - Problem 2.2 is not suitable for beginners. It is a challenging problem that requires a solid understanding of quantum field theory and electromagnetism. It is recommended for advanced students or those with a strong background in physics and mathematics.

## 5. Are there any resources available to help solve QFT for Gifted Amateur - Problem 2.2?

Yes, there are resources available to help solve QFT for Gifted Amateur - Problem 2.2. The book itself provides detailed explanations and solutions for the problem, and there are also online forums and study groups where readers can discuss and collaborate on solving the problem together.

• Advanced Physics Homework Help
Replies
2
Views
488
• Advanced Physics Homework Help
Replies
2
Views
1K
• Advanced Physics Homework Help
Replies
24
Views
1K
• Advanced Physics Homework Help
Replies
13
Views
2K
• Advanced Physics Homework Help
Replies
1
Views
2K
• Quantum Physics
Replies
3
Views
490
• Advanced Physics Homework Help
Replies
4
Views
1K
• Advanced Physics Homework Help
Replies
4
Views
1K
• Advanced Physics Homework Help
Replies
1
Views
1K
• Advanced Physics Homework Help
Replies
3
Views
1K