L1.3 Calculating the energy corrections

TL;DR

The energy corrections in perturbation theory can be calculated by finding the expectation value of the perturbation in the unperturbed state.

Transcript

PROFESSOR: The left hand side here is 0 because H0 and N0 by taking the eigenstate equation is equal to En0 times N0. So this really evaluates to En0 and therefore the two cancel, so this thing is 0. The right hand side, this is a number, En1. It's a number. It's in an expectation value. N0 has a unit expectation value, so this is En1. That's the n... Read More

Key Insights

• ❓ The energy corrections in perturbation theory can be found by evaluating the expectation value of the perturbation operator in the unperturbed state.
• 💱 The first correction to the energy does not depend on how the state changes but only on the original state and the perturbation.
• 🪈 The kth order energy correction can be calculated if the (k-1)th correction to the state is known.
• 👻 Perturbation theory allows for systematic analysis of perturbed systems by determining energy corrections.
• ✋ The generalization formula provides a simple and powerful tool for calculating higher-order energy corrections.
• 🆘 Knowing the energy corrections in perturbation theory can help in understanding the behavior of physical systems under perturbations.
• 🦾 Perturbation theory is widely used in various domains of physics, providing insights into quantum mechanics and other fields.

Questions & Answers

Q: How is the left-hand side of the equation zero in perturbation theory?

The left-hand side of the equation is zero because the eigenstate equation evaluates to zero, simplifying the calculation of the energy correction.

Q: How can the first correction to the energy be calculated?

The first correction to the energy can be obtained by finding the expectation value of the perturbation operator in the unperturbed state, without needing to know how the state changes.

Q: What is the significance of the generalization formula in perturbation theory?

The generalization formula allows for calculating higher-order corrections to the energy by knowing the previous correction, providing a systematic approach to analyzing perturbed systems.

Q: How does the generalization formula relate to the first correction in perturbation theory?

The generalization formula builds upon the concept of the first correction, stating that the kth order energy correction can be determined if the (k-1)th correction to the state is known.

Summary & Key Takeaways

• The left-hand side of the equation is zero because the eigenstate equation evaluates to zero. The right-hand side is a number representing the expectation value of the perturbation operator.

• The first correction to the energy can be obtained by calculating the expectation value of the perturbation in the unperturbed state without needing to know how the state changes.

• The generalization of the formula involves calculating higher-order corrections to the energy by knowing the previous correction.