L9.1 The interaction picture and time evolution

TL;DR

Time-dependent perturbation theory is introduced as a method to study systems with time-dependent perturbations, and the interaction picture is presented as a way to separate the effects of the time-independent and time-dependent parts of the Hamiltonian.

Transcript

PROFESSOR: We're finished with WKB. In recitation, you saw some transmission across the barrier and that's also included in the notes. That's an important application of WKB and should look at it. And today we're going to start with a new topic. It's time-dependent perturbation theory. And time-dependent perturbation theory is going to keep us busy... Read More

Key Insights

• ⌛ Time-dependent perturbation theory is a valuable tool for studying systems with time-dependent perturbations, offering insights into various practical applications.
• 👶 Energy eigenstates cannot be defined for a time-dependent Hamiltonian, presenting new challenges in understanding the behavior of quantum systems.
• ⌛ The interaction picture provides a way to separate the effects of the time-independent and time-dependent parts of the Hamiltonian, allowing for a more manageable analysis.

Questions & Answers

Q: Why is time-dependent perturbation theory important and what types of problems can it be applied to?

Time-dependent perturbation theory is important because it allows us to study systems with time-dependent perturbations, such as radiation problems and ionization problems. It provides a method to calculate transition probabilities and understand the behavior of quantum systems under changing conditions.

Q: How is the Hamiltonian divided into a time-independent part and a time-dependent perturbation?

The Hamiltonian is divided into H0, which represents the time-independent part of the system, and delta H, which represents the time-dependent perturbation. H0 corresponds to the known Hamiltonian before the perturbation is applied, while delta H represents the changes induced by the perturbation.

Q: Why can't energy eigenstates be defined for a time-dependent Hamiltonian?

Energy eigenstates are defined for time-independent Hamiltonians because their eigenvalues remain constant over time. However, for a time-dependent Hamiltonian, the eigenvalues are no longer constant, making it impossible to define energy eigenstates for the perturbed system.

Q: How does the interaction picture separate the effects of the time-independent and time-dependent parts of the Hamiltonian?

In the interaction picture, the time-dependent perturbation is treated in the Schrödinger picture, where the wave function varies in time, and the time-independent part is treated in the Heisenberg picture, where the operators acquire additional time-dependence due to the interaction. This separation allows for a clearer understanding of how the system evolves under the influence of both parts of the Hamiltonian.

Summary & Key Takeaways

• Time-dependent perturbation theory is a powerful tool for studying systems with time-dependent perturbations, such as radiation problems and ionization problems.

• The Hamiltonian is decomposed into a time-independent part (H0) and a time-dependent perturbation (delta H).

• Energy eigenstates, which were used in time-independent perturbation theory, cannot be defined for a time-dependent Hamiltonian.

• The interaction picture is introduced, where the time-dependent perturbation is treated in the Schrödinger picture and the time-independent part is treated in the Heisenberg picture.