Time-Independent Perturbation Theory

In summary, the speaker is working on a physics project that involves using perturbation theory to calculate corrections to the eigenvalues and eigenvectors of a perturbed matrix. The unperturbed matrix is real and symmetric, while the perturbed matrix is complex and non-Hermitian. The speaker is unsure if they can use the standard matrix perturbation theory for Hermitian Hamiltonians, but after reviewing a relevant chapter, they do not see any mention of the perturbed matrix needing to be Hermitian. Therefore, they believe they can use the standard approach.
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Homework Statement


I am working on a physics project for which I need to use perturbation theory to calculate the first- and second-order corrections to the eigenvalues and eigenvectors of a perturbed matrix. The unperturbed matrix is real and symmetric, and the eigenvalues and eigenvectors are easy to calculate. However, the perturbed matrix is complex and non-Hermitian (the perturbation introduces complex components on the main diagonal). I am new to perturbation theory. My question is whether I can use the standard matrix perturbation theory for Hermitian Hamiltonians, as explained in Chapter 6 of "Introduction to Quantum Mechanics" by Griffiths. Clearly, the unperturbed matrix needs to be Hermitian, but it doesn't seem as if the perturbed one has to be. I would just like to double-check this.

Homework Equations


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The Attempt at a Solution


I went through the derivation in Griffiths, and didn't see the assumption that the perturbed matrix has to be Hermitian being used anywhere.
 
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  • #2
I've never used perturbation theory with non-Hermitian matrices, but I don't foresee any problem in using the standard approach.
 

FAQ: Time-Independent Perturbation Theory

What is Time-Independent Perturbation Theory?

Time-Independent Perturbation Theory is a mathematical technique used in quantum mechanics to approximate the energy levels and wave functions of a system in the presence of a small perturbation, or disturbance, to the original system. It allows for the calculation of the effects of the perturbation on the system without the need for solving the full Schrödinger equation.

How does Time-Independent Perturbation Theory work?

The theory works by expanding the Hamiltonian (the operator that represents the total energy of a system) into a series of terms, with the unperturbed Hamiltonian as the first term. These additional terms represent the perturbation to the system. By solving the resulting equations, one can calculate the energy levels and wave functions of the perturbed system.

What are the assumptions of Time-Independent Perturbation Theory?

There are three main assumptions of Time-Independent Perturbation Theory. First, the perturbation must be small compared to the unperturbed system. Second, the perturbation must be turned on and off gradually. Third, the perturbation must be time-independent, meaning it does not vary with time.

When is Time-Independent Perturbation Theory applicable?

Time-Independent Perturbation Theory is applicable in situations where the perturbation is small and does not change with time. It is commonly used in quantum mechanics to calculate the effects of electromagnetic fields on atomic and molecular systems, as well as in solid-state physics to study the effects of impurities on electronic states.

What are the limitations of Time-Independent Perturbation Theory?

While Time-Independent Perturbation Theory is a useful tool for approximating the effects of a perturbation on a system, it has some limitations. It is only accurate for small perturbations and can only be applied to systems with discrete energy levels. It also assumes that the perturbation does not change with time, which may not always be the case in real-world systems.

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