Two-state perturbation problem (QM)

In summary, the conversation discusses the calculation of eigenvalues and eigenstates for a hamiltonian H0 with two eigenvalues and corresponding eigenstates. A distortion W is then added to the hamiltonian and the conversation delves into how to calculate the new eigenvalues and eigenstates. The solution is found by defining a constant k and solving for it, which is the canonical method of solving such problems.
  • #1
Tim87
1
0

Homework Statement



Given is a hamiltionan H0:

E1 0
0 E2

with E1 and E2 being eigenvalues of two eigenstates phi1 and phi2

A distortion W, with W12 = W21* (complex conjugate):

0 W12
W21 0

Calculate the eigenvalues and eigenstates of H = H0 + W


Homework Equations



See for most info:
http://farside.ph.utexas.edu/teaching/qm/lectures/node50.html

it's a two-state perturbation problem


The Attempt at a Solution



I found out most things like how to calculate the eigenvalues. A lot of it is explained at the mentioned site (http://farside.ph.utexas.edu/teaching/qm/lectures/node50.html) The thing I don't get is how they got from the eigenvalues to the eigenstates.

I have tried the following:

( E1 - E1' W12 )(a) (0)
( W21 E2 - E1' )(b) = (0)

with E1' being the first new eigenvalue: E1'=E1 + |W12|^2/(E1-E2)

But I do not get to the answer provided on the site.


I hope I have been clear about my question, so someone can explain it to me. Thanks in advance!

Tim
 
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  • #2
Note that since eigenvectors are only defined up to a scaling, your matrix only represents one equation. So define a=kb, and solve for k. (This is the canonical way to proceed. In practice people just stare at it and then write down the solution.)
 

1. What is the Two-state perturbation problem in quantum mechanics?

The Two-state perturbation problem in quantum mechanics is a mathematical model used to describe the behavior of a two-state system under the influence of an external perturbation. This problem is often encountered in quantum mechanics when studying the behavior of atoms, molecules, and other quantum systems.

2. How is the Two-state perturbation problem solved?

The Two-state perturbation problem is solved using the time-independent perturbation theory, which involves expanding the system's Hamiltonian in a series of terms and finding the corrections to the unperturbed energy levels and wavefunctions. This method allows for the calculation of the perturbation effects on the system's energy levels and wavefunctions.

3. What are the applications of the Two-state perturbation problem?

The Two-state perturbation problem has various applications in different fields of physics, such as quantum chemistry, solid-state physics, and nuclear physics. It is used to study the behavior of atoms and molecules in the presence of external fields, the effects of impurities on the electronic properties of solids, and the interactions between different nuclear states.

4. What are the limitations of the Two-state perturbation problem?

The Two-state perturbation problem assumes that the perturbation is small compared to the unperturbed system, making it unsuitable for studying strong perturbations. It is also limited to two-state systems, and more complex systems require different approaches, such as the time-dependent perturbation theory.

5. How does the Two-state perturbation problem relate to quantum superposition?

The Two-state perturbation problem is closely related to quantum superposition, as it involves the combination of two or more quantum states to form new states. In the Two-state perturbation problem, the unperturbed system is in a superposition of two states, and the perturbation causes transitions between these states, resulting in new superposition states.

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