Find the eigenvector with zero eigenvalues at any time t from the Hamiltonian

In summary, the conversation discusses a question about a 3 level system and its Hamiltonian. The Hamiltonian is given by a formula with cos^2 and sin^2 terms, and the person has been asked to find an eigenvector with zero eigenvalues at any time. They are encouraged to attempt the problem and have some knowledge about eigenvectors and eigenvalues for assistance.
  • #1
Jack_11
1
0
Homework Statement
A Hamiltonian of 3 level system is given by:
Relevant Equations
H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|)

I have been asked to find that H has an eigenvector with zero eigenvalues at any time t, but I don't know where to start
I have a question relates to a 3 levels system. I have the Hamiltonian given by:

H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|)

I have been asked to find that H has an eigenvector with zero eigenvalues at any time t
 
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  • #2
Jack_11 said:
Homework Statement: A Hamiltonian of 3 level system is given by:
Homework Equations: H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|)

I have been asked to find that H has an eigenvector with zero eigenvalues at any time t, but I don't know where to start

I have a question relates to a 3 levels system. I have the Hamiltonian given by:

H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|)

I have been asked to find that H has an eigenvector with zero eigenvalues at any time t

You have to make your best attempt. You must know something about eigenvectors and eigenvalues.

We can't help you if you simply know nothing about the material at all.
 

1. What is an eigenvector and why is it important in the Hamiltonian?

An eigenvector is a vector that, when multiplied by a matrix, results in a scalar multiple of itself. In the Hamiltonian, eigenvectors represent the possible states of a system, and their corresponding eigenvalues represent the energy associated with that state. Finding the eigenvectors of the Hamiltonian allows us to understand the behavior of a system and make predictions about its evolution over time.

2. How do you find the eigenvector with a zero eigenvalue in the Hamiltonian?

To find the eigenvector with a zero eigenvalue in the Hamiltonian, we need to solve the eigenvalue equation H|ψ⟩=λ|ψ⟩, where H is the Hamiltonian, |ψ⟩ is the eigenvector, and λ is the eigenvalue. This can be done by setting λ=0 and solving for |ψ⟩. The resulting vector will be the eigenvector with a zero eigenvalue.

3. Can the eigenvector with a zero eigenvalue change over time in the Hamiltonian?

Yes, the eigenvector with a zero eigenvalue can change over time in the Hamiltonian. This is because the Hamiltonian itself can change over time, and thus the eigenvectors and eigenvalues can also change. In general, the eigenvalues and eigenvectors of a time-dependent Hamiltonian are functions of time.

4. How does the eigenvector with a zero eigenvalue affect the dynamics of a system in the Hamiltonian?

The eigenvector with a zero eigenvalue plays a crucial role in the dynamics of a system in the Hamiltonian. It represents a stationary state of the system, meaning that if the system is initially in this state, it will remain in that state indefinitely. This is because the Hamiltonian operator does not change the state of an eigenvector with a zero eigenvalue. Additionally, the other eigenvectors of the Hamiltonian can be expressed as linear combinations of the eigenvector with a zero eigenvalue, making it an important component in the overall behavior of the system.

5. Are there any practical applications of finding the eigenvector with a zero eigenvalue in the Hamiltonian?

Yes, there are several practical applications of finding the eigenvector with a zero eigenvalue in the Hamiltonian. One example is in quantum mechanics, where the eigenvectors and eigenvalues of the Hamiltonian are used to calculate the probabilities of different outcomes in a system. Additionally, understanding the eigenvectors with zero eigenvalues is important in fields such as chemistry and materials science, where it can help predict the behavior of complex systems.

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