Eigen functions/values for many-body Hamiltonian with creation/annihilation operators

In summary, the problem at hand is understanding how to find Eigen functions/values for a Hamiltonian with creation/annihilation operators in many-body problems. The suggested procedures are to set up a simple case, obtain creation/annihilation field operators, and second-quantize all elements of the Hamiltonian. Then, the next step would be to find the normal modes of the system and use quantized harmonic oscillators to find the states for each normal mode. If this is not possible, perturbation theory may be used.
  • #1
ranytawfik
11
0
Problem:
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I’m trying to understand how to generally find Eigen functions/values (either analytically or numerically) for Hamiltonian with creation/annihilation operators in many-body problems.

Procedures:
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1. I setup a simple case of finite-potential well with two in-distinguishable fermions (ignore spin for the moment).
2. I got the creation/annihilation field operators.
3. I second-quantized all the elements of the Hamiltonian (kinetic energy, well/external potential, and electron-electron electrostatic interaction).

My question is how to proceed next? I have the Hamiltonian which is now function of the creation/annihilation operators. How can I solve for the many-body Eigen function/values after that?

Thanks so much.
 
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  • #3


In general, the procedure is as follows:

1) Try to find the normal modes of your system, that is, transform your creation/annihilation operators to a basis in which the Hamiltonian is diagonal: H = ∑ ak*ak.

2) Find the states for each normal mode as a quantized harmonic oscillator. The ground state is |0k> such that ak|0k> = 0. The excited states are |nk> = (ak*)n|nk>.

If you can't do step (1), you'll have to use perturbation theory.
 
  • #4


Thanks strangerep and Bill. I'll try to do what you suggested, Bill, and post a follow up with the detailed Latex equations and procedures.
 

1. What are eigenfunctions and eigenvalues in the context of many-body Hamiltonian with creation/annihilation operators?

Eigenfunctions and eigenvalues are mathematical concepts used to describe the behavior of a quantum system. In the context of many-body Hamiltonian with creation/annihilation operators, eigenfunctions represent the possible states of the system, while eigenvalues represent the energy associated with each state.

2. How are eigenfunctions and eigenvalues calculated for a many-body Hamiltonian with creation/annihilation operators?

Eigenfunctions and eigenvalues for a many-body Hamiltonian with creation/annihilation operators can be calculated using a variety of mathematical techniques, such as diagonalization or perturbation theory. These methods involve solving the Schrödinger equation for the Hamiltonian and identifying the eigenvectors and eigenvalues.

3. What is the significance of eigenfunctions and eigenvalues in quantum mechanics?

Eigenfunctions and eigenvalues play a crucial role in quantum mechanics as they provide information about the possible states and energies of a quantum system. They also serve as a basis for constructing more complex wavefunctions and making predictions about the behavior of the system.

4. Can eigenfunctions and eigenvalues for a many-body Hamiltonian with creation/annihilation operators change over time?

Yes, the eigenfunctions and eigenvalues for a many-body Hamiltonian with creation/annihilation operators can change over time. This is due to the time-dependent nature of quantum systems and the possibility of the system transitioning between different states.

5. How are eigenfunctions and eigenvalues related to observables in quantum mechanics?

Eigenfunctions and eigenvalues are related to observables in quantum mechanics through the concept of measurement. The eigenvalues represent the possible outcomes of a measurement, while the eigenfunctions describe the probability of obtaining each outcome. This relationship is known as the Born rule and is a fundamental aspect of quantum mechanics.

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