Many particle physics - Hamiltonian for Fermi system

In summary, the conversation discusses a question from Mahan's 'Many Particle Physics' book about a fermion system with three energy states and corresponding eigenvalues and matrix elements. The question asks for the Hamiltonian in terms of creation and annihilation operators and determination of eigenvalues. The proposed solution includes writing the standard Hamiltonian for a quantum SHO and considering the unchanged energy eigenvalues due to possible transitions.
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Plaetean
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Homework Statement


Working through problems in Mahan's 'Many Particle Physics' book, and at the end of the 1st chapter there's a question where we're asked to consider a fermion system with three energy states with eigenvalues E1, E2, E3, and matrix elements M12, M23, M13 which connect them and allow transitions between them.

The question asks us to write down a Hamiltonian for the system in terms of creation and annihilation operators, and then determine the eigenvalues for the system.

Homework Equations

The Attempt at a Solution


I'm really a bit lost as to where to start for this, and all I can really think of doing is writing the standard Hamiltonian for a quantum SHO as a sum over states, but I'm not confident this is remotely right.
[tex]
H=\hbar\sum_{n=1}^{3}\omega_n(a_n^\dagger a_n + \frac{1}{2})+M_{12}+M_{23}+M_{13}
[/tex]

Is it the case that the energy eigenvalues will just be E1, E2 and E3, as the fact that transitions can occur doesn't change the actual eigenvalues of the system?

Thanks as always!
 
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FAQ: Many particle physics - Hamiltonian for Fermi system

What is the Hamiltonian for a Fermi system?

The Hamiltonian for a Fermi system is a mathematical operator that describes the total energy of a system of fermions, which are particles with half-integer spin. It takes into account the kinetic energy and interactions between the particles, and is used to solve for the system's quantum states.

How does the Hamiltonian differ for a Fermi system compared to a classical system?

The Hamiltonian for a Fermi system takes into account the quantum mechanical nature of fermions, while the Hamiltonian for a classical system only considers the classical mechanical properties of particles. This means that the Hamiltonian for a Fermi system includes terms for the Pauli exclusion principle and anti-symmetry of fermion wavefunctions.

What are the applications of the Hamiltonian for Fermi systems?

The Hamiltonian for Fermi systems is used in a wide range of fields, including condensed matter physics, nuclear physics, and astrophysics. It is essential for understanding the behavior of fermionic systems, such as electrons in a solid or neutrons in a nucleus, and can be used to predict and analyze various physical phenomena.

How is the Hamiltonian for a Fermi system solved?

The Hamiltonian for a Fermi system is typically solved using various mathematical techniques, such as perturbation theory, mean-field theory, and Monte Carlo simulations. These methods allow for the calculation of the system's energy levels and wavefunctions, which can then be used to make predictions about the system's behavior.

What are some open questions in the field of many particle physics and the Hamiltonian for Fermi systems?

Some current open questions in this field include the behavior of strongly interacting fermions, the emergence of superconductivity and superfluidity, and the role of topology in fermionic systems. Additionally, there is ongoing research into the development of new techniques for solving the Hamiltonian for Fermi systems, such as tensor network methods and machine learning approaches.

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