How Do I Find the Energy of a Non-Hermitian Hamiltonian with Bosonic Operators?

In summary, the conversation discusses the Hamiltonian of a quantum harmonic oscillator in terms of bosonic operators and its energy. The energy is determined by real constants ω, alpha, and β, and is given by E=(n+1/2)*epsilon where epsilon is ω^2-4*alpha*β. The conversation also addresses the difficulty in finding the energy due to the non-Hermitian nature of the Hamiltonian.
  • #1
ozlemathph
2
0
Hi all,

There is a Hamiltonian in terms of "a" and "a^{dagger}"bosonic operators H=ω*(a^{dagger}a+1/2)+alpha*a^2+β*a^{dagger}^2 and ω, alpha and β are real constants and its energy is E=(n+1/2)*epsilon where epsilon is ω^2-4*alpha*β. Now, I tried to find this energy but I couldn't. Would you help me please? Thanks.
 
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  • #2
When you say "its energy" what do you mean specifically?

For the usual quantum harmonic oscillator, the number states |n> are eigenstates of the Hamiltonian. In this case, they are not. So what states are you trying to find the energies of?
 
  • #3
ozlemathph, The first thing you might wonder about is whether your Hamiltonian is correct. It's not Hermitian!
 

What is a Quadratic Hamiltonian Energy?

A Quadratic Hamiltonian Energy is a mathematical representation of the energy of a system that follows the laws of classical mechanics. It is described by a quadratic function known as the Hamiltonian, which takes into account the position and momentum of particles within the system.

How is Quadratic Hamiltonian Energy used in physics?

Quadratic Hamiltonian Energy is used to study and analyze various physical systems, such as simple harmonic oscillators and coupled oscillators. It also plays a crucial role in quantum mechanics, where it is used to describe the energy of particles in a system.

What is the difference between Quadratic Hamiltonian Energy and Total Energy?

Quadratic Hamiltonian Energy is a specific type of energy that is based on the Hamiltonian of a system, while Total Energy is a more general term that refers to the sum of all forms of energy (kinetic, potential, etc.) in a system. Quadratic Hamiltonian Energy is often used in classical mechanics, while Total Energy is used in both classical and quantum mechanics.

Can Quadratic Hamiltonian Energy be negative?

Yes, Quadratic Hamiltonian Energy can be negative. This typically occurs when there is a potential energy term in the Hamiltonian that is negative. In terms of physical systems, this could mean that the particles in the system have a lower energy state, or that they are in a bound state.

How is Quadratic Hamiltonian Energy related to the Principle of Least Action?

The Principle of Least Action states that the path a system takes between two points is the one that minimizes the action (or energy) of the system. Quadratic Hamiltonian Energy is used in the Lagrangian formulation of this principle, where the Hamiltonian is used to calculate the action of the system. This allows for the prediction of the most probable path a system will take between two points, based on its energy.

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