# Physical system for this two level Hamiltonian?

• Peeter
In summary, the conversation discusses a two-level quantum system with a Hamiltonian of a specific form, which is also found in the orientation of an ammonia atom. The factor of i and different signs do not affect the diagonalization of the system.
Peeter
An old QM exam question asked for consideration of a two-level quantum system, with a Hamiltonian of

$$H = \frac{\hbar \Delta}{2} ( {\lvert {b} \rangle}{\langle {b} \rvert}- {\lvert {a} \rangle}{\langle {a} \rvert})+ i \frac{\hbar \Omega}{2} ( {\lvert {a} \rangle}{\langle {b} \rvert}- {\lvert {b} \rangle}{\langle {a} \rvert})$$

where $\Delta$ and $\Omega$ are real positive constants.

I did the question itself, but was left wondering what sort of physical system has a Hamiltonian of this form? Reading Feynman he reasons that the "up" or "down" orientation of the ammonia atom has a Hamiltonian with a similar form. Namely,

$$H = E_0 ( {\lvert {b} \rangle}{\langle {b} \rvert}+ {\lvert {a} \rangle}{\langle {a} \rvert})- A( {\lvert {a} \rangle}{\langle {b} \rvert}+ {\lvert {b} \rangle}{\langle {a} \rvert})$$

That's a fairly close match, but doesn't have the imaginary factor and different signs.

The factor of $i$ and different signs do nothing. You have a match.

Ah, yes. They both have the same diagonalization. Thanks.

## 1. What is a physical system for this two level Hamiltonian?

A physical system for a two level Hamiltonian is any system that can be described by a two level quantum mechanical model. This can include atoms, molecules, and other quantum systems that have only two distinct energy levels.

## 2. What is the Hamiltonian operator in this context?

The Hamiltonian operator is a mathematical operator that represents the total energy of a system. In the context of a two level Hamiltonian, it is a 2x2 matrix that describes the energy levels and transitions between them.

## 3. How is the Hamiltonian matrix determined for a specific physical system?

The Hamiltonian matrix is determined by the properties of the physical system, such as the energy levels and transition probabilities. These properties can be measured experimentally or calculated theoretically using quantum mechanical principles.

## 4. How does the two level Hamiltonian model explain the behavior of physical systems?

The two level Hamiltonian model describes the behavior of physical systems by quantizing the energy levels and transitions between them. This allows for a more accurate and detailed understanding of how these systems interact with their environment and evolve over time.

## 5. What are some real-world applications of the two level Hamiltonian model?

The two level Hamiltonian model has many practical applications, including in quantum computing, spectroscopy, and quantum communication. It is also used in fields such as chemistry, biology, and material science to study the behavior of complex physical systems at the molecular and atomic level.

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