# Physical system for this two level Hamiltonian?

## Main Question or Discussion Point

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.