Physical interpretation of a Hamiltonian with a constraint

[Alex Cros]

Dear physics forums,

What is the physical interpretation of imposing the following constrain on a Hamiltonian:
$$Tr(\hat H^2)=2\omega ^2$$
where $\omega$ is a given constant. I am not very familiar with why is the trace of the hamiltonian there.

Thanks in advance,
Alex

 

[Demystifier]

The physical interpretation requires a physical context, and you didn't explain the physical context. At the very least, you should provide the reference in which this constraint is introduced.

 

[Demystifier]

Without the context, I would guess that the Hamiltonian describes a two-state system (hence the factor 2) both of which have the same energy $\omega$.

 

[stevendaryl]

Without the context, I would guess that the Hamiltonian describes a two-state system (hence the factor 2) both of which have the same energy $\omega$.

In case the OP didn't know this already---for any operator, the trace is equal to the sum of the eigenvalues. So if the trace of $H^2$ is $2\omega^2$, then it means that the energy eigenvalues are such that $E_1^2 + E_2^2 + ... = 2\omega^2$. Demystifier's example is one of the simplest: $E_1 = E_2 = \omega$. Or it could be $E_1 = \omega, E_2 = -\omega$. Or it could be $E_1 = \omega, E_2 = \omega/\sqrt{2}, E_3 = \omega/\sqrt{4}, E_4 = \omega/\sqrt{8} ...$

The sum of the eigenvalues doesn't seem to me to be a "constraint" on the Hamiltonian, it's just a fact about it.