Photon energy in Heisenberg picture

[zonde]

I would like to ask a quick (I suppose) question.
Does a photon have definite energy in Heisenberg picture?
My motivation for this question comes from reading that Hamitonian is generator of time evolution. But in Heisenberg picture time evolution is associated with operator not quantum system. Therefore question about energy of quantum system that is not undergoing time evolution.

 

[blue_leaf77]

Well, the energy operator in Heisenberg picture is time independent:
$$
H_H(t) = e^{iH_St}H_Se^{-iH_St} = H_S.
$$

 

[jfizzix]

I would like to ask a quick (I suppose) question.
Does a photon have definite energy in Heisenberg picture?
My motivation for this question comes from reading that Hamitonian is generator of time evolution. But in Heisenberg picture time evolution is associated with operator not quantum system. Therefore question about energy of quantum system that is not undergoing time evolution.

A one-photon state of the electromagnetic field may be in a superposition of many different energy and momentum modes, and so it needn't have a definite energy.

The Hamiltonian of the electromagnetic field is a sum over energy and momentum modes. If the one-photon state were only in one mode, then you could have a definite energy value, but that may not be physically realizeable when you consider the energy-time uncertainty principle.

 

[blue_leaf77]

Does a photon have definite energy in Heisenberg picture?

The comment from jfizzix made me realize that it was possible that the OP mixed the definition between "indefinite" and "time-dependent". Definite/indefinite in QM is usually associated with whether there is nonzero uncertainty in measuring certain physical quantity, it has nothing to do with the time dependency of that quantity.

In the example of photon state being a superposition of multiple modes (multiple frequencies) as given as an example by jfizzix above, there will be more than one possible results of energy measurement, i.e. the energy measurement yields indefinite outcomes. But so long as there is no light source and/or absorbing material in the system, the energy is conserved, i.e. it's time independent.

 

[zonde]

The comment from jfizzix made me realize that it was possible that the OP mixed the definition between "indefinite" and "time-dependent". Definite/indefinite in QM is usually associated with whether there is nonzero uncertainty in measuring certain physical quantity, it has nothing to do with the time dependency of that quantity.

Yes, I was thinking about the same. What was it that I meant by my question? As it can be understood in two different ways.
I suppose I was thinking whether energy of photon can be modeled as local hidden variable of photon. So it is more related to the time dependency than uncertainty across ensemble.