What is the energy of a hydrogen atom in a mixed state?

[beman]

Suppose a single hydrogen atom is in mixed state.
Ψ=(1/√2) Ψ_100+(1/√2) Ψ_200
Then energy will be E=(1/2)*13.6+(1/2)*(3.4)=8.5 eV.
But there is no spectral line at 8.5 eV.

 

[blue_leaf77]

The state you have there is a superposition state, which is not the eigenstate of energy. Therefore, if you run a series of energy measurement to the H atoms all prepared to be in that state, you won't get a fixed value during the repeated measurements. The possible values you can get is 13.6 eV and 3.4 eV. Furthermore, if you take the average of these measurements, given large number of measurements, you will get what is called the expectation value of energy, whose value has been calculated by you to be 8.5 eV. To emphasize, 8.5 eV is not among the possibly measured energies, and hence the corresponding frequency will not appear in your spectrometer.