Can a photon be a superposition of multiple frequency states? Kind of similar to how an electron can be a superposition of multiple spin states.
Answer
Yes. Consider quantizing electromagnetic fields in a box. This corresponds to photons being trapped inside of said box since photons are just the mode quanta of the EM fields. The Hilbert space (called Fock space in this case) of the quantized radiation is found to be spanned by states $$ |\mathbf k_1, \mu_1; \dots, ; \mathbf k_N, \mu_N\rangle, \qquad N=1,2,\dots $$ which represents a state with $N$ photons in the box with momenta $\mathbf p_i = \hbar \mathbf k_i$ and polarizations $\mu_i$ plus the vacuum state $|0\rangle$ containing no photons. Now suppose that at some point in time, the state of the system is $$ |\psi\rangle = \frac{1}{\sqrt 2}\left(|\mathbf k_1,\mu_1\rangle+|\mathbf k_2,\mu_2\rangle\right) $$ This represents a state in which there is a single photon in the box that is in a superposition of states; the vector $|\mathbf k_1, \mu_1\rangle$ represents a state with a single photon having momentum $\hbar \mathbf k_1$ and polarization $\mu_1$ while the vector $|\mathbf k_2, \mu_2\rangle$ represents a state with a single photon having momentum $\hbar \mathbf k_2$ and polarization $\mu_2$. In particular, recall that the frequency of a photon is related to $\mathbf k$; $E =\hbar\omega = \hbar c|\mathbf k|$ so that this state represents a photon in the box that is in a superposition of states corresponding to different frequencies.
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