# energy of a photon for a complex wave

If I use a plane wave of light given by

$$E=E_o \sin(\omega t+\phi)$$

for photoelectric effect, then the energy of photon associated is given by $$h\nu$$ where $$\nu=\frac{\omega}{2\pi}$$

But suppose if I have two light waves of different frequencies given by

$$E_1 \sin(\omega_1 t+\phi _1)$$ and $$E_2 \sin(\omega_2 t+\phi _2)$$ and use their resultant light for photoelectric experiment. What will be the energy associated with a photon in such an experiment

• Please do not make significant changes to your question after it has received one or more relevant answers. Minor edits are fine, but not edits that invalidate the existing answers! – PM 2Ring Dec 1 '18 at 14:50

You are combining two electromagnetic waves that have the same frequency but different amplitude and phase.

Photons detected in any measurement of the combined wave will have that same frequency and thus the same energy as photons detected in a measurement of the waves separately.

However, the detection rate of the photons (the number of photons striking a detector per second) depends on the squared amplitude of the combined wave.

• What if the frequencies are different? – ATHARVA Dec 1 '18 at 7:13
• Sorry it should have been $\omega_1$ and $\omega_2$ – ATHARVA Dec 1 '18 at 7:14

The energy of a photon is always $$h\nu$$ in quantum mechanics as I know it.

• What if frequencies are different? @my2cts – ATHARVA Dec 1 '18 at 7:15
• Light has an extrinsic and an intrinsic bandwidth . Extrinsic bandwidth means that photons of different energy are present, due to for example doppler shift or different transitions contributing to it. The intrinsic bandwidth is the bandwidth of the individual photons due to lifetime broadening. A transition takes a finite time and this broadens the resonance. In your case two different kinds of photons is the most likely. – my2cts Dec 1 '18 at 8:39

If you have two such waves, then each correspond to a single photon. Thus the resulting energy is $$2h\nu$$ By the way, in quantum mechanics, we talk about the wavefunction of a particle (photon) which is related to how probable it is to find that particle in an infinitesimal length $$dx$$.