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The reason why two independent bulbs cannot be used to create the young's two slit interference pattern is that the phase difference between those two sources varies very rapidly and therefore, the interference pattern cannot be sustained and shifts rapidly producing general illuminance. But by using a single source and two nearby slits illuminated by the common source eliminates this problem, because the phase of the light being emitted may change as rapidly as possible, but the phase difference between the two slits is "locked" and remains constant (only geometric path difference).

  • Considering the generation of light by the common incandescent bulb, it generates light as the electric energy is converted to heat and at a high enough temperature, it emits a characteristic light due to excitation and de-excitations of the constituent atoms emitting photons. All the photons which constitute the emitted light have varying phases (all of the emitted photons are not in phase). Why then does the phase relation get fixed between the slits, since it is not necessary that a single photon must simultaneously pass from both the slits (I appreciate the quantum mechanical property of a single photon to pass through both the slits simultaneously)? Two different photons with an arbitrary phase might pass through not allowing the interference pattern to form.
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You're making unwarranted assumptions. In general, you will have a very difficult time getting an interference pattern from an incandescent source. If you had an incandescent point source it would work (with careful attention to path length matching, as there is little temporal phase stability in addition to spatial). However, since a single bulb is an extended source, it's practically the same as using two separate lamps.

Try putting a 5-micron pinhole between your lamp and the slits, and you may have some luck getting an interference pattern.

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