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Imagine an atom with a valence electron becomes excited and the electron moves to a higher orbital, is this transition between ground state and excited state at speed of light?

If so, since it is not instantaneous, then wouldn't it emit photons of various wavelengths during the change between ground and excited state?

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You are very correct, this transition is not instantaneous! These transition occurs with a typical time-scale of $\Delta t \sim \frac{\hbar}{\Delta E}$. For example, the Hydrogen 1s to 2p transition occurs in the time-scale of a many hundreds of atto-seconds.

You can understand this by thinking about energy photon you need for doing this transition from 1s->2p. Well, you get about 10.2eV, which is the extreme ultraviolet wavelength. And what is the period of this light? It is about 400 atto-seconds as expected.

Mathematically, we find this transition rate from time-dependent perturbation theory, which says

$$\frac{dc_1}{dt} \sim V(t) e^{i(E_1-E_0)t}c_0(t)$$

Here $V(t)$ is the potential causing the transition. What is important is that the potential will have maximum probability to get into the excited state ($c_1$) if it has the form $\mathrm{cos}((E_1-E_0)t)$ or $\mathrm{sin}((E_1-E_0)t)$. In that case, it takes just about 1 cycle of the cosine to get into the excited state, or about $1/(E_1-E_0)$ as mentioned.

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