Introduction
If a light source ('the atom') is in the excited state with energy, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy, releasing the difference in energy between the two states as a photon. The photon will have angular frequency and energy (=, where is the Planck constant and is the frequency):
where is the reduced Planck constant. The phase of the photon in spontaneous emission is random as is the direction in which the photon propagates. This is not true for stimulated emission. An energy level diagram illustrating the process of spontaneous emission is shown below:
If the number of light sources in the excited state is given by, the rate at which decays is:
where is the rate of spontaneous emission. In the rate-equation is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the Einstein A coefficient, and has units . The above equation can be solved to give:
where is the initial number of light sources in the excited state, is the time and is the radiative decay rate of the transition. The number of excited states thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to 36.8% of its original value (-time). The radiative decay rate is inversely proportional to the lifetime :
Read more about this topic: Spontaneous Emission
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