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 an energy : where is the reduced Planck constant. Note: , where is the Planck constant and is the linear frequency. 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 at time 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 s. 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 : Theory Spontaneous transitions were not explainable within the framework of the Schrödinger equation, in which the electronic energy levels were quantized, but the electromagnetic field was not. Given that the eigenstates of an atom are properly diagonalized, the overlap of the wavefunctions between the excited state and the ground state of the atom is zero. Thus, in the absence of a quantized electromagnetic field, the excited state atom cannot decay to the ground state. In order to explain spontaneous transitions, quantum mechanics must be extended to a quantum field theory, wherein the electromagnetic field is quantized at every point in space. The quantum field theory of electrons and electromagnetic fields is known as quantum electrodynamics. In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the QED vacuum, which can mix with the excited stationary states of the atom. As a result of this interaction, the "stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from