Why does an electron in motion not radiate energy and spiral into the nucleus?

An electron in an atom is a quantum mechanical systems where dynamical variables such as position, momentum, and energy are not numbers with the appropriate units but are instead differential operators. Classical mechanics and classical electrondynamics do not apply to electron in an atom. This is because electron in an atom emits radiation when a charge particle moves at high speed.

It’s wave function, which is the solution to Schrodinger equation with admissibility or boundary conditions, can be used to study electron’s behavior. These conditions allow for quantization of energy eigen value of Schrodinger equation. The electrons can now exist in different stable quantum states, with different probabilities.

Time dependent perturbation of the atom results in electrons transitioning to higher energy quantum states with a certain probability

The life time of an electron in high-energy quantum state is 10-8s. It can spontaneously transition to a lower energy state, emitting the photon equivalent to the difference in energy between two states.

It is very unlikely that an electron will ever exist. But, K electrons are very close to the nucleus. Now, nucleus may may capture K electron. This is more energetic than the positron emissions. It depends also on the overlap between K electron wave function and nuclear wave function. The characteristic X-rays are produced when electron is captured from K state by nucleus.

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