I Why does the energy level of an electron in an atom have a width?

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Electron in atom at higher level(excited state) of energy has tendency jump to lower level of energy.Then the state of electron is unstale,so there is uncertainty in energy of the electron(the level of energy has a width).What cause the level being a band of energy but not a thin line of energy?When we solve Schrodinger equation of electron in a potential well we receive the quantization of energy with each of level having a certainty of value.
 
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Likely you use time-independent Schrodinger equation. The energy line-width is just reverse proportional to state lifetime, and this effect is not covered by time-independent solution.
With time dependence, the infinite duration oscillation of wavefunction at unstable electron orbit (periodic stationary solution) converts to wavelet (transient solution), which has finite line-width.
 

vanhees71

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Any "atomic state" except the ground state have a finite width alone from the spontaneous emission contribution of QED. What you calculate as "atomic states" is an approximation, where you take into account the static contributions of electromagnetic interactions in the Coulomb gauge only. That's not the whole truth, because the electromagnetic field is a dynamical entity too and has to be quantized. Since the em. field as a spin-1 field is necessarily a boson field, as follows from the fundamental assumptions of local QFT demanding a stable ground state, there's also spontaneous emission in addition to induced emission and absorption which also is described in semi-classical electromagnetics, where the em. field is kept classical.

The possibility of spontaneous emission, i.e., the transition of an excited "atomic state" to a lower one under emission of one (or sometimes more than one) photon(s) has a finite probability (except if selection rules, i.e., conservation laws forbid some specific transition), the "atomic states" are not true energy eigenstates anymore as soon as you take into account the "radiation corrections" with a quantized em. field. This leads to the conclusion that in reality these "atomic states" are (quite narrow) resonances of finite width and the corresponding finite lifetime. It's also called "natural linewidth".
 
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There are several factors that made the width of the lines, I would like to mention the red/blue shift due to doppler effect due to the movement** of the center of mass of the atom.

**(unavoidable, even in principle due to uncertainty in the momentum)
 

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