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I have been told in a lecture that the broadening of peaks detected from photon emission in a hydrogen lamp is due to the uncertainty principle, but can’t work out how it does or find any equation involving the width of peaks.

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I am copying this because it has the main reasons of line broadening:

A spectral line emitted by an atom or molecule ....has a finite width and there are three distinct physical processes which contribute to this broadening.

This gives the connection between the uncertainty principle that you are asking:

The energy levels between which the transition takes place have a natural width, which arises from the reaction of the radiation field on matter. This means that an atom has a finite time in any level before spontaneous decay takes place.

So it uses the time energy Heisenberg uncertainty.

Also the emitting atom may be moving in the line of sight, which produces an apparent change in the line frequency through the Doppler effect. Finally, broadening of the spectral line is caused by the interaction of the emitting atom with surrounding particles in the plasma. Thus a detailed analysis of line shapes can yield much information on the density, temperature and motion of plasmas, whether they are high density plasma created in the laboratory, or the lower density plasma of interest in astrophysics, such as the solar corona or the interstellar medium.

Usually it is the other reasons that give the broadening, as the natural line widths are very small:

Here is a description of how this natural width comes about, at page 3.

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The equation you are looking for, that is linking the emission linewidth to the lifetime of a state, is the time-energy uncertainty relation. This relation can be written as

$$ \sigma_E \frac{\sigma_B}{\left|\frac{d\langle\hat{B}\rangle}{dt}\right|} \ge \frac{\hbar}{2} $$

where $\sigma_E$ is the standard deviation of the energy (Hamiltonian) operator and the fraction, that has the dimension of a time, is the lifetime of the considered state with respect to a certain observable, that is defined as the time interval in which the expectation value of the given observable changes in an appreciable way.

As an example, consider an atom that is somehow excited to an electronic excited state: after some time, it will return to its ground state and, in doing this, a photon will be emitted. According to the previous equation, the smaller the time interval in which the said atom will remain in its excited state, the larger will be the uncertainty on the energy of this excited state and, as a consequence, the higher will be the uncertainty on the energy of the emitted photon. This actually links the width of the considered emission peak to the lifetime of the excited state involved. Another consequence is that only quantum states with an infinite lifetime will have a precise value of the energy.

In spectroscopy, we are thus talking of natural (line) broadening when the emission spectrum is broadened as a consequence of the energy-time uncertainty relation. However, this is only one possible source of line broadening and sometimes it can only give rise to negligible effects.

Sources:

  1. Wikipedia page for "Uncertainty principle" and links therein
  2. Wikipedia page for "Spectral line", in particular with respect to natural broadening
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