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Name: Michael
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In photoelectron spectroscopy, the uncertainty principle for energy and time seems to be important in determining the width of spectral lines. I have read that the natural width of an energy level is inversely proportional to the lifetime of the hole created by photoelectron ejection. Why is the energy inversely proportional to the lifetime of the vacancy and not of the state in which the electron fills the vacancy. From my understanding, the uncertainty in the energy of system is inversely proportional to the amount of time for which it remains in that state. Since the energy of the emitted electrons are measured based on the energy of the "filled vacancy," why would the uncertainty in energy be based on the amount of time the vacancy exists?


The energy is NOT inversely proportional to the lifetime of the hole. The WIDTH of the energy level is inversely proportional to this lifetime. Should such a photoelectron be emitted in an energy level, there is a range of energies contained in this level. The electron in that state will tend to be between two energies rather than at one specific energy. This range is the uncertainty of the energy level.

The lifetime is not how long it will take for the hole to be filled. You cannot know exactly how long this will take. The hole might be filled immediately. The hole might be filled after one lifetime. The hole might take three lifetimes to fill. An empty state with a short lifetime will not stay open for much time. An empty state with a long lifetime might stay open for a long time. This gives us the uncertainty for how long the state might exist.

A state with a very 'tight' energy level must be able to exist for a long time. A state with an exact energy must be stable. A state that cannot exist very long will have a large uncertainty regarding its energy level. If you know exactly how much energy, you have no idea when. If you know exactly when, you have no idea how much energy is needed to make it happen.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

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