Question:
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?
Replies:
Michael,
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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