# Vacuum fluctuation

sanjibghosh
recently i listened something about vacuum fluctuation.i don't know what exactly is it.but from that fluctuation is it possible to say that there is some width of the energy eigenvalue of a system where the particles are interact with em field.i don't know anything about field theory.

Bob_for_short
It is easy. The vacuum in quantum mechanics is the lowest energy state of a quantum mechanical system (not of free or empty space). Think of atomic ground state. In this state there is no excited states (no excitations) so it is kind of vacuum. The electron and nucleus coordinates "fluctuate" in this state: the charges are quantum mechanically "smeared". The same is valid for the quantized electromagnetic field which is a set of independent harmonic oscillators. In their ground states the oscillator variables (field tensions) "fluctuate".

The quantized electromagnetic field is always coupled to charges and vice versa. In the ground (vacuum) state no energy exchange between an atom and the quantized EMF is possible (there is no energy available, everything is at its energy minimum). But if you prepare an excited atomic state, the atomic electron will have additional energy and will be still interacting with the quantized EMF (oscillators). This interaction makes it possible to transfer the atomic excitation energy to the resonant oscillator. Thus the atom returns to its ground state and the field oscillator gets excited. This is how a photon created. It takes some time T and determines the spectral line width 1/T.

Bob.

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sanjibghosh
thanks Bob,
is there any explanation behind this T.i mean, we have to calculate this T.what i am trying to say is that ,this fluctuation may be related to this T.
if there is such fluctuation,then it can be treat as the external perturbation (time dependent) in that atom,and
therefore we are able to calculate some width in the energy.like energy separation in a hydrogen atom in presence of an external electric field or magnetic field.

Bob_for_short
Yes, there is an explanation of the time T. It is calculated. The simplest is a dipole approximation when the electron-field interaction term j*A is replaced with r*E. These calculations are given in many textbooks. Try, for example, H. Bethe, Salpeter, "Quantum mechanics of one- and two-electron atoms". Or take any book on quantum mechanics where radiation of atoms is treated.

Bob.

sanjibghosh
oh..but, thank you very much.
can i neglect the j*A term (for simplicity)? because it is much weaker than electric field.
actually i want to calculate it without looking any book.

Bob S
The Heisenberg Uncertainty Principle permits fluctuations in the instantaneous energy and time duration of a vacuum fluctuation, as long as delta E delta t <= h-bar. For a delta E > 1.02 MeV, a virtual electron positron pair can be created. Virtual pair production is the lowest order correction to a photon or Coulomb field.

Bob_for_short
oh..but, thank you very much.
can i neglect the j*A term (for simplicity)? because it is much weaker than electric field.
actually i want to calculate it without looking any book.

Without looking in a book, I myself cannot perform calculations. Unfortunately, I do not have a proper book on this subject.

Bob.

sanjibghosh
The Heisenberg Uncertainty Principle permits fluctuations in the instantaneous energy and time duration of a vacuum fluctuation, as long as delta E delta t <= h-bar. For a delta E > 1.02 MeV, a virtual electron positron pair can be created. Virtual pair production is the lowest order correction to a photon or Coulomb field.

where this energy comes from? the virtual particles could be electron-positron or any thing else, why is it electron-positron pair?

Bob S
From Bob S
The Heisenberg Uncertainty Principle permits fluctuations in the instantaneous energy and time duration of a vacuum fluctuation, as long as delta E delta t <= h-bar. For a delta E > 1.02 MeV, a virtual electron positron pair can be created. Virtual pair production is the lowest order correction to a photon or Coulomb field.
where this energy comes from? the virtual particles could be electron-positron or any thing else, why is it electron-positron pair?
The energy fluctuation delta-E is the minimum uncertainty (Heisenberg Unvertainty principal) in determining the energy within a time interval delta-t. This is a quantum-mechanical version of the electrical engineer's delta-w and delta-t in the Fourier transform from time to frequency domain. If we write the uncertaity principal for a photon of energy E = h v = h-bar w we get

delta E delta t = h-bar delta w delta-t = h-bar (physics version)
or dividing by h-bar we get
delta-w delta t = 1 (engineer's version)

Electron-positron virtual particle pairs are the most dominant of the possible virtual pairs, because the electron and positron are the lightest charged particles, with a threshold of 1.02 MeV. Next in line are muon-anti-muon pairs, with a threshold of about 210 MeV.

There are two very precise fundamental physics measurements, "g-2" of electrons, and "g-2" of muons (g = gyromagnetic ratio) that have virtual particle pairs as the lowest order correction to QED. Electron pairs for the electron g-2, and muon pairs for the muon g-2. are identical corrections to the respective g-2 measurements. But a big change for the muon g-2 is that there is another much larger correction to the muon measurement; virtual electron pairs in the muon g-2.