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== A special case to show applications ==
A special case occurs where a system oscillates between two unstable levels that have the same life time <math>\tau</math>.<ref>Page-449, Quantum Mechanics, Vol.1, Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe</ref> If atoms are excited at a constant, say n/time, to the first state, some decay and the rest have a probability <math>P_{12}</math> to transition to the second state, so in the time interval between ''t'' and (''t'' + ''dt'') the number of atoms that jump to the second state from the first is <math>n(1-e^{-t/\tau})P_{12}dt</math>, so at time t the number of atoms in the second state is
[[File:Lorentz curve.png|thumbnail|Plot of Rate of Decay with Variation of Uniform Magnetic Field <math>B_0</math>]]
[[File:Half-Width plot for Life time.png|thumbnail|Plot of Half-Width of Lorentz Curve with Changing <math>B_1</math>]]
: <math>dN=n.e^{-t/\tau}.(1-e^{-t/\tau})P_{12}dt</math>
: = <math>n. e^{-t/\tau} .P_{12} dt</math>
The rate of decay from state two depends on the number of atoms that were collected in that state from all previous intervals, so the number of atoms in state 2 is <math display="inline">\int_{-\infty}^{0} ne^{-t/\tau}P_{12}\ dt</math>; The rate of decay of atoms from state two is proportional to the number of atoms present in that state, while the constant of proportionality is decay constant <math>\lambda</math> . Performing the integration rate of decay of atoms from state two is obtained as:
: <math>(n/2)\omega^2/(\delta\omega^2+\omega_1^2+1/\tau^2)</math>
From this expression many interesting points can be exploited, such
* Varying uniform magnetic field <math>B_0</math> so that <math>\omega_0</math> in <math>\delta\omega</math> produces a Lorentz curve (see [[Cauchy–Lorentz distribution]]), detecting the peak of that curve, the [[abscissa]] of it gives <math>\omega_0</math>, so now <math>\omega</math> (angular frequency of rotation of <math>\mathbf{B}_1</math> = <math>\gamma</math>  <math>(B_0)_\text{max}</math>, so from the known value of <math>\omega</math> and <math>(B_0)_\text{max}</math>, the gyromagnetic ratio <math>\gamma</math> of the dipole can be measured; by this method we can measure [[Nuclear spin]] where all electronic spins are balanced. Correct measurement of [[nuclear magnetic moment]] helps to understand the character of nuclear force.
* If <math>\gamma</math> is known, by varying <math>\omega</math>, the value of <math>B_0</math> can be obtained. This measurement technique is precise enough for use in sensitive magnetometers. Using this technique, the value of magnetic field acting at a particular lattice site by its environment inside a crystal can be obtained.
* By measuring half-width of the curve, ''d'' = <math>\sqrt{\omega_1^2+1/\tau^2}</math>, for several values of <math>\omega_1</math> (i.e. of <math>B_1</math>), we can plot ''d'' vs <math>\omega_1</math>, and by extrapolating this line for <math>\omega_1</math>, the lifetime of unstable states can be obtained from the intercept.