An example of the use of TDPT is a spin 1/2 system in an oscillating magnetic field.

This can be:

states in absence of magnetic field is eigenstate of tex2html_wrap184 , tex2html_wrap185 .

In the interaction picture, tex2html_wrap193

Now apply the results of TDPT. Initially tex2html_wrap194 so tex2html_wrap195 .

This corresponds to a probability of transition of:


near resonance, this breaks down. Away from resonance this goes as tex2html_wrap197 . Near resonance and with short times, you get: tex2html_wrap198 .

The exact solution is solvable. Go into rotated frame. tex2html_wrap199 , then you get: tex2html_wrap200 . eigenvalues of diagonalization are: tex2html_wrap201 . Eigenvectors can be written as tex2html_wrap202 and similarly for tex2html_wrap203 . where tex2html_wrap204 .

Time dependence in nonrotating frame is: tex2html_wrap205 tex2html_wrap206 .

The probability of changing state is then: tex2html_wrap207 . At resonance, the probability is 1. Around the peak, you have lorentzian behavior.

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