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

This can be:

• electron bound in an atom
• nucleon bound in nucleus
• nucleus (NMR)
• free electron

states in absence of magnetic field is eigenstate of , .

• spin states are ,
• Then the the hamiltonian with a z magnetic field will be: . Splitting of degeneracy occurs. Let = magnitude of split in degeneracy
• A rotating magnetic field perpendicular to the z direction will give where

In the interaction picture,

Now apply the results of TDPT. Initially so .

This corresponds to a probability of transition of:

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

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

Time dependence in nonrotating frame is: .

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

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TDPT