text: Given the above definitions the interaction Hamiltonian is
  as stated.  The next step is to find the Hamiltonian in the interaction picture, . The required unitary transformation is
  where the 3rd step can be proved by using a Taylor series expansion, and using the orthogonality of the states  and . Note that a multiplication by an overall phase of  on a unitary operator does not affect the underlying physics, so in the further usages of  we will neglect it. Applying  gives:
  Now we apply the RWA by eliminating the counter-rotating terms as explained in the previous section:
  Finally, we transform the approximate Hamiltonian  back to the Schrödinger picture:
  The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is