In time-dependent perturbation theory the main goal is to determine the time-evolution of a perturbed quantum system, with particular emphasis on calculating transition probabilities and modeling the irreversible decay of probability from a small quantum system coupled to a very large quantum system. The interaction picture will be used to discuss time-dependent perturbation.
For the operator H0 itself, the interaction picture and Schrödinger picture coincide
This is easily seen through the fact that H0 and eℏiH0St commute. From now on, we will omit the superscript representing pictures of H0 .
For the density operator ρ(t)，it has no difference from any other operators.
There are two ways to comprehend the additional term iℏU˙U† in the expression of transformed Hamiltonian H~. The first way is to compare the situation with that in classical mechanics.
In classical mechanics, if an object undergoes circular motion with an angular frequency ω , then its centrifugal potential energy is E=21mω2R2. When switching to a reference frame rotating at the same angular frequency of ω , an observer in this reference frame will find that the object is completely stationary. Then this observer will consider its energy as 0. The centrifugal potential energy disappears when performing rotating frame transformation.
In quantum mechanics, the energy is described by the Hamiltonian. iℏU˙U† is that disappeared centrifugal potential energy. For example, in a two-level system H=ℏωg∣g⟩⟨g∣+ℏωe∣e⟩⟨e∣ , for a unitary transformation like
one can obtain with some calculation
therefore, the stationary part is transformed into
We could draw a conclusion that the effect of U is to shift down the energy level of eigenstates by ω1 and ω2 .
The second way is to look back on Schrodinger equation. Still in our two-level system, supposing the initial state is
The state at arbitrary time t is then determined by iℏdtd∣ψ(t)⟩=H(t)∣ψ(t)⟩, thus
where the energy of eigenstates, ℏωe and ℏωg, appear on the exponential of phase factor.
The unitary transformation ∣ψ(t)⟩→∣ψ~(t)⟩=U∣ψ(t)⟩ actually shift the kets to
And to examine whether the interaction part of the Hamiltonian is now time-independent under your transformation, just substitute the transformed ∣a~⟩,∣b~⟩,∣c~⟩ into the original expression and see if the oscillating factor of electric field is cancelled.