Formalism

Summary: Time-dependent perturbation theory applies to cases where the perturbing ﬁeld changes with time, but also to processes such as absorption and emission of photons, and to scattering.

In time-dependent perturbation theory, we typically consider a situation where, at the beginning and end, the only Hamiltonian acting is the time-independent

{\hat{H}}^{(0)}

, but for some time in between another, possibly time-dependent, eﬀect acts so that for this period

\hat{H} = {\hat{H}}^{(0)} + {\hat{H}}^{(1)} (t)

. If a system starts oﬀ in an eigenstate of

{\hat{H}}^{(0)}

it will have a certain probability of ending up in another, and that transition probability is generally what we are interested in.

There are many similarities in approach to the time-independent case, but it is worth noting that our interest in the states

| n^{(0)} ⟩

is slightly diﬀerent. In the time-independent case these were only approximations to the true eigenstates of the system. In the time-dependent case they remain the actual eigenstates of the system asymptotically (a phrase that means away from the inﬂuence of the perturbation, either in space or, as here, in time). Furthermore while the time-dependent perturbation is acting the familiar connectionbetween the TISE and TDSE breaks down, so the eigenstates of the Hamiltonian, while instantaneously deﬁnable, are not of any real signiﬁcance anyway. For that reason we will drop the label

^{(0)}

on the states, since we won’t be deﬁning any other set of states, and similarly we will refer to their unperturbed energies just as

E_{n}

Because the states

| n ⟩

are a complete set, at any instant we can decompose the state of the system

| ψ (t) ⟩ = \sum_{n} c_{n} (t) | n ⟩ \equiv \sum_{n} d_{n} (t) e^{- i E_{n} t ∕ ℏ} | n ⟩

In deﬁning the

d_{n} (t)

we have pulled out the time variation due to

{\hat{H}}^{(0)}

. So in the absence of the perturbation, the

d_{n}

would be constant and equal to their initial values. Conversely, it is the time evolution of the

d_{n}

which tells us about the eﬀect of the perturbation. Typically we start with all the

d_{n}

except one equal to zero, and look for non-zero values of the others subsequently.

where we used

m

as the dummy index in the summation and took the inner product with

e^{i E_{n} t ∕ ℏ} ⟨ n |

, and we have deﬁned

ω_{n m} \equiv (E_{n} - E_{m}) ∕ ℏ

(and

ḋ_{n}

represents the time derivative of

d_{n}

So far this is exact, but usually impossible to solve, consisting of an inﬁnite set of coupled diﬀerential equations! Which is where perturbation theory comes in. If we start with

d_{i} (t_{0}) = 1

and all others zero (

i

for initial), and if the eﬀect of

{\hat{H}}^{(1)}

is small, then

d_{i}

will always be much bigger than all the others and, to ﬁrst order, we can replace the sum overall states by just the contribution of the

i

th state. Furthermore to this order

d_{i}

barely changes, so we can take it out of the time integral to get

d_{n} (t) = - \frac{i}{ℏ} \int_{t_{0}}^{t} e^{i ω_{n i} t} ⟨ n | {\hat{H}}^{(1)} (t) | i ⟩ d t

Having obtained a ﬁrst-order expression for the

d_{n}

, we could substitute it back in to get a better, second-order, approximation and so on.

4.1 Formalism