QuantumPhysics.dvi

13.2 The interaction picture

We now turn to time dependent problems in the general case whereH=H 0 +V(t). We shall
assume that the eigenvalues and eigenstates ofH 0 are known as follows,

H 0 |n〉=En|n〉 (13.14)

For simplicity, we shall assume here that the spectrum ofH 0 is discrete, but it is not hard to
generalize to the continuous case, something we shall do later on. We prepare an initial state|φa〉
att= 0, which we may decompose in the basis of eigenstates|n〉by

|φa〉=

∑

n

cn(0)|n〉 (13.15)

In the absence of interaction potential,V(t) = 0, this state would evolve as follows,

|φa;t〉=

∑

n

cn(0)e−itEn/ ̄h|n〉 (13.16)

On the other hand, once the interaction potential is turned on, the time evolution will be more
complicated, giving time dependence also to the coefficientscn. The corresponding state will be
denoted|ψa;t〉. It satisfies the full time dependent Schr ̈odinger equation,

i ̄h

d

dt

|ψa;t〉= (H 0 +V(t))|ψa;t〉 (13.17)

and has the following decomposition,

|ψa;t〉=

∑

n

cn(t)e−itEn/ ̄h|n〉 (13.18)

Notice that we can also write this time evolution as

|ψa;t〉=e−itEn/ ̄h|ψa;t〉I (13.19)

where

|ψa;t〉I=

∑

n

cn(t)|n〉 (13.20)

The subscriptI has been added to indicate that this state is now considered in theinteraction
picture, a method due to Dirac. The advantage is that the coefficientscnnow have time depen-
dence only through the effects of the time-dependent interaction potential, the effects of the free
Hamiltonian having been factored out.

More generally, if|ψa;t〉Sis a general state in the Schr ̈odinger picture, satisfying the Schr ̈odinger
equation,

i ̄h

d

dt

|ψa;t〉S=

(

H 0 +V(t)

)

|ψa;t〉S (13.21)