Categories:
Physics,
Quantum mechanics.
Interaction picture
The interaction picture or Dirac picture
is an alternative formulation of quantum mechanics,
equivalent to both the Schrödinger picture
and the Heisenberg picture.
Recall that Schrödinger lets states ∣ψS(t)⟩ evolve in time,
but keeps operators L^S fixed (except for explicit time dependence).
Meanwhile, Heisenberg keeps states ∣ψH⟩ fixed,
and puts all time dependence on the operators L^H(t).
However, in the interaction picture,
both the states ∣ψI(t)⟩ and the operators L^I(t)
evolve in t.
This might seem unnecessarily complicated,
but it turns out be convenient when considering
a time-dependent “perturbation” H^1,S
to a time-independent Hamiltonian H^0,S:
H^S(t)=H^0,S+H^1,S(t)
With H^S(t) the full Schrödinger Hamiltonian.
We define the unitary conversion operator:
U^(t)≡exp(iℏH^0,St)
The interaction-picture states ∣ψI(t)⟩ and operators L^I(t)
are then defined to be:
∣ψI(t)⟩≡U^(t)∣ψS(t)⟩L^I(t)≡U^(t)L^S(t)U^†(t)
Equations of motion
To find the equation of motion for ∣ψI(t)⟩,
we differentiate it and multiply by iℏ:
iℏdtd∣ψI⟩=iℏ(dtdU^∣ψS⟩+U^dtd∣ψS⟩)=iℏ(iℏH^0,S)U^∣ψS⟩+U^(iℏdtd∣ψS⟩)
We insert the Schrödinger equation into the second term,
and use [U^,H^0,S]=0:
iℏdtd∣ψI⟩=−H^0,SU^∣ψS⟩+U^H^S∣ψS⟩=U^(−H^0,S+H^S)∣ψS⟩=U^(H^1,S)U^†U^∣ψS⟩
Which leads to an analogue of the Schrödinger equation,
with H^1,I=U^H^1,SU^†:
iℏdtd∣ψI(t)⟩=H^1,I(t)∣ψI(t)⟩
Next, we do the same with an operator L^I
to find a description of its evolution in time:
dtdL^I=dtdU^L^SU^†+U^L^SdtdU^†+U^dtdL^SU^†=ℏiU^H^0,S(U^†U^)L^SU^†−ℏiU^L^S(U^†U^)H^0,SU^†+(dtdL^S)I=ℏiH^0,IL^I−ℏiL^IH^0,I+(dtdL^S)I=ℏi[H^0,I,L^I]+(dtdL^S)I
The result is analogous to the equation of motion in the Heisenberg picture:
dtdL^I(t)=ℏi[H^0,I(t),L^I(t)]+(dtdL^S(t))I
Time evolution operator
Recall that an alternative form of the Schrödinger equation is as follows,
where a time evolution operator or
generator of translations in time KS(t,t0)
brings ∣ψS⟩ from time t0 to t:
∣ψS(t)⟩=K^S(t,t0)∣ψS(t0)⟩K^S(t,t0)≡exp(−iℏH^S(t−t0))
We want to find an analogous operator in the interaction picture, satisfying:
∣ψI(t)⟩≡K^I(t,t0)∣ψI(t0)⟩
Inserting this definition into the equation of motion for ∣ψI⟩ yields
an equation for K^I, with the logical boundary condition K^I(t0,t0)=1:
iℏdtd(K^I(t,t0)∣ψI(t0)⟩)iℏdtdK^I(t,t0)=H^1,I(t)(K^I(t,t0)∣ψI(t0)⟩)=H^1,I(t)K^I(t,t0)
We turn this into an integral equation
by integrating both sides from t0 to t:
iℏ∫t0tdt′dKI(t′,t0)dt′=∫t0tH^1,I(t′)K^I(t′,t0)dt′
After evaluating the left integral,
we see an expression for K^I as a function of K^I itself:
KI(t,t0)=1+iℏ1∫t0tH^1,I(t′)K^I(t′,t0)dt′
By recursively inserting K^I once, we get a longer expression,
still with K^I on both sides:
KI(t,t0)=1+iℏ1∫t0tH^1,I(t′)dt′+(iℏ)21∫t0tH^1,I(t′)∫t0t′H^1,I(t′′)K^I(t′′,t0)dt′′dt′
And so on. Note the ordering of the integrals and integrands:
upon closer inspection, we see that the nth term is
a time-ordered product T
of n factors H^1,I:
K^I(t,t0)=1+∫t0tH^1,I(t1)dt1+21∫t0t∫t0t1T{H^1,I(t1)H^1,I(t2)}dt1dt2+...=1+n=1∑∞n!1(iℏ)n1∫t0t⋯∫t0tnT{H^1,I(t1)⋯H^1,I(tn)}dt1⋯dtn=n=0∑∞n!1(iℏ)n1T{(∫t0tH^1,I(t′)dt′)n}
This construction is occasionally called the Dyson series.
We recognize the well-known Taylor expansion of exp(x),
leading us to a final expression for K^I:
K^I(t,t0)=T{exp(iℏ1∫t0tH^1,I(t′)dt′)}
References
- H. Bruus, K. Flensberg,
Many-body quantum theory in condensed matter physics,
2016, Oxford.