The interaction representation

Obviously, any unitary transformation can be applied to the wavefunctions and operators and used to our advantage. In particular, for any Hamiltonian that is written as 

The time evolution equations in the interaction representation are easily derived from these definitions 

Equations (2.72) and ll.Pi indicate that in the interaction representation the time evolution of the operators is carried by Hq, while that of the wavefunctions is determined by the interaction V, or rather by its interaction representation that is itself a time-dependent operator. 

---

The use of c instead of a is said to constitute the interaction representation it is designed to remove H from the Schrodinger equation for c. Passage from a to c is equivalent to a change from ua to e immug, and in the special case where ua is an eigenfunction of H, the latter may be written e liln)B< tua. Hence, all matrices with components PIm(it) in the basis are transformed to P, m by the rule... 

If H has a small time-dependent part, as in Interaction Representation, Section 7-3, it is advisable to use the interaction representation... 

We now regard Eq. (8-233) as analogous to Schrodinger s equation, and proceed to carry out the transformation to the interaction representation described in Chapter 7, Section 7.7. We define the transformed density matrix R and the transformed potential U by... 

In the interaction representation, the occupation number of atomic orbital 0), at time t, is... 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

Similarly, in equation (6.9) for the covariances the matrices A and B are now time-dependent. We define the interaction representation by setting... 

The starting point is equation (3.4), in which the interaction representation (3.5) has not yet been introduced. This equation can also be rewritten in a condensed form using time ordering by means of the following additional trick. Although W does not depend on time we write W(t), where the argument t merely serves to indicate its position in the chronological order. Similarly we write B(t) for the constant operator B. Then (3.4) is the same as... 

Exercise. When A0 is also a function of t (although not stochastic) the same method can be used provided that the definition of the interaction representation is modified [as in (VIII.6.15)]. 

Consider again the linear stochastic differential equation (2.1). There is no need now to assume At(t) stationary, nor to eliminate its average as was done in (2.2). Transform (2.1) to the interaction representation (2.3). According to (XV.3.9) a formal solution can be written by means of the time-ordered exponential... 

Thirdly one needs a drastic step to turn this integral equation into a differential equation. This is the Markov approximation , which comes in two varieties. The first variety consists in replacing ps(t — t) with ps(t). The error is of relative order rc/rm (where l/rm is the unperturbed rate of change due to S s) and of absolute order a2T2/rm. In this approximation one may as well omit the S s in the exponent of (4.13) and the result is the same as (3.19). The second variety takes the zeroth order variation of ps into account by setting ps(t — r) = e T sps(t). The result is the same as was obtained in (3.14) by means of the interaction representation, and the only requirement is arc <[Pg.444]

We will consider only zero temperature. It is convenient to switch [19] to the interaction representation H —> H — (.irNr — iirNl- This transformation induces time dependence in the electron creation and annihilation operators. As a result, 2U2nkF cos(n g

[Pg.152]

The issue of a gauge invariant calculation can now be formulated succinctly the interaction representation perturbation V(t) obtained from (65) is a functional of the arbitrary Green s function g(x,x ) because Hnil (56) is, and thuifj(t, (the time de-... 

Finally, we need the expression of a Heisenberg operator, defined by the full Hamiltonian H = Ho + V(t), through an operator in the interaction representation. The transformation, corresponding to (258), is given by... 

In the interaction representation one should repeat the calculations performed before and given the expressions (275), (280), and then replace usual times by contour times r, so we obtain... 

We found that Green functions can be written in the interaction representation with a help of the S -operator. For example, time-ordered fermionic Green function is... 

The computation becomes somewhat simpler if we work in the interaction representation [61]. Defining... 

This operator acts only on the electron-positron and electromagnetic field variables. The normal ordered form of Hm taken in the interaction representation must be added to the interaction Hamiltonian. It gives additional elements to the... 

An operator in the Heisenberg representation can be expressed in the interaction representation as follows ... 

The ordering operator Tc places the operators in the Taylor expansion of the Tc exponent to the left with a later-in-time variable on the contour y. The operators are taken in the interaction representation on the Keldysh contour. 

In Eq. (57), HmJ in the interaction representation is defined as the interaction between the lead and bridge electrons ... 

---