ABSTRACT
The analysis from the final paragraph of chapter 5 is based upon the view that the effective interaction potential Vˆ disappears at large negative and positive values of time (t → ±∞). This is because in the Dirac picture of quantum mechanics, even when Vˆ does not depend explicitly upon time, the probability at |t| → ∞ becomes predominantly high for finding the particle far away from the scattering centre where Vˆ ≈ 0ˆ. Therefore, the physical results should remain unaltered if we assume that the interaction Vˆ is ‘adiabatically turned on and off’. This can be conveniently accomplished by the exponential screening of the interaction via the replacement of Vˆ by Vˆe−ε|t|. Here, ε > 0 is an infinitesimally small positive number, such that the limit ε → 0+ should be taken at the end of the calculations. In fact, by using |t| we can carry out the analysis for both Møller wave operators Ωˆ+ and Ωˆ−. This concept is known as the adiabatic theorem. The meaning of this exponential screening of the potential Vˆ via attenuation or damping is seen by observing that for every proper normalizable physical state, there exists a sufficiently small number ε such that the function exp (−ε|t|) is significantly different from unity only for the values of time t for which the interaction Vˆ vanishes. For small values of |t| and infinitesimally small ε > 0, it is obvious that Vˆ and Vˆ exp (−ε|t|) coincide with each other. In contrast to this, for large values of |t| the difference between the operators Vˆ and Vˆ exp (−ε|t|) becomes appreciable, since in this case the damping factor exp (−ε|t|) tends to zero, thus also causing Vˆ exp (−ε|t|) to vanish. Thus, Vˆ and Vˆ exp (−ε|t|) will again coincide with each other only if the potential Vˆ itself tends to zero for large values of |t|. This is precisely the case for short-range potentials that are, by definition, negligible at long distances at which the probability is high to find the colliding particle for large values of time |t|. In this context, a very important question emerges: is the adiabatic theorem
applicable to scattering states in view of the fact that they are improper i.e. non-normalizable? The answer is in the affirmative. Namely, as long as we keep ε 6= 0, the adiabatic theorem is also valid for improper states, such as plane waves, provided that they are averaged via an integration over a weight
ION-ATOM
function such that (5.9) holds true. This additional procedure of averaging within improper delocalized state vectors by forming localized wave packets is a consequence of the fact that the improper states are normalized to the Dirac δ−function. As is well-known, the δ−function is not a function in the strict meaning of the word, but rather it is a distribution which can be interpreted physically only underneath an integral [24]. After these remarks, we are in a position to formulate the adiabatic theorem as follows1: Theorem 4: If the standard time-dependent Schro¨dinger equation
i(∂/∂t)Ψ(t, r ) = (Hˆ0 + Vˆ)Ψ(t, r ) is solved for the auxiliary interaction Vˆ(r ) × exp (−ε|t|) instead of the original potential operator Vˆ ≡ Vˆ(r ) with the boundary condition at large distances Ψ(t, r ) =⇒
t→−∞ exp (iki · r − iEt) , then for any finite time t the limit ε → 0+ yields a stationary solution of the Schro¨dinger equation HˆΨ(r ) = EΨ(r ). A similar reasoning is also valid if we consider the other time asymptote, t→ +∞. The physical meaning of this theorem is that when the interaction Vˆ (=
Hˆ − Hˆ0) is turned on adiabatically, then any eigen-vector of the operator Hˆ0 is transformed into an eigen-vector of Hˆ with the same eigen-energy for both limits t → −∞ and t → +∞. From here, the following two important consequences emerge: (a) Only the perturbation potential Vˆ is responsible for the transition of
the system between its two unperturbed states that are the initial state i from the remote past (t→ −∞) and the final state f from the distant feature (t→ +∞). (b) The transition i −→ f is physically measurable, since it preserves the
energy E of the whole system via the conservation law Ei = Ef ≡ E. We say that the transition takes place on the energy shell [7]. This is reflected in the fact that the scattering Sˆ−operator commutes with the Hamiltonian Hˆ0 as symbolized by [Sˆ, Hˆ0] = 0ˆ.
