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Propagators in Quantum Chemistry, 2nd Edition by Yngve Öhrn, Jan Linderberg

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Chapter 3

Propagators and Second Quantization

The Schrödinger equation for an electron or for N noninteracting electrons is

Image

with units such that Planck’s constant is 2π and the electron mass is 1, and with ξ = (Image, ζ) being a combined space-spin variable. The wave function or Schrödinger amplitude Image(ξ, t) can be expressed in an orthonormal basis {us(ξ)} as

Image

giving the Schrödinger equation in discrete form

Image

The notation

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has been employed.

Let x be a unitary transformation to energy eigenstates: xx = xx = 1, xhx = ∊ (diagonal). A formal solution to Eq. (3.3) may be written as

Image

The system may be prepared such that |ar(t′)|2 = 1, and |as(t′)|2 = 0 for sr. Then the quantity

Image

is the probability ...

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