**Chapter 3**

**Propagators and Second Quantization**

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

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

giving the Schrödinger equation in discrete form

The notation

has been employed.

Let x be a unitary transformation to energy eigenstates: x^{†}x = xx^{†} = **1**, x^{†}hx = ∊ (diagonal). A formal solution to Eq. (3.3) may be written as

The system may be prepared such that |*a*_{r}(*t*′)|^{2} = 1, and |*a*_{s}(*t*′)|^{2} = 0 for *s* ≠ *r*. Then the quantity

is the probability ...

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