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8). e. the exclusion principle; the state "02 already contains an electron in the state u hence we get zero when we 0 operate on it with aR ). One energy integration is performed immediately using the function. 6)) "C(E)" "C(E)" 1 1 ! "! dE"C(E)" . J(0)" dE ! 12) The sign of the current is consistent with a current of (previously bound) electrons to the left. In the limit we are interested in (a large) "C" should be averaged over a thus yielding 1 dE ¹ (E) . J(0)"! 2 58 N. Dombey, A.
DE"C(E)" . J(0)" dE ! 12) The sign of the current is consistent with a current of (previously bound) electrons to the left. In the limit we are interested in (a large) "C" should be averaged over a thus yielding 1 dE ¹ (E) . J(0)"! 2 58 N. Dombey, A. Calogeracos / Physics Reports 315 (1999) 41}58 Note that whereas for a Klein step we describe the particle emission in terms of electron}positron pair production while we say that a supercritical barrier will spontaneously emit electrons. But an observer just inside the barrier will see a corresponding current which he will interpret from its sign as a positron current.
21) and (22). According to our conventions " "! uR (E, x) u (E, x)"(2 / )/( #1) , * * W * (23) N. Dombey, A. (2 / )/( #1) . 2. The dexnition of the vacuum and the vacuum expectation value of the current Now expand the wavefunction in terms of creation and annihilation operators which refer to our left- and right-travelling solutions: (x, t)" dE +a (E)u (E, x)e\ #R#a (E)u (E, x)e\ #R#bR (E)v (E, x)e #R#bR (E)v (E, x)e #R, * * 0 0 * * 0 0 (25) with R given by the Hermitian conjugate expansion.