Sophus Lie: Gesammelte Abhandlungen by Engel F., Heegaard P. (eds.)

By Engel F., Heegaard P. (eds.)

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6. A generalized formalism is discussed in Chap. 4 on “Imperfect Quantum Operations”. 36 CHAPTER 2. QUANTUM COMPUTATION AND SYSTEMS The matrix Uxy (t) must be unitary to preserve the normalization of the probability amplitudes and hence to be compatible with the probabilistic interpretation of the state vector. , in atomic, molecular and optical systems. 11) i ∂t where ψ, the wave function, depends on both time t ∈ R and space r = (x1 , x2 , . . 11) is an operator, called the Hamiltonian, and is the Planck constant.

Independent of , 2n (2 +1) where Ln+ 0, 1, 2, . . 28) are the associated Laguerre polynomials such that for m, n = (m) (m) (m) xLm+n + (m + 1 − x)Lm+n + (m + n)Lm+n = 0, (m) Lm+n (x) = ex x−(m+n) dm+n −x 2m+n (e x ), (m + n)! 2. QUANTUM MECHANICAL SYSTEMS 41 (0) (when m = 0, Ln (x) is simply denoted as Ln (x)). The electron state of the hydrogen can thus be characterized by three “quantum numbers” n, , m: ⎧ |n m : Rn (r)Θ m (θ)Φm (φ) is its spatial wave function, with a ⎪ ⎪ ⎪ ⎨ proper normalization factor, n = 1, 2, .

It can be described by a 6-tuple (Q, A, B, δ, q0 , qa ), where Q = {q0 , q1 , q2 , . . , qm } is a finite set of control states; A = {α1 , α2 , α3 , . . 1. TURING MACHINES AND BINARY LOGIC GATES 29 δ : Q × A → Q × A × {L, R} is the transition function which describes the rules for the moves of the TM, where “L” and “R” stand for the tape head moving left and right by one cell, respectively; q0 ∈ Q is the initial state; qa ∈ Q is the accepting state. Initially, a finite string of symbols from A (the input) is written on the tape.

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