C. Cohen-Tanoudji, Quantum Mechanics, Wiley press, New York (1977)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is left open. 9 2 Basics 2.1 Quantum mechanics 2.1.1 Background In this section we shall try to give a brief description of the the way Quantum Mechanics Theory views a simple system. This section is not intended to teach Quantum Physics, for this purpose the reader should refer to [15]. In the first subsection we give the postulates on which the mathematical description of a quantum system is based. These postulates should be treated as axioms. They are the foundations of Quantum Mechanics, and they come from observing physical phenomena and help us to formalize the ....

.... v 2 we denote by v 1 Omega v 2 their tensor product. 7.2 Tensor Product In the following section we describe briefly the notion of tensor product of vector spaces. The description is not intended to be an axiomatic definition of tensor product,rather a description of its properties, see [15]) Tensor product spaces are used in quantum mechanics for the description of many particles systems . For the description of a single particle i we use a vector space H i . For the description of a n particles system we use the vector space H which is the tensor product H 1 Omega Delta Delta ....

Cohen Tanudji. Quantum Mechanics. Springer-Verlag, 1984. 49

....form the projection postulate is known as Luders rule , which says that upon a measurement yielding result b corresponding to a projector Pi b an initial density operator ae evolves to ae 0 b j Pi b ae Pi b p b ; 4. 1) where p b = Trae Pi b is the probability of obtaining result b [35, 36]. Hence the aftermeasurement unconditional density operator becomes ae 0 j P b Pi b ae Pi b . But in fact this postulate describes only one of the many possible ways in which a physical process of measurement may affect a system. I will call measurements in which the effect on the system is ....

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, John Wiley and Sons, New York, 1977.

....situation. Luders rule says that upon a measurement yielding result b corresponding to a projector Pi b (onto a degenerate eigenspace of the observable) an initial density operator ae evolves to ae 0 b : Pi b ae Pi b p b ; 1) where p b = Trae Pi b is the probability of obtaining result b [4], 5] Hence the aftermeasurement unconditional density operator becomes ae 0 : P b Pi b ae Pi b . But in fact this 2 postulate describes only one of the many possible ways in which a physical process of measurement may affect a system. I will call measurements in which the effect on the ....

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, John Wiley and Sons, New York, 1977.

....of an ion. This will be accomplished by insuring that the internal states are nondegenerate and by using resonant excitations to couple only two levels at a time. We will find it convenient to represent a two level system by its analogy with a spin 1 2 magnetic moment in a static magnetic field [93, 94]. In this equivalent representation, we assume that a (fictitious) magnetic moment # = mMS, where S is the spin operator (S = 1 2) is placed in a (fictitious) magnetic field B = B 0 z . The Hamiltonian can therefore be written H internal = v 0 S z , 11) where S z is the operator for the z ....

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, John Wiley and Sons, New York (1977) p. 405.

....worst case play an important role. The success of the scientific method in the natural sciences even where deductive results would be possible in principle is a further hint that such hypotheses may play an increasingly important role in algorithmics. For example, Cohen Tannoudji et al. [4] (after 1095 pages of deductive results) state that in all fields of physics, there are very few problems which can be treated completely analytically. Even a simple two body system like the hydrogen atom cannot be handled analytically without making simplifying assumptions (like handling the ....

C. Cohen-Tannoudji, B. Diu, and F. Laloe. Quantum Mechanics, volume 2. John Wiley & Sons, Inc., 1977.

.... 1 2 ( L t r Theta r Gamma r Theta L t r ) j p f p ex ; 18) where p f = r( Gammai r ) Gamma r Theta L r ; 19) p ex = 1 2 [ p t Gamma L t r ) L Theta r Gamma r Theta L(p t Gamma L t r ) 20) Here we have defined a fictitious particle momentum p f [11] such that r Theta p = L r Theta p ex ; 21) and hence L = r Theta p f ; 22) where L is the system (particle plus tube) orbital angular momentum. By (18) and L = L q L t we see that p f = p q 1 2 [ L t r L Theta r Gamma r Theta L L t r ] 23) We also find from (19) ....

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, John Wiley and Sons, Inc., 1977. 17

....which is much simpler. At the end of the chapter, I present the model of quantum Turing machines, for completeness. For background on basic quantum mechanics such as Hilbert spaces, Schrodinger equation and measurements I recommend to consult the books by Sakurai[167] and by Cohen Tanoudji[71]. As for more advanced material, the book by Peres[161] would be a good reference. However, I will give here all the necessary definitions. A quantum circuit is a system built of two state quantum particles, called qubits. We will work with n qubits, the state of which is a unit vector in the ....

C. Cohen-Tanoudji, Quantum Mechanics, Wiley press, New York (1977)

....the frequency range of these pulses can be made selective for single spins. For example, a selective pulse of sufficient duration and intensity to rotate the net magnetization of the th spin by and in phase with the imaginary component of the carrier corresponds to the unitary matrix , [12] where the matrix occurs as the th factor of the Kronecker product. k p 2 1 . 1 U p 2 1 . 1 U p 2 1 2 1 1 1 1 = U p 2 k 14 For a two spin system, we have the energy level diagram shown in Fig. 1. The four dashed two headed arrows are the ....

Cohen-Tannoudji, C., Diu, B. and Laloë, F. (1977) Quantum Mechanics (J. Wiley & Sons, New York, NY), vols. 1 & 2.

....j2pmlf g x ( e j2pf x i 1 = N = 45 of the input and the output which is given below. 3 11 where is the order of diffraction. In order to determine the KL distance between the input and output plane wave distributions, mathematical objects known in quantum mechanics as density matrices [31, 32, 33] have to be introduced. The elements of a density matrix are all possible bilinear products of the coefficients of an expansion in some basis. As an example, consider Eq 3 15, where the input to the grating is represented as an expansion in a plane wave basis. The density matrix corresponding to ....

Cohen-Tannoudji, C., Quantum Mechanics, Wiley, (1977). 76

....of the matrix (see section 4.1) The derivatives of the eigenvalues and eigenvectors of a matrix valued function of a scalar argument have been rediscovered many times, see e.g. 12] 17] and references therein. In physics, the eigenvalue derivatives are implicit in the Hellmann Feynman theorem [18], but the eigenvector derivatives appear to be much less widely known. For this reason we h 1 y 1 r T g 1 y 1 , y 1 S g 1 g 2 , r g 2 y 2 , h 2 y 1 r T g 1 y 1 , y 1 r g 2 ....

C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics (J. Wiley & Sons, New York, 1977), p. 1192. 81

....full use of quantum superposition. Bernstein and Vazirani [12] gave the rst eOEcient construction of a universal quantum computer which can simulate any quantum computer with only polynomial overhead. 1. 2 Quantum Mechanical Basis In this section we explain the basic rules of quantum mechanics [32, 21, 13]. The quantum state of any physical system is characterized by a state vector belonging to an abstract Hilbert space H called the state space of the system. Any element or vector of H is called a ket vector or more simply a ket and is represented by a symbol #. Any element of the dual space H ....

C. Cohen-Tannoudji, B. Diu, and F. Lalo#, Quantum mechanics, Hermann, Paris, 1977.

....on distance. Therefore, the potential function must have a K Gamma1 dependence for short distance repulsion, but drop to zero faster than K Gamma1 for large distances. Also, the function and its derivative must be continuous. A function that has these attributes is the Yukawa potential [5]: U(K) A e GammaffK K (28) Figures 8 and 9 show this function with ff = 1 and A = 1 for a rectangle and a triangle. The parameter ff determines how rapidly the potential rises near the object and falls off away from the object. This rate must be related to the rate at which the n ness of ....

B. Cohen-Tannoudji, C. Diu, and F. Laloe. Quantum Mechanics, volume 2. John Wiley and Sons, New York, 1977.

....important and difficult problems concerning the eigenvalues and eigenvectors of a certain sequence of Hermitian matrices that arise in the quantummechanical study of antiferromagnetic Heisenberg chains. Let V be the three dimensional inner product space C 3 . The spin 1 operators (see, e.g. [8]) are the operators on V defined by S x = 1 p 2 0 0 1 0 1 0 1 0 1 0 1 A ; S y = i p 2 0 0 1 0 Gamma1 0 1 0 Gamma1 0 1 A ; S z = 0 Gamma1 0 0 0 0 0 0 0 1 1 A ; with respect to an orthonormal basis which we denote by fv Gamma1 ; v 0 ; v 1 g. The choice of subscripts on the v ....

....7. Other spin values. The vector space V and spin operators S x ; S y , and S z discussed in this paper correspond to atoms with spin 1. In general, spin values may be any element of f 1 2 ; 1; 3 2 ; 2; g. In this section, we discuss the spin operators for all spin values (see, e.g. [8]) and Haldane s more general conjecture ( 9] 10] For spin s 2 f 1 2 ; 1; 3 2 ; 2; g, the vector space required is V s = C 2s 1 with orthonormal basis fv Gammas ; v Gammas 1 ; v s Gamma1 ; v s g. The operator S z on V s is defined by S z (v j ) jv j , extended by ....

Cohen-Tannoudji, C., Diu, D., Laloe, F.: Quantum Mechanics, vol. 1 (Chapter VI). Paris: Hermann 1977

.... t 1 2 ( L t r Theta r Gamma r Theta L t r ) j p f p ex ; 18) where p f = r( Gammai r ) Gamma r Theta L r ; 19) p ex = 1 2 [ p t Gamma L t r ) L Theta r Gamma r Theta L(p t Gamma L t r ) 20) Here we have defined a fictitious particle momentum p f [10] such that r Theta p = L r Theta p ex ; 21) and hence L = r Theta p f ; 22) where L is the system (particle plus tube) orbital angular momentum. By (18) and L = L q L t we see that p f = p q 1 2 [ L t r L Theta r Gamma r Theta L L t r ] 23) We also find from (19) ....

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, John Wiley and Sons, Inc., 1977.

....kinetic energy, and is ensured by setting the magnitude of the potential at the surface to infinity. Away from the surface, the energy values behave like natural potentials (e.g. electrostatic, gravitational, etc. in their inverse dependence on distance. This is done with the Yukawa potential [9] which has K Gamma1 dependence for short distance repulsion, but exponential decay at larger distances: U(K) A e GammaffK K (3) Figure 3 shows this function with ff = 1 and A = 1 for a rectangle. The parameter ff determines how rapidly the potential rises near the object and falls off ....

B. Cohen-Tannoudji, C. Diu, and F. Laloe. Quantum Mechanics, volume 2. John Wiley and Sons, New York, 1977.

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C. Cohen-Tanoudji, Quantum Mechanics, Wiley press, New York (1977)

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C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (Herman/Wiley, Paris, 1977).

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Cohen-Tannoudji, C., Diu, B., LaLoe, F., Quantum mechanics, 1977, pp. 108--181.

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Cohen-Tannoudji, C., B. Diu, and F. Lalo e, Quantum Mechanics, Volume I, J. Wiley & Sons, 1977.

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C. Cohen--Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, John Wiley & Sons, New York (1977)

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Cohen-Tannoudji, C., Diu , B., & Laloe. F. 1977, Quantum Mechanics, New York : John Wiley

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