R. Feynman and A. Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....continuum limits of r and t, s M (r, t) s dM dt, G,7 1 ) r 7 1 r ) r M . 66) The previous development of mesocolumnar interactions via nearest neighbor derivative couplings permits the regional short time propagator P to be developed in terms of the Lagrangian L [110]: P( M) 2 6) d Mg 5 2 exp[ N S( M) P( M ) S = min dt L[ M(t ) M(t ) L =58 rL, 67) where 8 is the area of the region considered, and 58 r =58 dx dy = 0 5 . 68) The Euler Lagrange (EL) equations, giving the extrema M , are obtained from ....

R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

....# ) 4.3) The integral on the right hand side diverges. This is a consequence of the fact that we have not used Feynman s original time slicing procedure for defining the path integral on a temporal lattice of spacing a. As in the case of an ordinary harmonic oscillator discussed in detail in [2, 6] this would lead to a finite integral in which # # is replaced by # # (2 2 cos a# # ) a 1 # #2 ] log # #2 #(# ) 4.4) where #(# ) # # is the sign of # . For a derivation see Eqs. 2.319) and (2.346) in Ref. 6] This finite result can equally ....

R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals," McGraw--Hill, New York, 1965.

....for the rate w i = # i h of direct transitions from the state i into the continuum (a factor counting the density of final states has been included in the definition of the matrix elements U and W ) Eq. 8) ii) resembles the expression for the transition rate in second order perturbation theory [12], but in contrast to the perturbative expression the denominator in Eq. 8) ii) has a finite imaginary part. We note that the width arising from direct transitions between regular states and the continuum evolves smoothly under variations of an external parameter. Any strong fluctuations in the ....

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York, 1964).

....he addresses different questions than the ones we consider. After obtaining a formula for the amplitudes of the walk as a sum of binomial coefficients (which we state as Lemma 4) Meyer proceeds to analyzing the continuoustime limit of QLGA and shows that this limit is given by the Dirac equation [13]. The results about the continuous time limit apparently do not imply anything for the discrete case that we study in this paper. Farhi and Gutmann [12] and Childs, Farhi and Gutmann [6] analyze quantum walks on trees and exhibit collections of graphs on which the quantum process hits one ....

R. Feynman and A. Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, 1965.

....Suppose that .i ci ci = 1 (summing over i) with c i ci =1 for each i. Then a b = a b = a S ci ci b = S a ci ci b = S a c i c i b . Iterating this principle of expansion over a complete set of states leads to the most primitive form of the Feynman integral [FH]. Imagine that the initial and final states a and b are points on the vertical lines x=0 and x=n 1 respectively in the x y plane, and that (c(k)i(k) k) is a given point on the line x=k for 0 i(k) m. Suppose that the sum of projectors for each intermediate state is complete. That is, we assume ....

R. Feynman and A.R. Hibbs. Quantum Mechanics and Path Integrals. McGraw Hill (1965).

....take from A to B, and the amplitude for each path is known. If x and y are the amplitudes for the two paths, then the amplitude of going from A to B is x y, so the probability of going from A to B is jx yj 2 rather than jxj 2 jyj 2 which is the sum of the probabilities for each path [2]. For our purposes here we define the amplitude to be the positive square root of the probability. Let x j = fi(p; a j ) and y j = fi(q; a j ) We have seen that x j is the probability that a random chain from p goes through a j , and that y j is the probability that a random chain from q goes ....

Feynman, R. P., and A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, 1965.

....but he addresses di erent questions than the ones we consider. After obtaining a formula for the amplitudes of the walk as a sum of binomial coecients (which we state as Lemma 4) Meyer proceeds to analyzing the continuoustime limit of QLGA and shows that this limit is given by the Dirac equation [13]. The results about the continuous time limit apparently do not imply anything for the discrete case that we study in this paper. Farhi and Gutmann [12] and Childs, Farhi and Gutmann [6] analyze quantum walks on trees and exhibit collections of graphs on which the quantum process hits one ....

R. Feynman and A. Hibbs. Quantum Mechanics and Path Integrals. McGraw{Hill, 1965.

.... [1999] and Jalnapurkar, Leok, Marsden and West [2001] Discrete mechanics also has some intriguing links with quantization, since Feynman himself first defined path inte REFERENCES 10 grals through a limiting process using the sort of discretization used in the discrete action principle (see Feynman and Hibbs [1965]) Conclusions. As we hope this brief history shows, reduction theory remains very much an active and exciting area of research with many significant applications to problems of interest in both Engineering and Physics. This is rather remarkable since this theory in its modern form is nearly 30 ....

Feynman, R.p. and A. R. Hibbs [1965], Quantum Mechanics and Path Integrals, McGraw-Hill.

....with geometric quantization. From the point of view of canonical quantization, if the value of action W : R phase space = R phase space d a dq a of a system on the phase space of a k dimensional configuration space M is of order h, i.e. W h, then this system is a quantum system [9] and the quantization structure can be described by 6 W = Nh ; N 2 Z or by [ a ; q b ] Gammaihffi b a ; a; b = 1; k. From the point of view of geometric quantization [7] this quantization is due to the integrality of the first Chern class ch 1 2 Omega 2 (M ; U(1) flat ) on the ....

R. P. Feynman and A. R. Hibbs, Quantum mechanics and Path Integrals (McGraw-Hill 1965).

....solutions, we also get the solutions of non relativistic equations. The mathematical procedures are very much alike. Only the physical meanings of the variables need to be modi ed. We take the propagator in constant B eld for example. We can compare our result with others in some text books. [2] They solved it using the path integral method. m 2 iT 3 2 T=2 sin T=2 exp im 2 (z z 0 ) 2 T 2 cot T 2 (x x 0 ) 2 (y y 0 ) 2 (x 0 y xy 0 ) 6 13) T = t t 0 ; eB m (6 14) We list all modi cations and basic elements in the ....

R. P. Feynman & A. R. Hibbs, \Quantum Mechanics and Path Integrals", McGraw-Hill, New York, 1964.

....that he asks are quite di erent from ours. The only overlapping result is the formula for the amplitudes as the sum of binomial coecients (our Lemma 6) After obtaining this result, he proceeds to analyzing the continuous time limit of QLGA and shows that this limit is given by the Dirac equation [5]. The results about the continuous time limit does not imply anything for the discrete case which we study in this paper. Farhi and Gutmann [4] analyze quantum random walks on trees and exhibit a collection of trees on which the quantum process hits one particular leaf exponentially faster than ....

R. Feynman and A. Hibbs. Quantum Mechanics and Path Integrals. McGraw{Hill, 1965.

.... are connected with springs of zero natural length and the object function is defined as S PEB ( R 1 ; RP Gamma1 ) P X i=0 V ( R i ) P X i=1 Pk 2 ( R i Gamma R i Gamma1 ) 2 (1) then the chain is mathematically analogous to a Feynman path integral [21] for an offdiagonal element of a density matrix describing a quantum particle. Kuki and Wolynes carried out thermal sampling of such paths to identify important tunneling paths of electrons in proteins [22] In the context of finding MEPs for classical systems, one could envision minimizing the ....

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw Hill, New York, 1965).

....(0) 1 2 ) 1 N ln(1 Gamma exp( GammaN ) 1 N ln[ 1 X n=0 exp( GammaN (n 1 2) Omega Gamma ) 6) 4 with Omega Gamma ) the effective frequency Omega Gamma ) 2asinh( 2) 7) where is a function of n i , N i and c i . This of course goes back to Feynman [11] and is nothing but the exact solvability of the harmonic oscillator path integral with a finite step size. For an Abelian zero momentum background field this dependence of on c i collapses to a very simple expression and leads to the so called vacuum valley effective potential [3,12] c i = ....

....for the Rayleigh Ritz analysis of the effective Hamiltonian. The kinetic term will be discussed after we have dealt with the issue of discrete time. 4. The effective Hamiltonian There is an elegant way of incorporating the discrete time when one realizes that the path integral is given by [8,11] Z = Tr( e GammaK e GammaV ) N 0 ) Tr(T N 0 ) 18) where T is the transfer matrix, and the trace is taken over a complete set of quantum states. Here K is the (SU(2) or standard S 3 ) kinetic and V the potential term. The masses obtained from lattice Monte Carlo calculations, in ....

R.P. Feynman and A.R. Hibbs, "Quantum Mechanics and Path Integrals", (McGrawHill, New York, 1965).

....measure given formally by Z Gamma1 exp( GammafiH) 19) where fi is an inverse temperature and Z is a normalization constant. We choose arbitrarily fi = 1 and choose the data from the corresponding distribution. The differentiated terms in the expression of H characterize the Wiener process [3], and thus the sample functions drawn from this probability measure are almost nowhere differentiable [4] We do not wish to discuss here how such solutions are to be approximated 6 by finite differences. We start instead with a set of finite difference equations that are formally consistent ....

R. Feynman and A. Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, New York, 1965.

....equation on the unit circle with a constant potential, iu t = Gammau xx m 2 u (35) where u is a complex valued function on the circle and m 2 is a constant. This equation is the Hamilton equation of motion for the Hamiltonian H[u] Z 2 0 Theta ju x j 2 m 2 juj 2 dx: 36) see [11]) Equation (35) preserves the density of the canonical ensemble, f 0 [u] e GammaH [u] 37) where the temperature has been chosen equal to one. The measure defined by equation (37) is absolutely continuous with respect to a Wiener measure [15] and its samples are, with probability one, ....

R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, NY, 1965.

....= 1:0 Theta 10 Gamma6 . 13 with p t 0 = R dtk t G tt 0 . So long as the t k s are independent, we can write OE(p t ) exp(i Z dtp t Delta F t ) Y k 1 X N k =0 PN k ( Z dt k R k (t k ) exp(i Z dt p t f k (t Gamma t k ) k ) N k where k = R dt k R k (t k ) see [19, 20, 21]) Here PN k is the probability of N k events in the time interval of interest, and R k the average rate of these events. To derive x t 2 ffi(x(t 2 ) Gamma x 2 )ffi( x(t 2 ) Gamma x 2 ) for the integral differential equation in the text, we must calculate at j = 0: Gammai j Z d ....

Feynman, R.P. and A.R. Hibbs, Quantum Mechanics and Path Integrals, New York, McGraw Hill, 1965.

....convergence properties when applied to model problems. Key Words: Feynman Kac Path Integrals, Quantum Statistical Mechanics, Functional Integration, Path Integral Methods 1. INTRODUCTION The path integral approach provides a powerful method for studying properties of quantum manybody systems [1]. When applied to statistical mechanics [2] each element of the quantum density matrix is expressed as an integral over all curves connecting two con gurations: a:b D [x ( exp h [x ( 1) The symbol D [x ( indicates that the integration is performed over the set of ....

....x : 0; h] R , with x (0) a and x ( h) b. The integer d re ects the dimensionality, with d = 3N for a system of N particles in 3 dimensional space. The functional can be derived from the classical action by introducing a relationship between temperature and imaginary time (it = h) [1]. In this paper, we will restrict our attention to the quantum many body system, for which takes the following form: x ( m i x i ( V [x ( d : 2) Calculating the path integral in (1) is a challenging task, which in general cannot be performed analytically. It is ....

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

....i.e. by an integral over the space of continuous functions. In general, S is non linear. As already mentioned in the introduction, for H(u) exp(u) 9) is the famous FeynmanKac formula for the solution of the heat equation with the initial condition v and the potential function V , see e.g. [1, 8, 12]. This is why we call (9) a Feynman Kac path integral independently of the choice of the function H. We wish to approximate S with (worst case) error at most for various v and V . In what follows, we assume that we know some upper bounds on their norms. For positive and V, let and V denote ....

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path-Integrals (McGraw-Hill, New York, 1965).

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R. Feynman and A. Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, 1965.

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Feynman R.P. and Hibbs A.R., Quantum Mechanics and Path Integrals, New York, McGraw-Hill, 1965.

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Feynman R.P., & Hibbs A.R. (1965). Quantum mechanics and path integrals, McGraw-Hill Book Company, 365 p.

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R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGarw Hill, NY, (1965)

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R.P. Feynman and H.R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, 1965).

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R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965).

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Feynman, R. P. and Hibbs, A. R., Quantum Mechanics and Path Integrals, New York, McGraw-Hill, 1965, p. 245.

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