Order <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><mi>N</mi></math></span> corrections to the time-dependent Hartree approximation for a system of <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>+</mo><mn>1</mn><mn></mn></math></span> oscillators

Bogdan Mihaila; John F. Dawson; Fred Cooper

doi:10.1103/PhysRevD.56.5400

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Order $1 / N$ corrections to the time-dependent Hartree approximation for a system of $N + 1$ oscillators

Bogdan Mihaila and John F. Dawson

Fred Cooper

Department of Physics, University of New Hampshire, Durham, New Hampshire 03824

Theoretical Division, MS B285, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

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Phys. Rev. D 56, 5400 – Published 1 November, 1997

DOI: https://doi.org/10.1103/PhysRevD.56.5400

Abstract

We solve numerically to order $1 / N$ the time evolution of a quantum dynamical system of $N$ oscillators of mass $m$ coupled quadratically to a massless dynamic variable. We use Schwinger’s closed time path formalism to derive the equations. We compare two methods which differ by terms of order ${1 / N}^{2} .$ The first method is a direct perturbation theory in $1 / N$ using the path integral. The second solves exactly the theory defined by the effective action to order $1 / N .$ We compare the results of both methods as a function of $N .$ At $N = 1,$ where we expect the expansion to be quite innacurate, we compare our results to an exact numerical solution of the Schrödinger equation. In this case we find that when the two methods disagree they also diverge from the exact answer. We also find at $N = 1$ that the $1 / N$ corrected evolutions track the exact answer for the expectation values much longer than the mean field $(N = \infty)$ result.

References (11)

H. E. Stanley, Phys. Rev. 176, 718 (1968); K. Wilson, Phys. Rev. D 7, 2911 (1973); S. Coleman, R. Jackiw, and H. D. Politzer, 7, 2911 (1973).
F. Cooper, J. Dawson, S. Habib, Y. Kluger, D. Meredith,and H. Shepard, Physica D 83, 74 (1995).
Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola, Phys. Rev. D 45, 4659 (1992).
F. Cooper, J. M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky, Phys. Rev. D 48, 190 (1993).
F. Cooper, Y. Kluger, E. Mottola, and J. P. Paz, Phys. Rev. D 51, 2377 (1995); M. A. Lampert, J. F. Dawson, and F. Cooper, 54, 2213 (1996); M. A. Lampert, Ph.D. dissertation, University of New Hampshire, 1996.
F. Cooper, J. F. Dawson, D. Meredith, and H. Shepard, Phys. Rev. Lett. 72, 1337 (1994).
F. Cooper, J. Dawson, S. Habib, and R. D. Ryne, quant-ph/9610013.
F. Cooper, S. Habib, Y. Kluger, E. Mottola, J. P. Paz, and P. R. Anderson, Phys. Rev. D 50, 2848 (1994).
J. Schwinger, J. Math. Phys. (N.Y.) 2, 407 (1961); P. M. Bakshi and K. T. Mahanthappa, 4, 1 (1963); 4, 12 (1963); L. V. Keldysh, Zh. Éksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)]; G. Zhou, Z. Su, B. Hao, and L. Yu, Phys. Rep. 118, 1 (1985).
C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
S. E. El-gendy, Comput. J. (UK) 12, 282 (1969).

[c1] H. E. Stanley, Phys. Rev. 176, 718 (1968); K. Wilson, Phys. Rev. D 7, 2911 (1973); S. Coleman, R. Jackiw, and H. D. Politzer, 7, 2911 (1973).

[c2] F. Cooper, J. Dawson, S. Habib, Y. Kluger, D. Meredith,and H. Shepard, Physica D 83, 74 (1995).

[c3] Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola, Phys. Rev. D 45, 4659 (1992).

[c4] F. Cooper, J. M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky, Phys. Rev. D 48, 190 (1993).

[c5] F. Cooper, Y. Kluger, E. Mottola, and J. P. Paz, Phys. Rev. D 51, 2377 (1995); M. A. Lampert, J. F. Dawson, and F. Cooper, 54, 2213 (1996); M. A. Lampert, Ph.D. dissertation, University of New Hampshire, 1996.

[c6] F. Cooper, J. F. Dawson, D. Meredith, and H. Shepard, Phys. Rev. Lett. 72, 1337 (1994).

[c7] F. Cooper, J. Dawson, S. Habib, and R. D. Ryne, quant-ph/9610013.

[c8] F. Cooper, S. Habib, Y. Kluger, E. Mottola, J. P. Paz, and P. R. Anderson, Phys. Rev. D 50, 2848 (1994).

[c9] J. Schwinger, J. Math. Phys. (N.Y.) 2, 407 (1961); P. M. Bakshi and K. T. Mahanthappa, 4, 1 (1963); 4, 12 (1963); L. V. Keldysh, Zh. Éksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)]; G. Zhou, Z. Su, B. Hao, and L. Yu, Phys. Rep. 118, 1 (1985).

[c10] C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

[c11] S. E. El-gendy, Comput. J. (UK) 12, 282 (1969).

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Order $1 / N$ corrections to the time-dependent Hartree approximation for a system of $N + 1$ oscillators

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Order 1/N corrections to the time-dependent Hartree approximation for a system of N+1 oscillators

Abstract

References (11)

Order $1 / N$ corrections to the time-dependent Hartree approximation for a system of $N + 1$ oscillators