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TECHNICAL PAPERS

Quantum-theoretic Shapes of Constituents of Systems in Various States

[+] Author and Article Information
Elias P. Gyftopoulos

Massachusetts Institute of Technology, Room 24-111, 77 Massachusetts Avenue, Cambridge, MA 02139

Michael R. von Spakovsky

Virginia Polytechnic Institute and State University Energy Management Institute, Department of Mechanical Engineering, Blacksburg, VA 24061-0238

J. Energy Resour. Technol 125(1), 1-8 (Mar 14, 2003) (8 pages) doi:10.1115/1.1525245 History: Received January 01, 2002; Revised March 01, 2002; Online March 14, 2003
Copyright © 2003 by ASME
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References

Gyftopoulos, E. P., and Beretta, G. P., 1991, Thermodynamics: Foundation and Applications, Macmillan, New York.
Hatsopoulos,  G. N., and Gyftopoulos,  E. P., 1976, “A Unified Quantum Theory of Mechanics and Thermodynamics. Part I: Postulates,” Found. Phys., 6(1), pp. 15–31.
Gyftopoulos,  E. P., and Cubukcu,  E., 1997, “Entropy: Thermodynamic Definition and Quantum Expression,” Phys. Rev. E, 55(4), pp. 3851–3858.
Leighton, R. B., 1959, Principles of Modern Physics, McGraw-Hill, New York.
Brandt, S., and Dahmen, H. D., 1995, The Picture Book of Quantum Mechanics, Springer-Verlag, New York.
Slater, J., 1963, Quantum Theory of Molecules and Solids, McGraw-Hill, New York.
Brandt, S., and Dahmen, H. D., op.cit., p. 106.
Beretta,  G. P., Gyftopoulos,  E. P., Park,  J. L., and Hatsopoulos,  G. N., 1984, “Quantum Thermodynamics. A New Equation of Motion for a Single Constituent of Matter,” Nuovo Cimento, 82B(2), pp. 169–191.
Hatsopoulos, G. N., and Gyftopoulos, E. P., 1979, “Thermionic Energy Conversion, Volume II: Theory, Technology, and Applications,” MIT Press, Cambridge, MA, pp. 144–145.
Shankar, R., 1994, “Principles of Quantum Mechanics,” 2nd Ed., Plenum Press, New York.
von Neumann, J., 1995, “Mathematical Foundations of Quantum Mechanics,” Princeton University Press, New Jersey.
Brandt, S., and Dahmen, H. D., op.cit., p. 249.

Figures

Grahic Jump Location
Schematic of a linear box.
Grahic Jump Location
Shape—probability density function versus position and time—of a particle in a linear box (infinitely deep potential well) in a zero entropy nonequilibrium state. Arbitrary dimensionless units (Brandt and Dahmen 7).
Grahic Jump Location
Shape—probability density function versus position— of a particle in a linear box (infinitely deep potential well) in a nonequilibrium state of energy E (Eq. 13). Arbitrary dimensionless units of the spatial coordinate and the probability density function.
Grahic Jump Location
Shape—probability density function versus position— of a particle in a linear box (infinitely deep potential well) in an unstable equilibrium state of energy E (Eq. 13). Arbitrary dimensionless units of the spatial coordinates and the probability density function.
Grahic Jump Location
Shape—probability density function versus position— of a particle in a linear box (infinitely deep potential well) in a stable equilibrium state having the same energy E (Eq. 13) as the unstable equilibrium state in Fig. 4, and evaluated for 30 terms on the right hand side of Eq. 21 Arbitrary dimensionless units of the spatial coordinate and the probability density function.
Grahic Jump Location
Shape—probability density function versus position— of a particle in a linear box (infinitely deep potential well) in a stable equilibrium state that corresponds to α=0.1. The shape is evaluated for 30 terms on the right hand side of Eq. 21 Arbitrary dimensionless units of the spatial coordinate and the probability density function.
Grahic Jump Location
Shape—probability density function versus positions— of a particle in a square box (infinitely deep potential well) in a nonequilibrium state of energy E=94h2/8Md2. Arbitrary dimensionless units of the two spatial coordinates and the probability density function.
Grahic Jump Location
Shape—probability density function versus positions— of a particle in a square box (infinitely deep potential well) in a stable equilibrium state of the same energy E as the nonequilibrium state in Fig. 7. Arbitrary dimensionless units of the two spatial coordinates and the probability density function.
Grahic Jump Location
Shape—probability density function p(ξ) versus the dimensionless position ξ—of a harmonic oscillator in a stable equilibrium state evaluated for α=0.1 and N=5 (see text). Arbitrary dimensionless units.
Grahic Jump Location
Shape—probability density function versus the dimensionless position ξ—of a harmonic oscillator in a stable equilibrium state evaluated for α=0.1 and N=30 (see text). Arbitrary dimensionless units.
Grahic Jump Location
Shape—probability density function p(ξ) versus the dimensionless position ξ—of a harmonic oscillator in a stable equilibrium state evaluated for α=0.1 and N=108 (see text). Arbitrary dimensionless units.
Grahic Jump Location
Shape—probability density function p(ξ) versus the dimensionless position ξ—of a harmonic oscillator in a stable equilibrium state, evaluated for α=1 and N=148 (see text). Arbitrary dimensionless units.
Grahic Jump Location
Shape—probability density function p(ξ) versus the dimensionless position ξ—of a harmonic oscillator in a stable equilibrium state evaluated for α=10 and N=148 (see text). Arbitrary dimensionless units.
Grahic Jump Location
Shape—probability density function p(ξ) versus the dimensionless position ξ—of a harmonic oscillator in a stable equilibrium state evaluated for α=100 and N=148 (see text). Arbitrary dimensionless units.
Grahic Jump Location
Subensemble surfaces—shapes—of the constant probability density function ρ3lm=0.0002 in full x, y, z-space of the heterogeneous ensemble of the electron in a hydrogen atom in a stable equilibrium state (Brandt and Dahmen 12). The shapes for m=−1 and −2 are identical to the ones shown for m=1 and 2, respectively.
Grahic Jump Location
Surface—shape—of the constant probability density function in full x,y,z-space of the homogeneous ensemble of the electron in a hydrogen atom in a stable equilibrium state (arbitrary units).

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