Partition Function for the Harmonic Oscillator

In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In Secs. ?? and ??, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in Sec. ??, the comparison of results is provided with reference to other existing research.

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Partition Function for the Harmonic Oscillator

Classical and Quantum Dynamics ◽

10.1007/978-3-642-97921-7_23 ◽

1992 ◽

pp. 257-262

Author(s):

Walter Dittrich ◽

Martin Reuter

Keyword(s):

Partition Function ◽

Harmonic Oscillator

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On the quantum mechanical solutions with minimal length uncertainty

International Journal of Modern Physics A ◽

10.1142/s0217751x16501013 ◽

2016 ◽

Vol 31 (18) ◽

pp. 1650101 ◽

Cited By ~ 13

Author(s):

Homa Shababi ◽

Pouria Pedram ◽

Won Sang Chung

Keyword(s):

Heat Capacity ◽

Partition Function ◽

Harmonic Oscillator ◽

Internal Energy ◽

Free Particle ◽

Minimal Length ◽

Quantum Mechanical ◽

Uncertainty Principles ◽

Trigonometric Functions ◽

Eigenvalues And Eigenfunctions

In this paper, we study two generalized uncertainty principles (GUPs) including [Formula: see text] and [Formula: see text] which imply minimal measurable lengths. Using two momentum representations, for the former GUP, we find eigenvalues and eigenfunctions of the free particle and the harmonic oscillator in terms of generalized trigonometric functions. Also, for the latter GUP, we obtain quantum mechanical solutions of a particle in a box and harmonic oscillator. Finally we investigate the statistical properties of the harmonic oscillator including partition function, internal energy, and heat capacity in the context of the first GUP.

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On the Complementarity of the Harmonic Oscillator Model and the Classical Wigner–Kirkwood Corrected Partition Functions of Diatomic Molecules

Entropy ◽

10.3390/e22080853 ◽

2020 ◽

Vol 22 (8) ◽

pp. 853

Author(s):

Marcin Buchowiecki

Keyword(s):

Phase Space ◽

Partition Function ◽

Harmonic Oscillator ◽

High Temperatures ◽

Low Temperatures ◽

Diatomic Molecules ◽

Partition Functions ◽

Harmonic Oscillator Model ◽

Harmonic Oscillator Approximation ◽

Classical Partition Function

The vibrational and rovibrational partition functions of diatomic molecules are considered in the regime of intermediate temperatures. The low temperatures are those at which the harmonic oscillator approximation is appropriate, and the high temperatures are those at which classical partition function (with Wigner–Kirkwood correction) is applicable. The complementarity of the harmonic oscillator and classical integration over the phase space approaches is investigated for the CO and H2+ molecules showing that those two approaches are complementary in the sense that they smoothly overlap.

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Rotational Cut-Off and Anharmonicity Effects on Thermodynamic Properties of Gases at High Temperatures

Aeronautical Quarterly ◽

10.1017/s0001925900007071 ◽

1974 ◽

Vol 25 (4) ◽

pp. 287-292

Author(s):

N M Reddy

Keyword(s):

Partition Function ◽

Thermodynamic Properties ◽

Harmonic Oscillator ◽

Specific Heat ◽

High Temperatures ◽

Anharmonic Oscillator ◽

Rigid Rotator ◽

Temperature Range ◽

Vibrational Levels ◽

Harmonic Oscillator Approximation

SummaryThe exact expressions for the partition function Q and the coefficient of specific heat at constant volume Cv for a rotating-anharmonic oscillator molecule, including coupling and rotational cut-off, have been formulated and values of Q and Cv have been computed in the temperature range of 100°K to 100 000°K for O2, N2 and H2 gases. The exact Q and Cv values are also compared with the corresponding rigid-rotator harmonic-oscillator (infinite rotational and vibrational levels) and rigidrotator anharmonic-oscillator (infinite rotational levels) values. The rigid-rotator harmonic-oscillator approximation can be accepted for temperatures up to about 5000°K for O2 and N2. Beyond these temperatures the error in Cv will be significant, owing to anharmonicity and rotational cut-off effects. For H2, the rigid-rotator harmonic-oscillator approximation becomes unacceptable even for temperatures as low as 2000°K.

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GENERALIZED FERMI PARTITION FUNCTION OBTAINED VIA COHERENT STATE REPRESENTATION OF THE DENSITY OPERATOR

Modern Physics Letters A ◽

10.1142/s0217732399002571 ◽

1999 ◽

Vol 14 (35) ◽

pp. 2471-2479 ◽

Cited By ~ 3

Author(s):

HONG-YI FAN ◽

HUI ZOU ◽

YUE FAN

Keyword(s):

Partition Function ◽

Coherent State ◽

Harmonic Oscillator ◽

Density Operator ◽

Fermi System ◽

State Representation

We generalize the well-known partition function of Fermi harmonic oscillator (H =ω(f+f + ½))[Formula: see text] to [Formula: see text] for Fermi system described by the n-mode fermionic Hamiltonian [Formula: see text]. This result is derived via constructing the fermionic coherent state representation of the density operator [Formula: see text] in which [Formula: see text] is identified as exp {-β Γ Π}.

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