Semiclassical path integral theory of a double‐well potential in an electric field

Douvropoulos, Theodosios G.; Nicolaides, Cleanthes A.

doi:10.1002/qua.20865

Cited by 1 publication

(5 citation statements)

References 21 publications

(36 reference statements)

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“…In the present work we extend the model proposed for the ammonia molecule in Ref. 6, in order to demonstrate the existence of the above‐mentioned indeterminacies. In this way, the overall positive imaginary part for each pole is proved to contain a hidden negative quantity, corresponding to the decay rate of the indeterminacies.…”

Section: Introductionmentioning

confidence: 54%

“…In each case, it is obvious that the coefficients of the real energy poles are nearly equal to while the coefficients of the complex energy poles are nearly equal to From the above relations, it is clear that the two types of coefficients differ by a factor of exp(−κ)/2, which results as a quite strong difference. This factor approaches the value of 1.3 × 10 −3 for the ammonia model of 6. Therefore, it is to be understood that the terms related to the complex energy poles, play a significant role only in the case where the real energy differences E − z

_{italicR}^{±}

approach zero.…”

Section: Comments Relative To the Character Of The Complex Energy Polesmentioning

confidence: 82%

“…We will follow the same approach as we did in Ref. 6, where we used a specific type of analytic potential for the description of the ammonia molecule. In particular, we used the polynomial form and fixed parameters D , C , and V O to get the correct experimental value of 0.8 cm −1 for the inversion frequency, when we imposed the semiclassical path integral theory.…”

Section: Quantities γ O+ and γ O−: Magnitude And Sign In Ammonia Molementioning

confidence: 99%

“…In fact, we can give a numerical example concerning the above quantities, for the model describing the ammonia molecule in Ref. 6. We calculated the ratio of the quantities R ± , to the energy difference of the first two levels, Δε, and the ratio of the quantities ħ/ I ± , to the inversion period of the molecule, T inv , and found correspondingly: It is then clear from the above results that not only is the rate with which the energy indeterminacies decay very small compared with the period with which the nitrogen atom oscillates between the two wells, but these indeterminacies are themselves very small quantities, and so can hardly be understood.…”

Section: Comments Relative To the Character Of The Complex Energy Polesmentioning

confidence: 99%

“…The appearance of a negative imaginary part for each complex energy pole, indicates a possible decaying activity, meaning that the system does not remain bound and stationary with time. Indeed, this is the case when the molecule is disturbed by a static electric field imposed on one side of the hydrogen plane only 6. The imaginary part, then, which is called the width function, plays the role of the system's decay rate if the exponential law is assumed for its time evolution.…”

Section: Introductionmentioning

confidence: 99%

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Positive width function and energy indeterminacies in ammonia molecule

Douvropoulos

2006

Int J of Quantum Chemistry

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ABSTRACT:A recently published methodology based on the semiclassical path integral theory was applied in a double well structure and gave the analytic form of the system's Green's function. This type of potential can describe the ammonia molecule as far as the motion of the nitrogen atom perpendicular to the hydrogen plane is discussed. Because of the fact that a double well describes a bound system and correspondingly stationary states (constructed by the symmetric and antisymmetric superposition of the eigenstates of the two unperturbed wells), it was expected that the energy spectrum would be real, in a form of doublets due to the splitting effect that takes place. However, the result was a pair of complex poles, which had a clearly positive imaginary part for each member. The present work explains the role of the imaginary parts of the complex poles as the decay rate of quantities defined as the energy indeterminacies, which are directly related to the fact that energy is not well determined in a classically forbidden region of motion. These quantities come as a function of (d)/dE, which is the derivative of the classical action inside the potential barrier, with respect to energy. The major contribution comes from the turning points, and then the imaginary parts are responsible, not only for the conservation of energy, but for the correct sign of time as well. In this way, a different approach for the tunneling process is adopted, in which the entry or exit of the particle from the potential barrier takes place inside a neighborhood of the turning point, as though the latter was broadened and fluctuating. The magnitude of the previously mentioned decay rate is equal to /, where is the frequency of the classical oscillations inside one well. In contrast, the inversion frequency is generated by the part of the complex pole that is unrelated to (d)/dE and is much smaller in magnitude than the classical frequency, since it is given as / exp(Ϫ). In this way, the period of the energy fluctuations is much smaller than the internal period of the system produced by the oscillating communication of the two classically allowed regions of motion.

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Section: Introductionmentioning

confidence: 54%