Advanced embedding details, examples, and help! Current analytical satellite vulnerability planning in the U. Since their first operational use, both propagators have incorporated updated theory and mathematical techniques to model additional forces in the space environment, causing their calculation methods to diverge over time.
The aggregate effects of these diverging mathematical techniques cause calculation differences for perturbations of an orbit over time, resulting in differences in future predicted positions from PPT3 and SGP4, as well as differences in their accuracy. The atmospheric model within each propagator is determined to be the most effective component of each propagator to test, as the theoretical atmospheric drag calculation methods of PPT3 and SGP4 differ greatly.
PPT3 and SGP4 both perform well within the expected accuracy limits inherent with analytical models, with neither propagator demonstrating an accuracy rate decay that was significantly better or worse than the other. In more precise words, a generalization of Heisenberg uncertainty principle, called generalized uncertainty principle GUP , should be considered at energies of order Planck scale [ 4 — 7 ].
This generalization corresponds to a generalization of wave equation of quantum mechanics. Till now, various wave equations of quantum mechanics, different interactions, and other related mathematical aspects and physical concepts have been considered in this framework [ 8 — 18 ]. In our paper, we are going to combine these two subjects.
In Section 2 , we review the essential concepts of GUP and write the generalized Hamiltonian for free particle. In Section 3 , we obtain the propagator for this system in which the details of calculations are brought. At low energies, that is, energies much smaller than the Planck mass, the second term on the right hand side of 1 vanishes and we recover the well-known Heisenberg uncertainty principle. The GUP of 1 corresponds to the generalized commutation relation where , and.
The limits and correspond to the standard quantum mechanics and extreme quantum gravity, respectively. Equation 2 gives the minimal length in this case as. If the wave function is known at a time , we can explicitly write the wave function at a later time using the propagation relation as [ 20 ] For a small time interval , we have Therefore, the quantum mechanical propagator for small time interval , corresponding to this nonlocal Hamiltonian, can be written as in which the Lagrangian is given by [ 20 ] Therefore, the propagator appears as or with In a more explicit form, the propagator for free particle under minimal length is Now, if we assume , then the one-dimensional free particle propagator is given by In order to obtain free particle propagator for a finite time interval we divide the interval into subintervals of equal length such that.
Now, the propagator of a finite time interval is written as The integral in 17 can be calculated as [ 20 ] Substituting 18 into 17 , the propagator is obtained as where. Replacing and by and , respectively, and using , we obtain the final expression as Now, if we calculate the probability of detecting the particle at a finite region , enclosing final point , from 20 , we get In the limit , the final form of propagator is given by which is the result in ordinary quantum mechanics.
We considered the nonrelativistic free particle propagation problem in an analytical manner in minimal length formalism. We first transformed arising differential equation into a second-order differential equation which included a modified effective potential. We next calculated the propagator. Apart from the application of the study, the work is of pedagogical interest in graduate physics. Ghobakhloo and H. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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