اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدComplex Trajectories and Dynamical Origin of Quantum Probability
149
0
0.0
(
0
)
تأليف
Moncy V. John
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Complex quantum trajectories, which were first obtained from a modified de Broglie-Bohm quantum mechanics, demonstrate that Borns probability axiom in quantum mechanics originates from dynamics itself. We show that a normalisable probability density can be defined for the entire complex plane, though there may be regions where the probability is not locally conserved. Examining this for some simple examples such as the harmonic oscillator, we also find why there is no appreciable complex extended motion in the classical regime.
قيم البحث
اقرأ أيضاً
It is shown that a normalisable probability density can be defined for the entire complex plane in the modified de Broglie-Bohm quantum mechanics, which gives complex quantum trajectories. This work is in continuation of a previous one that defined a
conserved probability for most of the regions in the complex space in terms of a trajectory integral, indicating a dynamical origin of quantum probability. There it was also shown that the quantum trajectories obtained are the same characteristic curves that propagate information about the conserved probability density. Though the probability density we now adopt for those regions left out in the previous work is not conserved locally, the net source of probability for such regions is seen to be zero in the example considered, allowing to make the total probability conserved. The new combined probability density agrees with the Borns probability everywhere on the real line, as required. A major fall out of the present scheme is that it explains why in the classical limit the imaginary parts of trajectories are not observed even indirectly and particles are confined close to the real line.
This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical tr
ajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories are those conceived in a modified de Broglie-Bohm scheme and we note that identical classical and quantum trajectories for coherent states are obtained only in the present approach. The study is extended to Gazeau-Klauder and SUSY quantum mechanics-based coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller (PT) potential by solving for the trajectories numerically. For the coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the PT potential, though classical trajectories are not regained, a periodic motion results as t --> infty.
Complex quantum trajectory approach, which arose from a modified de Broglie-Bohm interpretation of quantum mechanics, has attracted much attention in recent years. The exact complex trajectories for the Eckart potential barrier and the soft potential
step, plotted in a previous work, show that more trajectories link the left and right regions of the barrier, when the energy is increased. In this paper, we evaluate the reflection probability using a new ansatz based on these observations, as the ratio between the total probabilities of reflected and incident trajectories. While doing this, we also put to test the complex-extended probability density previously postulated for these quantum trajectories. The new ansatz is preferred since the evaluation is solely done with the help of the complex-extended probability density along the imaginary direction and the trajectory pattern itself. The calculations are performed for a rectangular potential barrier, symmetric Eckart and Morse barriers, and a soft potential step. The predictions are in perfect agreement with the standard results for potentials such as the rectangular potential barrier. For the other potentials, there is very good agreement with standard results, but it is exact only for low and high energies. For moderate energies, there are slight deviations. These deviations result from the periodicity of the trajectory pattern along the imaginary axis and have a maximum value only as much as $0.1 %$ of the standard value. Measurement of such deviation shall provide an opportunity to falsify the ansatz.
Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in
the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without loosing coherence, as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.
Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framewo
rk. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
