In this article we have developed a formalism to obtain the Schrödinger equation for a particle in a frame undergoing an uniform acceleration in an otherwise flat Minkowski space-time geometry. We have presented an exact solution of the equation and obtained the eigenfunctions and the corresponding eigenvalues. It has been observed that the Schrödinger equation can be reduced to an one dimensional hydrogen atom problem. Whereas, the quantized energy levels are exactly identical with that of an one dimensional quantum harmonic oscillator. Hence considering transitions, we have predicted the existence of a new kind of quanta, which will either be emitted or absorbed if the particles get excited or de-excited respectively.

In the conventional scenario, the Hawking radiation is believed to be a tunneling process at the event horizon of the black hole. In the quantum field theoretic approach the Schwinger's mechanism is generally used to give an explanation of this tunneling process. It is the decay of quantum vacuum into particle anti-particle pairs near the black hole surface. However, in a reference frame undergoing a uniform accelerated motion in an otherwise flat Minkowski space-time geometry, in the non-relativistic approximation, the particle production near the event horizon of a black hole may be treated as a kind of Fowler-Nordheim field emission, which is the typical electron emission process from a metal surface under the action of an external electrostatic field. This type of emission from metal surface is allowed even at extremely low temperature. It has been noticed that in one-dimensional scenario, the Schrödinger equation satisfied by the created particle (anti-particle) near the event horizon, can be reduced to a differential form which is exactly identical with that obeyed by an electron immediately after the emission from the metal surface under the action of a strong electrostatic field. The mechanism of particle production near the event horizon of a black hole is therefore identified with Schwinger process in relativistic quantum field theory, whereas in the non-relativistic scenario it may be interpreted as Fowler-Nordheim emission process, when observed from a uniformly accelerated frame.

In this article we have solved an hypothetical problem related to the stability and gross properties of two dimensional self-gravitating stellar objects using Thomas-Fermi model. The formalism presented here is an extension of the standard three-dimensional problem discussed in the book on statistical physics, Part-I by Landau and Lifshitz. Further, the formalism presented in this article may be considered as class problem for post-graduate level students of physics or may be assigned as a part of their dissertation project.

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The Saha equations for photo-ionization process of hydrogen atoms and the creation of electron-positron pairs at high temperature are investigated in a reference frame undergoing a uniform accelerated motion in an otherwise flat Minkowski space-time geometry or equivalently, in a rest frame in presence of a strong uniform gravitational field. It is known as the Rindler space.