Ph.D. Thesis Work (August 2019- Present)

In this study, we present a new analytical solution for the time-dependent Schrödinger equation for a free particle in one-dimensional case. The solution is derived by doing a non-linear transform to the linear Schrödinger equation and converting it into a Burger-like equation. We obtained an interesting non-stationary wave function where our soliton solution moves in time while maintaining its shape. The new solution is then analyzed for three different cases: a periodic box, a box with hard wall boundary conditions, and a periodic array of Dirac delta potentials. The resulting analytical solutions exhibit several interesting features, including quantized soliton velocity and velocity bands. The analytical soliton solution that has been proposed, in our opinion, makes an important contribution to the study of quantum mechanics, and we believe it will significantly contribute to our understanding of how particles behave in one-dimensional box potentials. Soliton solutions for a quantum particle in one-dimensional boxes.

This paper investigates a fascinating quantum mechanics phenomenon called quantum tunneling, for the case when the particle has a finite probability of approaching the barrier from both sides. This exciting and hitherto unrecognized component of quantum tunnelling, intrinsic to quantum systems, prompts us to redefine the conventional framework of quantum tunneling. A detailed, explicit calculation is given for the Dirac delta function and rectangular potential barrier cases. Our results extend the boundaries of quantum tunneling and encourage future research into the fundamentals of quantum particle behavior.

The Boltzmann formula describes how energy is distributed when a system is in thermal equilibrium. The time it takes for the system to reach thermal equilibrium when it is initially out of equilibrium depends on the system’s size, since thermal equilibrium is reached by a random search through all

potential energy distributions. The result of our research demonstrates that a small energy bias of the order of a few kT can significantly shorten the thermal relaxation time. We also give molecular dynamics simulation results to support our work.

The dynamics of the solvation of an instantaneously created dipole in a polar solvent is a very fundamental area of research. Earlier theoretical studies concluded that the transverse component of polarization contributes significantly to the energy and dynamics of this solvation. As the dynamics related to the transverse component are slower than the dynamics related to the longitudinal component, the overall solvation dynamics are slower. But recent theoretical studies conclude that the transverse component of polarization contributes nothing to the solvation due to the microscopic nature of the solvation of molecular-size solutes, which has been confirmed by computer simulation. Therefore, we propose an interesting model that can explain the recent theoretical and simulation results.

M.Sc. Thesis Work (Jun 2018- June 2019)

Conventionally lasing occurs in a system of a large number of atoms or molecule called gain medium, which is required in a laser to compensate for the resonator losses. It has been found that for lasers a four level gain medium is required. In my thesis work we predict lasing in a system of two two-level quantum dots inside a high-quality cavity. We show cooperative interaction between Qds and cavity leads to unconventional laser oscillations. First of all, we discussed the physics of quantum dots. We further, discussed the different aspects of coherent and incoherent pumping in the system. Lasers having a few emitters as a gain medium are important in the realization of micro-lasers. We used master equation formalism to study the dynamics of the system and bring out important role of phonon interaction.

Short Term Projects (Dec 2017 – Jun 2018)

A new method for solving the non-linear Schrodinger equation: A model for Bose-Einstein Condensation.

Dec. 2017 - Feb. 2018 (Winter Project)

Study of Seeback coefficient and Nernst effect in metallic and semi metallic compounds.

Feb.2018- May 2018

**Under the supervision of Dr. Aniruddha Chakraborty****Soliton solutions for a quantum particle in one-dimensional boxes**In this study, we present a new analytical solution for the time-dependent Schrödinger equation for a free particle in one-dimensional case. The solution is derived by doing a non-linear transform to the linear Schrödinger equation and converting it into a Burger-like equation. We obtained an interesting non-stationary wave function where our soliton solution moves in time while maintaining its shape. The new solution is then analyzed for three different cases: a periodic box, a box with hard wall boundary conditions, and a periodic array of Dirac delta potentials. The resulting analytical solutions exhibit several interesting features, including quantized soliton velocity and velocity bands. The analytical soliton solution that has been proposed, in our opinion, makes an important contribution to the study of quantum mechanics, and we believe it will significantly contribute to our understanding of how particles behave in one-dimensional box potentials. Soliton solutions for a quantum particle in one-dimensional boxes.

**Unraveling Quantum Tunneling for Bidirectional Approaching Waves to the Barrier**This paper investigates a fascinating quantum mechanics phenomenon called quantum tunneling, for the case when the particle has a finite probability of approaching the barrier from both sides. This exciting and hitherto unrecognized component of quantum tunnelling, intrinsic to quantum systems, prompts us to redefine the conventional framework of quantum tunneling. A detailed, explicit calculation is given for the Dirac delta function and rectangular potential barrier cases. Our results extend the boundaries of quantum tunneling and encourage future research into the fundamentals of quantum particle behavior.

**Unveiling the Role of Small Energy Bias in Shortening Thermal Relaxation Time**The Boltzmann formula describes how energy is distributed when a system is in thermal equilibrium. The time it takes for the system to reach thermal equilibrium when it is initially out of equilibrium depends on the system’s size, since thermal equilibrium is reached by a random search through all

potential energy distributions. The result of our research demonstrates that a small energy bias of the order of a few kT can significantly shorten the thermal relaxation time. We also give molecular dynamics simulation results to support our work.

**Understanding the dynamics of dipole solvation.**The dynamics of the solvation of an instantaneously created dipole in a polar solvent is a very fundamental area of research. Earlier theoretical studies concluded that the transverse component of polarization contributes significantly to the energy and dynamics of this solvation. As the dynamics related to the transverse component are slower than the dynamics related to the longitudinal component, the overall solvation dynamics are slower. But recent theoretical studies conclude that the transverse component of polarization contributes nothing to the solvation due to the microscopic nature of the solvation of molecular-size solutes, which has been confirmed by computer simulation. Therefore, we propose an interesting model that can explain the recent theoretical and simulation results.

M.Sc. Thesis Work (Jun 2018- June 2019)

**Under the supervision of Dr. Pradyumna Kumar Pathak****Lasing in two quantum dot system**Conventionally lasing occurs in a system of a large number of atoms or molecule called gain medium, which is required in a laser to compensate for the resonator losses. It has been found that for lasers a four level gain medium is required. In my thesis work we predict lasing in a system of two two-level quantum dots inside a high-quality cavity. We show cooperative interaction between Qds and cavity leads to unconventional laser oscillations. First of all, we discussed the physics of quantum dots. We further, discussed the different aspects of coherent and incoherent pumping in the system. Lasers having a few emitters as a gain medium are important in the realization of micro-lasers. We used master equation formalism to study the dynamics of the system and bring out important role of phonon interaction.

Short Term Projects (Dec 2017 – Jun 2018)

A new method for solving the non-linear Schrodinger equation: A model for Bose-Einstein Condensation.

Dec. 2017 - Feb. 2018 (Winter Project)

**Under the supervision of Dr. Aniruddha Chakraborty**Study of Seeback coefficient and Nernst effect in metallic and semi metallic compounds.

Feb.2018- May 2018

**Under the supervision of Dr. C.S. Yadav**