Swarnava Mitra

A manifestation of the Pauli Exclusion Principle is observed when fermions are trapped in the ground state of a 2D harmonic oscillator trap at very low temperatures. This non-interaction of fermions results in the formation of Pauli crystals. This work introduces a statistical mechanical interpretation of the principles that led to the observation of Pauli crystals by calculating the energy, radius and other parameters of the non-interacting arrangement of ultra-cold fermions. This model approaches the problem from two different directions, namely fermion degeneracy and harmonic oscillator treatment. Unifying the two different approaches gives a more comprehensive and robust description of the various parameters of Pauli crystals.

There are three commonly considered processes of heat transfer, namely, conduction, convection and radiation. But the cooling of a hot beverage does not take place through any of the above processes-it is mostly through the process of vaporization. We have estimated the order of magnitude of the heat lost for different processes and compared them to arrive at the above conclusion.

We will try to numerically solve the unidimensional time-independent Schrödinger equation for constant potentials using Numerov's method. The standard approach is to find the energy eigenvalues and infer the wave functions thereon. In contrast to that, we will try to find the generalized wave functions and then infer the energy eigenvalues. We start with a constant potential in infinite space, find the general solution then particularise the wave function to a potential well.

A manifestation of the Pauli Exclusion Principle is observed when fermions are trapped in the ground state of a 2D harmonic oscillator trap at very low temperatures. This non-interaction of fermions results in the formation of Pauli crystals. This work introduces a statistical mechanical interpretation of the principles that led to the observation of Pauli crystals by calculating the energy, radius and other parameters of the non-interacting arrangement of ultra-cold fermions. This model approaches the problem from two different directions, namely fermion degeneracy and harmonic oscillator treatment. Unifying the two different approaches gives a more comprehensive and robust description of the various parameters of Pauli crystals.

We will try to numerically solve the unidimensional time-independent Schrödinger equation for constant potentials using Numerov’s method. The standard approach is to find the energy eigenvalues and infer the wave functions thereon. In contrast to that, we will try to find the generalized wave functions and then infer the energy eigenvalues. We start with a constant potential in infinite space, find the general solution then particularise the wave function to a potential well.

There are three commonly considered processes of heat transfer, namely, conduction, convection and radiation. But the cooling of a hot beverage does not take place through any of the above processes-it is mostly through the process of vaporization. We have estimated the order of magnitude of the heat lost for different processes and compared them to arrive at the above conclusion.

This experiment is developed with the aim of designing a temperature-controlled sample holder by using a commonly available power transistor as the heating element. Most temperature-controlled sample holders use commonplace heaters, which are made of high resistance materials like nichrome 80/20 (80% nickel, 20% chromium) wire and similar materials. The fabrication of this temperature-controlled sample holder also leads to the usage of high power electronic components, like power transistors and power resistors which are, otherwise, neglected in most laboratory experiments. Moreover, to develop this system, Arduino Uno Rev3 and resistance temperature detector (RTD) were used for the purposes of data acquisition and temperature measurement, respectively. Arduino is a single board micro-controller and RTD functions as a temperature sensor. This experiment serves as a good example of application and unification of basic concepts of electronics, heat and thermodynamics and offers an insight into data acquisition. The experiment is non-proprietary, and the apparatus is entirely made from off-the-shelf items. Thus, reconstruction and use will be simple and inexpensive. The power transistor, along with the power resistors, generates enough heat to raise the temperature of the sample holder by about 100 K. Also, to exhibit the working of the sample-holder, the energy band gap of the material of a p–n junction diode (silicon) has been determined experimentally using the setup.