Wave mechanics of a particle in 1-D box (size
=d) is critically analyzed to reveal its untouched aspects. When the particle rests in its ground state, its zero-point force (
Fo) produces non-zero strain by modifying the box size from
d to
d′=d+Δd in all practical situations where the force (
Fa) restoring
d is not infinitely strong. Assuming that
Fa originates from a potential
∝x2 (
x being a small change in
d), we find that: (i) the particle and strained box assume a mutually bound state (under the equilibrium between
Fo and
Fa) with binding energy
ΔE=−εo′Δd/d′ (with
εo′=h2/8md′2 being the ground state energy of the particle in the strained box), (ii) the box size oscillates around
d′ when the said equilibrium is disturbed, (iii) an exchange of energy between the particle and the strained box occurs during such oscillations, and (iv) the particle, having collisional motion in its excited states, assumes collisionless motion in its ground state. These aspects have desired experimental support and proven relevance for understanding the physics of widely different systems such as quantum dots, quantum wires, trapped single particle/ion, clusters of particles, superconductors, superfluids, {\it etc.} It is emphasized that the physics of such a system in its low energy states can be truly revealed if the theory incorporates
Fo and related aspects.
This work is about the properties of several low dimensional, small systems of interacting particles. We demonstrate that interaction between particles in the low dimensional small systems can lead to many unexpected effects. We considered electrons in a Luttinger liquid, in a superconducting state, and atoms in a magneto-optical trap. Using bosonization techniques we calculated the thermopower of a Luttinger liquid wire with an impurity. We predicted the appearance of a phase dependent force and resonant phase dependent magnetization in the nanoscopic superconductor - normal metal superconductor (or superconductor - two dimensional electron gas - superconductor) junction. We also considered plasma oscillations inside thin superconducting tubes and rings and predicted that the velocities of the plasmons in these systems are periodic functions of the magnetic flux. By considering neutral atoms in a harmonic trap we discovered that strongly repelling atoms do not form Bose-Einstein condensate at zero temperature but tend to occupy different orbitals with small mutual overlap, forming crystallite structures similar to Wigner molecules of electrons inside a quantum dot.
Trajectories of a Bohmian particle confined in time-dependent cylindrical and spherical traps are computed for both contracting and expanding boxes. Quantum effective force is considered in arbitrary directions. It is seen that in contrast to the problem of a particle in an infinite rectangular box with one wall in motion, if particle initially is in an energy eigenstate of a tiny box the force is zero in all directions. Trajectories of a two-body system confined in the spherical trap are also computed for different statistics. Results of average separation of the particles confirm our expectation about the statistics.
Abstract The problem of a particle moving in an infinite well or one-dimensional box, defined by the potentialis one of the most familiar and easiest to solve of all problems in introductory quantum mechanics. It is the simplest case in which to study the phenomenon of quantized energy levels in bound states, one of the “smoking guns “ of wave mech- anics. In this chapter and the next, we use this problem (and some related ones) as a tool in understanding the formalism used in solving the time-independent Schrödinger equation for bound states in a potential, understanding their phys- ical interpretation, as well as some of the formal properties of its solutions. In Chapter 7, we also use the infinite well as a model system in which to examine the role that spin and the Pauli exclusion principle play in multiparticle quantum systems.
In this paper we have solved analytically the Schrodinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square of the linear momentum are equal to the classical time average of particle position and its square of the linear momentum, respectively.
One of the classic first problems in quantum theory textbooks is that of the particle trapped in a box. Despite the apparent artificiality of the situation, it has quite a few interesting features that we can quickly apply to some more physically plausible situations. The particle in a box is literally an object, such as an electron, trapped inside a ‘box.’ In a realistic physical example, this ‘box’ would be a volume of space bound by some constraining force acting on the particle. For example, an atom is a ‘box’ as far as the orbiting electrons are concerned, with the electrostatic force between the positive protons in the nucleus and the negative electrons providing the constraint. The first move is to assume that the wave function for any particle in the box will have separate spatial and temporal dependencies. One of the key steps in obtaining our wave function was the quantization of the momentum.