Investigation is made of the one-dimensional compressional vibrations of an ideal fluid o? variable density (in equilibrium under the action of an external force) confined in an infinite tube of bounded but variable cross-section. The non-homogeneity, is so chosen that the wave velocity is everywhere the same. It is assumed that the ordinary acoustic equations are applicable except as modified by the non-homogeneity and that the cross-section is small enough so that transverse motions may be neglected. It is then found that the problem of determining the stationary wave-pattern in such a tube satisfying the boundary condition that the condensation s=δρ/ρ 0 (where ρ 0 is the equilibrium density at any point and δρ the excess density) shall remain analytic, everywhere ≪ 1 and vanish at infinity, is formally equivalent to the problem of solving a one-dimensional Schrödinger wave equation subject to the usual boundary condition. The appropriate density variation and the shape of the tube are related to the potential energy of the particle whose motion is described by the Schrödinger equation. As illustrations the free particle, the simple harmonic oscillator, and the one-dimensional hydrogen atom are investigated. The motion of the first corresponds to the oscillations of a homogeneous fluid in an infinite tube of uniform cross-section. For the simple harmonic oscillator the corresponding tube is a surface of revolution obtained by revolving the curve representing the error function about the x axis. A method of generalizing the results to the case of three degrees of freedom is indicated.
Investigation is made of the one-dimensional compressional vibrations of an ideal fluid o? variable density (in equilibrium under the action of an external force) confined in an infinite tube of bounded but variable cross-section. The non-homogeneity, is so chosen that the wave velocity is everywhere the same. It is assumed that the ordinary acoustic equations are applicable except as modified by the non-homogeneity and that the cross-section is small enough so that transverse motions may be neglected. It is then found that the problem of determining the stationary wave-pattern in such a tube satisfying the boundary condition that the condensation s=δρ/ρ 0 (where ρ 0 is the equilibrium density at any point and δρ the excess density) shall remain analytic, everywhere ≪ 1 and vanish at infinity, is formally equivalent to the problem of solving a one-dimensional Schrödinger wave equation subject to the usual boundary condition. The appropriate density variation and the shape of the tube are related to the potential energy of the particle whose motion is described by the Schrödinger equation. As illustrations the free particle, the simple harmonic oscillator, and the one-dimensional hydrogen atom are investigated. The motion of the first corresponds to the oscillations of a homogeneous fluid in an infinite tube of uniform cross-section. For the simple harmonic oscillator the corresponding tube is a surface of revolution obtained by revolving the curve representing the error function about the x axis. A method of generalizing the results to the case of three degrees of freedom is indicated.
A general formulation of the eigenvalue condition is pointed out for a one-dimensional stationary part of a separable Schrodinger equation with finite bounds, i.e. where the wavefunction vanishes at two arbitrary values of the independent variable. The procedure is general and applies equally to other eigenvalue problems of second-order, homogeneous differential equations. Where appropriate a continuous density of eigenvalues can be defined and evaluated without first solving the eigenvalue problem. Illustrative examples are provided, including gravitational and harmonic oscillator potentials.
It is demonstrated that the probability density function, given by the square of a quantum mechanical wavefunction that is a real-valued eigenvector of a time-independent, one-body Schroedinger equation, satisfies a compressible-flow generalization of the Bernoulli equation, where the mass density is the probability density times the mass of the system; the pressure and velocity fields are defined by functions depending on the probability density, and the gradient and the Laplacian of the probability density, where there are two possible directions of the velocity on a streamline. The velocity given definition implies a generalization of the steady-flow continuity equation where mass is not locally conserved. The gradient of the Bernoullian equation is demonstrated to be equivalent to the steady flow Euler equation for variable mass and irrotational flow. A speed of sound quadratic equation is obtained from a spherical wave-pulse. One of the solutions indicates that the wave-pulse velocity on a streamline is equal in magnitude but opposite in direction of the fluid velocity on the streamline. The other solution is the focus of attention from that point on. It is proven that the extremums of the momentum per volume on a streamline occur at points that are Mach 1 speed. The developed formalism is applied to a particle in a one-dimensional box, the ground and first excited-states of the one-dimensional harmonic oscillator, and the hydrogen 1s and 2s states. Some behavior is repeated in all the applications examined. For example, an antinode, a point of local-maximum density, has zero velocity and zero Mach speed on the streamline, while a node, a point of minimum density, has infinite velocity and Mach 2. In between the node and antinode is an extremum of the momentum, and Mach 1. (This is a short version of the abstract.)
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
We address the paradoxical transformation of a classical-mechanical particle motion when the space and time scales of observation pass below the uncertainty principle threshold. This is analyzed in the language of classical statistical mechanics, considering specifically many-particle systems inhomogeneous along one spatial direction. We employ the density functional formalism in its square-gradient form and find: (i) The macroscopic solution is analogous to the classical trajectory of a particle under a potential of force given by (minus) the free energy density. Whereas, (ii) fluctuations around the solution in (i) are equal to the quantum-mechanical wave functions of a particle under a potential given by the curvature of the free energy density. We illustrate this situation with three textbook examples: A particle in a box, the harmonic oscillator, and the hydrogen atom. We show that their time-independent Schrödinger equation wave functions describe, respectively, the fluctuations of a fluid interface, of critical point fluctuations, and of a confined ideal gas. At large scales, sharp probability distributions make fluctuations irrelevant; the vanishing of the first variation yields the macroscopically observable statistical-mechanical non-uniformity, equivalent to the classical particle trajectory. But at sufficiently small scales, with necessarily very few particles, distributions appear much wider, fluctuations dominate, and one obtains the Schrödinger equation (for the microscopic potential).