Abstract We demonstrate that a certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the problem of a particle in a box. Twofold calculation of the mean value of energy for the polynomial wave function inside the well yields even argument p (p > 2) of Riemann zeta and related functions. This method can be applied to a wide class of exactly solvable quantum mechanical problems which may lead to different infinite sums. Besides, the analysis performed here provides deeper understanding of superposition principle and presents useful exercise for physics students.
Abstract We demonstrate that a certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the problem of a particle in a box. Twofold calculation of the mean value of energy for the polynomial wave function inside the well yields even argument p (p > 2) of Riemann zeta and related functions. This method can be applied to a wide class of exactly solvable quantum mechanical problems which may lead to different infinite sums. Besides, the analysis performed here provides deeper understanding of superposition principle and presents useful exercise for physics students.
Quantum particle bound in an infinite, one-dimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed energy spectrum for an arbitrary wave function often involves Riemann zeta function resulting in a π series. In this work, two "π formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, 8π2=∑k=1k=∞(2k−1)21, and the integral ∫−∞∞x2sin2xdx=π. The spectrum of the momentum operator appears to peak on classically allowed momentum values only for the states with even quantum number. The present article is inspired by another quantum mechanical derivation of π formula in \cite{wallys}.
Abstract Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.
A free particle is shown to possess any non-negative energy value. Its position is described by a localized wave function called a wave packet. The solution to the particle in infinite potential well problem displays how the concept of energy quantization appears in quantum mechanics. Like the particle in an infinite well, the particle in a finite well shows discretization of energy. But unlike the particle in an infinite well, the particle in a finite well exhibits a non-zero probability of being found outside the well leading to the concept of quantum-mechanical tunneling.
This solves quantum physics problems using the integral Rohit transform, including the scattering of low energy particles by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. The standard calculus approach is typically used to solve these quantum mechanics issues. To solve these quantum physics challenges, this research presents a new method: the integral Rohit transform. When compared to other approaches that are currently in the literature, the obtained solutions show how accurate the suggested method is. This demonstrates the potential and efficacy of the suggested approach to overcoming quantum mechanical issues, such as low-energy particle scattering by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. In this paper, the successful application of the integral Rohit Transform has been demonstrated in solving the one-dimensional time-independent Schrodinger's equation. This application has yielded results that include the determination of eigen energy values and eigen functions for a particle confined within an infinitely high potential well, as well as the calculation of the total scattering cross-section for low energy particles interacting with a perfectly rigid sphere. In the case of low energy limit, the total scattering cross-section for low energy particles due to a perfectly rigid sphere, as determined through quantum mechanics, is equivalent to the geometrical cross-section of said sphere. Additionally, the energy values that the particle can possess within a 1d infinitely high potential well demonstrate that the energy of said particle, when confined within this potential well, is quantized.
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.
If the valley encountered by the particle is infinitely deep, then the particle must be trapped inside. The best analogy is an infinitely deep well. Therefore, this potential is called the infinite square well , or the particle in a box (PIB). The TISE for the PIB is very easy to solve, again using the physical principles from Chapter 6 . The solutions are sine waves with an integral or half-integral number of wavelengths fitting the width of the well. This fact that only certain wavelengths are allowed is a perfect example of quantization of a trapped particle. From the allowed wavelengths, we can also derive the quantized momentum and kinetic energy of the particles. Since momentum is a vector quantity, care must be taken to consider the direction in which the particle might be moving. With this caveat, these simple sine wave solutions are ideal for illustrating principles and procedures such as eigenfunctions, normalization, orthogonality, orthonormality, and expectation values.