Abstract Modeling the hydrogen atom is often considered an essential milestone in an introductory quantum mechanics course. Mathematically, the instructional path is clear: from the square potential well, through the harmonic potential, the three-dimensional box, and the spherical well potential prior to the 1/r Coulomb potential. However, there are few experimental examples to help visualize and ground the key physical concepts. Here, we present an experiment where the light scattering of an evaporating, optically levitating water droplet can be used as an analogy to a quantum spherical well potential. We have distilled two important concepts in this analogy: angular momentum and tunneling. This approach is akin to the analogy between a square well potential and a Fabry–Pérot cavity. It can be used as a visual example on the path toward the hydrogen atom in undergraduate-level quantum mechanics courses.
Abstract Modeling the hydrogen atom is often considered an essential milestone in an introductory quantum mechanics course. Mathematically, the instructional path is clear: from the square potential well, through the harmonic potential, the three-dimensional box, and the spherical well potential prior to the 1/r Coulomb potential. However, there are few experimental examples to help visualize and ground the key physical concepts. Here, we present an experiment where the light scattering of an evaporating, optically levitating water droplet can be used as an analogy to a quantum spherical well potential. We have distilled two important concepts in this analogy: angular momentum and tunneling. This approach is akin to the analogy between a square well potential and a Fabry–Pérot cavity. It can be used as a visual example on the path toward the hydrogen atom in undergraduate-level quantum mechanics courses.
Common atomic physics courses jump from the square and harmonic well potentials straight to the hydrogen atom. However, there is a missing link in between, the spherical well potential. Although it is included in some textbooks, the lack of an experimental backing means the problem quickly becomes mathematically complex. Here we have built an optical toy atom using the scattering of an optically levitated, evaporating water droplet. We find a greatly simplified Mie scattering spectrum composed of a series of evolving Fano resonances organized in a set of combs. The whole spectrum can be intuitively explained through an analogy to a quantum spherical well potential. This produces a model of an atom including ground and excited states, quantized angular momentum, and tunneling.
Abstract In this chapter, the mathematical machinery of quantum mechanics is further developed in order to address real-world 3-dimensional physics. 3-dimensional vector notation is used for quantum mechanical operators and the Schrödinger equation is presented in this notation. The density of states of a particle in a box is considered. The angular momentum operators are defined. The eigenfunctions of the Laplacian are found. The Schrödinger equation with a spherical potential is analysed and solved for a Coulomb potential. The spectroscopy of the hydrogen atom is discussed. The spin operators are introduced. The Stern–Gerlach experiment and the Zeeman effect are discussed. The quantum mechanics of identical particles is considered and fundamental particles are shown to behave as either bosons or fermions depending on their spin. The action and the Feynman path integral are shown to offer an alternative approach to quantum mechanics that elucidates the connection between quantum and classical physics.
Our work utilizes the quantum model of the hydrogen atom which is based on the Schrödinger equation with Coulomb potential. Specifically, we concentrate on the angular components of the wave eigenfunctions derived from this model. We consider the quantum states with n ≤ 4. In order to visualize the orbital shapes of these states, we built in the spherical coordinates system their 3D geometric representations. Furthermore, we use the corresponding spherical harmonics, to calculate the θ nodal values that describe the configurations of these orbital states.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
We have developed an app for iOS-based smart-phones/tablets that allows a 3-D, complex phase-based colorful visualization of hydrogen atom wave functions. Several important features of the quantum behavior of atomic orbitals can easily be made evident, thus making this app a useful companion in introductory modern physics classes. There are many reasons why quantum mechanical systems and phenomena are difficult both to teach and deeply understand. They are described by equations that are generally hard to visualize, and they often oppose the so-called “common sense” based on the human perception of the world, which is built on mental images such as locality and causality. Moreover students cannot have direct experience of those systems and solutions, and generally do not even have the possibility to refer to pictures, videos, or experiments to fill this gap. Teachers often encounter quite serious troubles in finding out a sensible way to speak about the wonders of quantum physics at the high school level, where complex formalisms are not accessible at all. One should however consider that this is quite a common issue in physics and, more generally, in science education. There are plenty of natural phenomena whose models (not only at microscopic and atomic levels) are of difficult, if not impossible, visualization. Just think of certain kinds of waves, fields of forces, velocities, energy, angular momentum, and so on. One should also notice that physical reality is not the same as the images we make of it. Pictures (formal, abstract ones, as well as artists' views) are a convenient bridge between these two aspects.