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(Q-7)   Wave Mechanics


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Optional: Quantum Physics

Q1.Quantum Physics

Q2. Atoms

Q3. Energy Levels

Q4. Radiation from
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Q5.The Atomic Nucleus
        and Bohr's Model

Q6. Expansion of
        Bohr's Model

Q7.Wave Mechanics

Q8. Tunneling

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S-7.The Sun's Energy

S-7A. The Black Hole at
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LS-7A. Discovery
      of Atoms and Nuclei

S-8.Nuclear Power

S-9.Nuclear Weapons

    A new and successful approach to quantum mechanics started in 1925-7 and brought a complete revision of our notion of what a "particle" was. No, the atom was completely different from a miniature planetary system--the "planetary model" was at most a convenient visualization analogy. The first glimmers came through the work of Louis De Broglie in France and Werner Heisenberg in Germany, but the basic structure is mainly due to the Austrian Erwin Schrödinger (Schroedinger), with important contributions by Paul A.M. Dirac (England), Max Born (Germany), Wolfgang Pauli (Switzerland), Niels Bohr, Heisenberg and many others.

Einstein's fundamental relation

E = hν

seemed to imply a dual nature of electromagnetic radiation. As long as it was "in transit" it spread as a wave--was reflected, focused and reconfigured like one. Like a wave, it filled all space, although the probability of it "materializing" anywhere varied greatly with location--e.g., much higher in an illuminated region than in the shade.

    However, the moment the wave revealed itself by depositing its energy, it became sharply localized, maybe at the location of a single atom, and the amount of energy it transmitted, the photon, was fixed by the wave's frequency, according to Einstein's relation. Up to 1925, particles with mass--electrons, protons, atoms and molecules--seemed quite different. They were always localized, existing just around a definite point in space.

    The frustrating search for a theory of atomic behavior led to the idea that perhaps particles with mass behaved in some way like photons, too. Namely, they only acquired localized existence (so to speak) when some event caused them to exchange energy with the surrounding world, a process which revealed their presence. Until then there was no way to assign a definite position to the particle--the most one could do was to assign a higher probability to finding it in some given region.

    Calculations based on that idea indeed led to a theory consistent with observations. They suggested that the electron in a hydrogen atom should be properly viewed, not as a pin-point particle in a Keplerian orbit--that was, at most, an analogy to help one's intuition--but as a wave tied to the atom, confined to its vicinity, just as a sound wave inside a flute is confined to a finite column of oscillating air. A flute (or a tuned string) sounds certain notes and no others, because those frequencies (and not others) can resonate with its structure and build up there. Similarly, the wave representing an electron in hydrogen can stably oscillate in certain modes and not in others, and the calculation of those modes turned out to yield the same energy levels as the Bohr atom did. Unlike the Bohr-Sommerfeld atom, these calculations could be extended to more complex systems and to additional phenomena, e.g. the scattering of electrons or atoms following collisions. In any such process, until the electron actually underwent a transition, it gave no way of knowing "where it was" and at what energy.

    That was the basic idea of what was called "wave mechanics"--meaning, not the mechanics of waves, but a re-formulation, in terms of waves, of the branch of physics known as mechanics, which deals with motions of matter. Newtonian mechanics treats matter strictly as localized particles, or of bodies and fluids consisting of such particles. Wave mechanics asserts that when one gets down to the atomic level, particles sometimes need to be treated as waves, spread out in space, their location and momentum not known until they interact. Even then, as Heisenberg showed in his uncertainty principle, one can never extract full information.

Particles and Waves

What sort of strange wave was that? It may help if we first examine electromagnetic waves, and then compare.

    The concept of an electromagnetic wave evolved in steps. First came measurements of electric and magnetic forces (e.g. section 5 in "A Millennium of Geomagnetism") which led to the concept of constant electric and magnetic fields. A magnetic field originally was a region of space where a magnetic force could be detected, if a tiny magnet were placed there, and an electric field was where an electric charge could detect a local electric force. But in the absence of such probes, we have in principle no indication that a field exists--all we would see was empty space. At this stage there seemed to exist no compelling reason to claim such space was itself modified, even though Michael Faraday may have felt it was.

                    (1) Light as a Wave

    Maxwell took the next step by showing that oscillating linked electric and magnetic fields, propagated as a wave and could account for the properties of visible light and related radiations. "Propagating as a wave" here implies a certain mathematical representation, of oscillating and (in general) spreading electric and magnetic fields. Other waves in nature are usually disturbances spreading in some medium--e.g. sound in air is a disturbance in the ambient pressure. Earthquake waves inside the Earth also can propagate the way sound does, but a second separate mode of "transverse waves" exists there too, where the material shakes sideways, perpendicular to the direction of propagation, like the jiggling of gelatin jelly in a mold.

    Maxwell's electromagnetic waves were also "transverse" modes, with electric and magnetic forces perpendicular to the direction of spreading. But waves in what? What was the material in which they spread? For a while, physicists believed in the existence of an invisible all-pervading "aether," and electromagnetic waves were waves in the aether, which jiggled like a jelly.

    But the properties of this "aether" seemed strange. Sound spreading in air registers differently whether the observer approaches the source or recedes from it ("Doppler effect"), and so does light--that is how we tell distant galaxies are receding. However, if we take a beam of light in the lab and split it into two components ("Michelson-Morley experiment"), one parallel to the motion of the Earth around the Sun (at 1/10,000 the speed of light) and one perpendicular, no difference can be detected. If the universe is filled with "aether" in which light propagates, the motion of the Earth relative to it turned out to be umobservable. We therefore now regard those fields as properties of space, to which no motion can be ascribed, and the electromagnetic waves propagate "in space."

                    (2) Light as a Stream of Particles

    Light can also resemble a stream of particles, on two completely different levels. In elementary optics classes one often talks about rays of light, following straight lines like a stream of bullets. That was also how early research in optics viewed light. Later studies demonstrated that on small scales, of the order of a wavelength, light displayed a wave nature, e.g. in the dark and light lines on the boundary of a narrow slit (or on the boundaries of "floaters" often seen in one's own eye when it is half closed). Mathematics of wave motion showed that rays and waves were not contradictory: the behavior of light could be described by rays, if the scale of the viewed object was much larger than a wavelength.

    But on a scale of atoms light again behaved in a particle-like fashion. Einstein's relation

E = hν

showed that when light of frequency ν gave up or received energy, it did so in definite portions, in "photons" which could also be viewed as particles, since their action was localized. Like a material particle, the photon also has momentum, and the momentum of such photons makes solar sails possible. But unlike the particle aspect of ray motion, here the "particles of light" were linked to a definite constant of nature, Planck's constant h.

    While light is in transit, it is a wave, and its strength at any point gives us the likelihood, the probability, that a photon may be detected there. This analogy was also used by Schrödinger when he proposed to represent solid matter by a wave. The wave itself satisfies "Schrödinger's equation", a wave equation, and the value of the wave variable--usually represented by the Greek letter ψ (psi)--gives the probability of the particle will be observed there (Actually, the probability is given by the square of ψ, but that is similar to an electromagnetic wave, where the square of its amplitude determines the energy density).

    It is a strange kind of wave, indeed. For physicists whose intuition was trained on waves which spread energy in space, here was a wave which spread not energy but probability, the probability of finding the particle which it represented. It told them that science could no longer predict where (say) an electron may be found, the best it could do is give the probability of finding it at one spot or another.

    Larger objects also have their wave representation, but since the wavelength is very small, their behavior can be represented by ray motion, letting them (to a very good approximation) move in straight lines and satisfy Newton's equation.


    Only when we get down to atomic dimensions--as defined by Planck's constant h--does the wave nature dominate the behavior. In a hydrogen atom, for instance, the wave then can be stable only in certain resonant states--"eigenstates," a word cobbled together from German and English, with "eigenvalues" giving the energy levels. The atom is then like some musical instrument tuned to certain notes--e.g. it can sound a C or a D, but never an in-between false note. Until its note sounds, we only know the likelihood of it being this state or that.

    This is quite different from the Bohr-Sommerfeld atom where (at least initially) Kepler orbits served as a model to the motion of electrons. Those Kepler orbits were classified by total energy and angular momentum (corresponding to ellipticity), but they were all flat, two-dimensional. Even now, in the popular literature, atoms are often drawn as miniature planetary systems: but that is not the correct picture.

    In contrast, wave functions are 3 dimensional, spread over space. Yet their basic modes can also be classified, and interestingly, the scheme of modes turns out similar to the one based on Kepler motion, although the underlying concepts are quite different. (The mathematical tools are somewhat related to the ones used by Gauss to extract the main modes of the Earth's magnetic field--tools taken, in turn, from modes deduced for the Earth's gravity field). Nowadays the basic modes of atomic (or molecular) wave functions are often known as orbitals, patterns of regions in which the wave function is concentrated. The lowest orbitals are symmetric and spherical, but the more complex ones have multiple lobes, a bit like 3-dimensional cloverleaf patterns with various numbers of lobes.

    The basic classification is still by energy--n=1 levels, then n=2 levels, etc. In additions, the levels of asymmetry are denoted by values (0,1,2,3,...) of (lower case) L (representing angular momentum, related to asymmetry) or, traditionally, by letters (s,p, d, f....), as shown on the energy-level diagram in the preceding section. The "s" mode is symmetric, "p" has 2-lobed symmetry, then "d" , then "f", etc. A third number m exists, and visual descriptions of the peaks of the modes can be found here; of course, these are just the surfaces where the wave function is largest, from which it tapers down to the rest of space.

    Orbitals are important not only when the atom jumps from a high level to a lower (unoccupied) one, but also, in an atom at its lowest level ("ground state") it turns out electrons must occupy different orbitals (except that electrons with opposed spin may double up). That idea has led to an explanation of the periodic table of chemical elements. Orbitals can also be deduced for molecules, which have more complex spectra, usually in the infra-red.

Later Developments

    The above are just the basics, a preliminary reconnaissance of a new terrain. For detail and applications one must study quantum theory systematically--with all its math, its manipulations of angular momentum and spin and so forth, followed perhaps by scattering theory, Dirac's theory of the electron, quantum electrodynamics and more.

    The mathematical treatment has led to a fairly good understanding of atomic spectra, including their intensity variations--although solving for energy levels of complicated atoms requires tedious approximations. (It is the same in celestial mechanics--the tracking of a single planet around its sun is simple, but tracking multiple objects interacting with each other is hard). Quantum theory also led to an understanding of molecular spectra, of the chemical bond, of the periodic table of elements (see above), of the behavior of atoms arranged in crystals (including semiconductors, which made computers practical), of electric "superconductivity" in very cold materials, of magnetic effects on which both medical imaging (MRI) and modern magnetometry depend, of lasers and a lot more.

Questions from Users:   What keeps Electrons away from the Atomic Nucleus?
     ***       Double-slit diffraction of particles

Next Stop: (Q-8)   Quantum Tunneling

Or else, return to section #6 on physics related to the Sun: (S-6) Seeing the Sun in a New Light

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Author and Curator:   Dr. David P. Stern
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Updated: 13 February 2005     Re-formatted 27 March 2006     Edited 18 October 2016