## Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)

### Glossary

**absolute size**—An object is large or small not by comparison to another object, but rather by comparison to the intrinsic minimum disturbance that accompanies a measurement. If the disturbance is negligible, the object is large in an absolute sense. If the intrinsic minimum disturbance is nonnegligible, the object is absolutely small.

**absorption of light**—The process by which the amount light is reduced, and the energy of an object is increased. Light (photons, particles of light) of the proper frequency (color) will cause the quantum state of an object to go to a higher energy state. The increase in energy of the object exactly matches the decrease in energy of the light. Absorption of light by objects is responsible for their color.

**ångström**—A unit of length that is 10^{-10} m (one ten billionth of a meter). The ångström unit has the symbol, Å.

**anion**—A negatively charged atom or molecule, such as Cl^{-}, the chloride anion. An anion is formed by adding one or more negatively charged electrons to a neutral atom or molecule.

**atomic number**—The number of protons (positively charged particles) in an atomic nucleus. A neutral atom (not an ion) has the same number of electrons (negatively charged particles) as protons.

**atomic orbital**—The name given a wavefunction (probability amplitude wave) that describes the probability distribution of an electron about an atomic nucleus.

**black body radiation**—The light given off by a hot object. The colors of the light depend on the temperature of the object. Black body radiation is the first physical phenomenon described with the ideas that became quantum mechanics by Max Planck in 1900.

**Born interpretation**—The description of quantum mechanical wavefunctions as probability amplitude waves. The Born interpretation, also referred to as the Copenhagen interpretation, states that the quantum mechanical wavefunctions obtained from solving the Schrödinger equation describe the probability of finding a particle in some region of space.

**cation**—a positively charged atom or molecule, such as Na^{+}, the sodium cation. A cation ion is formed by removing one or more negatively charged electrons from a neutral atom or molecule.

**classical mechanics**—The theory of matter and light that was developed before the advent of quantum mechanics. It treats size as relative and cannot describe absolutely small particles (electrons, photons, etc.). It is a powerful theory that works extremely well for the description of large objects such as airplanes, the trajectory of a rocket, or bridges.

**classical waves**—Waves, such as water waves and sound waves, that can be described with classical mechanics. Electromagnetic waves, which are classical mechanics’ description of light, are also a type of classical wave. The classical description of light as waves works well for radio and other types of waves, but it cannot properly describe the particle nature of light (photons) responsible for such phenomena as the photoelectric effect.

**closed shell configuration**—An atom has the number of electrons associated with its nucleus that correspond to one of the noble gases, which comprise the right-hand column of the Periodic Table. A closed shell configuration is particularly stable. The noble gases, also called the inert gases, have the closed shell configuration, and are essentially chemically inert. An atom can obtain a closed shell configuration by gaining or losing electrons to become ions or by sharing electrons with another atom in a covalent bond.

**constructive interference**—Waves combine (add together) to make a new wave in a manner that increases the amplitude of the total wave. For waves with different wavelengths, constructive interference will occur only in some regions of space. The wave can be large in some region because of constructive interference but small elsewhere.

**Coulomb interaction**—The interaction between electrically charged particles that gets smaller as the distance increases. The interaction decreases in proportion to the inverse of the distance. The Coulomb interaction causes particles with opposite charges (positive and negative) to attract each other (an electron and a proton), and like charges to repel (two electrons or two protons).

**covalent bond**—A chemical bond that holds atoms together because the atoms share electrons.

**de Broglie wavelength**—The wavelength associated with a particle that has mass. All particles have de Broglie wavelengths. For large particles like baseballs, the de Broglie wavelength is so small that it is negligible. So large particles do not act like waves. For small particles (electrons, etc.), the wavelength is comparable to the size of the particle. Therefore, small particles can exhibit properties that are wavelike.

**destructive interference**—Waves combine (add together) to make a new wave in a manner that decreases the amplitude of the total wave. For waves with different wavelengths, destructive interference will occur only in some regions of space. The wave can be large in some regions because of constructive interference but small elsewhere because of destructive interference.

**Dirac’s assumption**—A minimum disturbance accompanies any measurement. This disturbance is not an artifact of the experimental method, but is intrinsic to nature. No improvement in technique can eliminate it. If the minimum disturbance is negligible, a particle is large in an absolute sense. If the disturbance is nonnegligible, the particle is absolutely small. Dirac’s assumption has been demonstrated by many experiments to be true and is central to quantum theory.

**double bond**—A chemical bond in which two pairs of electrons are shared between two atoms. A double bond is stronger and shorter than a single bond.

**eigenstate**—A pure state of a system associated with a perfectly defined value of an observable called an eigenvalue. A system in an energy eigenstate, such as a hydrogen atom, has a perfectly defined energy. The hydrogen atom has many different energy eigenstates, which have different energies (energy eigenvalues). A system in a momentum eigenstate has a perfectly defined value of the momentum. Each eigenstate has a wavefunction associated with it. Eigenstates are the fundamental states of quantum theory.

**electromagnetic wave**—A wave composed of electric and magnetic fields that oscillate at the same frequency and propagate at the speed of light. Electromagnetic waves are the classical mechanics description of light. The classical theory of electromagnetic waves is useful in describing many aspects of light and radio waves, but it cannot describe many phenomena, such as the photoelectric effect.

**electron**—A subatomic particle that has a negative charge. It is one of the basic constituents of atoms and molecules. Its negative charge is the same size but opposite in sign from the positive charge of a proton. An atom has the same number of electrons and protons, so it has no overall charge. Adding an electron to an atom makes an anion with one unit of negative charge. Removing an electron from an atom makes a cation with one unit of positive charge.

**energy levels**—In atoms, molecules, and other quantum absolutely small systems, energy is not continuous. Energy changes can only occur in discreet steps. Each distinct discreet energy is an energy level.

**excited state**—The state of an atom or molecule that has a higher energy than the minimum. An excited state is created when an atom or molecule that starts in its lowest energy state absorbs a photon of the right frequency to place the system in an energy level above the lowest energy, which is referred to as the ground state. Excited states can also be generated by heat and other mechanisms to put energy into an atom or molecule.

**free particle**—A particle that has no forces acting on it. A moving free particle will go in a straight line because no forces, such as gravity or air resistance, affect its motion.

**frequency**—The number of repetitions of a recurring event per unit of time. For a wave, the frequency is the number of wave peaks that pass by in a given time. For waves traveling at the same speed, a high frequency corresponds to a short wavelength and a low frequency corresponds to a long wavelength. The wavelength is the distance between peaks of a wave. For light waves, the frequency is the speed of light divided by the wavelength.

**ground state**—The lowest energy state of an atom or molecule. An excited state is created when an atom or molecule that starts in its lowest energy state absorbs a photon of the right frequency to place the system in an energy level above the lowest energy, the ground state. Excited states can also be generated by heat and other mechanisms to put energy into an atom or molecule.

**Heisenberg Uncertainty Principle**—The momentum and position of a particle cannot be known exactly simultaneously. If the momentum of a particle is known exactly, then the position is completely uncertain, that is, there can be no information on the position. If the position is known exactly, there can be no information on the magnitude of the momentum. In general, the principle states that the position and momentum can only be known within a certain degree of uncertainty. This is intrinsic to nature and not a consequence of experimental error.

**hybrid atomic orbitals**—Combinations (superpositions) of atomic orbitals that create new atomic orbitals with different shapes. Hybrid atomic orbitals are important in chemical bonding. Hybrid atomic orbitals will be formed to bond atoms together to produce a molecule with the lowest energy (most stable molecule). The shapes of the hybrid orbitals determine the shapes of molecules.

**hydrocarbon**—A molecule composed of only carbon and hydrogen, such as methane (natural gas) and oil.

**inert gases (noble gases).** Atoms such as helium, neon, argon, etc., that have closed electron shell configurations. They occupy the right-hand column of the Periodic Table of Elements. Because of closed shell configurations, they are essentially chemically inert. They do bond to other atoms to form molecules.

**interference of waves**—The combination of two or more waves to give a new wave. The waves can constructively interfere in some region of space to give an increased amplitude (larger wave) and destructively interfere in other regions of space to produce decreased or zero amplitude.

**Joule**—A unit of energy. One Joule (J) is a meter times kilograms squared divided by seconds squared. J = m kg^{2}/s^{2}.

**kinetic energy**—The energy associated with motion. A moving particle has kinetic energy equal to one half times the mass times the velocity squared, as in E_{ke} = 1/2mV^{2}.

**light quanta**—A single particle of light. A phonon.

**lone pair**—A pair of electrons in a molecule that occupies an atomic orbital but does not participate in bonding. Lone pair electrons are not shared between atoms.

**molecular orbital**—A wavefunction for a molecule composed of a combination of atomic orbitals (atomic wavefunctions) that span the molecule. Molecular orbitals can be bonding (bonding MO). Electrons in bonding MOs make the energy of the molecule lower. Molecular orbitals can be antibonding (antibonding MO). Electrons in antibonding MOs increase the energy of a molecule. To have a stable molecule, there must be more electrons in bonding MOs than in antibonding MOs.

**momentum eigenstate**—The state of a particle with perfectly defined momentum. A momentum eigenstate of a free particle, like a photon or electron, has a wavefunction that is delocalized over all space. The momentum can be known exactly but the position is completely uncertain. Momentum eigenstates can be superimposed (added together) to make a wave packet that has a more or less well-defined position.

**nanometer**—A unit of length that is one billionth of a meter, 10^{-9} m.

**node**—For a one-dimensional wave, a point where the amplitude of the wave is zero. For a three-dimensional wave, a node is a plane or other shaped surface where the wave amplitude is zero. The sign of the wavefunction changes when a node is crossed. In quantum mechanics, a node in a wavefunction describing a particle, such as an electron, is a place where the probability of finding the particle is zero.

**optical transition**—The change in state from one energy level to another in an atom or molecule caused by the absorption or emission of light.

**orbital**—Another name for the quantum mechanical wavefunction that describes an electron or pair of electrons in an atom or molecule. An atom has atomic orbitals, and a molecule has molecular orbitals.

**particle in a box**—A quantum mechanical problem in which a particle, such as an electron, is confined to a one-dimensional box with infinitely high and impenetrable walls. The energy levels of a particle in a box are quantized, that is, there are discreet energy levels. The particle in a box is the simplest quantum mechanical problem in which a particle is confined to a small region of space and has quantized energy levels.

**Pauli Exclusion Principle**—The principle that at most two electrons can be in an atomic or molecular orbital. If two electrons are in the same orbital, they must have opposite spins, that is, different electron quantum numbers s (one +1/2 and the other - 1/2). The Pauli Exclusion Principle is important in determining the structure of the Periodic Table of Elements and the properties of atoms and molecules.

**phase**—The position along one cycle of a wave. The peak of the wave (point of maximum positive amplitude) is taken to have a phase of 0 degrees (0°), then the first node (point where the amplitude is zero) is 90°. 90° is a quarter of a cycle of a wave. A phase of 180° is one half of a cycle. It is the point of maximum negative amplitude. Two waves of the same wavelength are said to be phase shifted if the peaks don’t line up.

**photoelectric effect**—The effect explained by Einstein in which a single particle of light, a photon, can eject a single electron from a piece of metal. Einstein’s explanation of the photoelectric effect showed that light is not a wave as described by classical electromagnetic theory.

**photon**—A particle of light.

**Planck’s constant**—The fundamental constant of quantum theory. It is designated by the letter h. It appears in many of the mathematical equations found in quantum mechanics. For example, E = hν says that the energy is the frequency ν (Greek letter nu) multiplied by Planck’s constant. Planck’s constant has the value h = 6.6 × 10^{-34} J s (Joule times seconds). Planck introduced the constant in 1900 in his explanation of black body radiation.

**potential energy well**—A region in space where energy is lowered because of some type of attractive interaction. A hole in the ground is a gravitational potential energy well. A ball will fall to the bottom, lowering its gravitational energy. It will require energy to lift the ball out of the hole. Electrons are held in atoms by a Coulomb potential energy well, that is, by the electrical attraction of negatively charged electrons for the positively charged nucleus. It requires the addition of energy to remove an electron from an atom. Enough energy can raise the electron out of the Coulomb potential energy well created by the attraction to the positively charged nucleus.

**probability amplitude wave**—Quantum mechanical wave (wavefunction) that describes the probability of finding a particle in some region of space. A probability amplitude wave can go positive and negative. The probability of finding a particle in some region of space is related to the square (actually the absolute value squared) of the probability amplitude wave. The greater the probability in some region of space, the more likely the particle will be found there.

**proton**—A subatomic particle that has a positive charge. It is one of the basic constituents of atoms and molecules. Its positive charge is the same size but opposite in sign from the negative charge of an electron. An atom has the same number of electrons and protons, so it has no overall charge. The number of protons in an atomic nucleus, called the atomic number, determines the charge of the nucleus. Different atoms have different numbers of protons in their nuclei.

**quantized energy levels**—Energy levels that come in discreet steps. The energy is not continuous. Atoms and molecules have quantized energy levels.

**quantum number—**A number that defines the state of a quantum mechanical system. There can be more than one quantum number to fully specify the state. In an atom, each electron has four quantum numbers, n, *l*, m, and s, which can only take on certain values. Quantum numbers arise from the mathematical description of quantum mechanical systems.

**radial distribution function**—The mathematical function that describes the probability of finding an electron a certain distance from the nucleus of an atom independent of the direction. It is obtained from the wavefunction for the electron in the atom.

**Rydberg formula**—The early empirical formula that gave the colors of light emitted or absorbed by hydrogen atoms.

**Schrödinger Equation**—A fundamental equation of quantum theory. Solution of the Schrödinger Equation for an atom or molecule gives the quantized energy levels and the wavefunctions that describe the probability amplitude of finding electrons in regions of space in an atom or molecule.

**single bond**—A chemical bond that holds two atoms together through the sharing of one pair of electrons.

**size, absolute**—An object is large or small depending on whether the intrinsic minimum disturbance that accompanies a measurement is negligible or nonnegligible. If the minimum disturbance is negligible, the object is large in an absolute sense. If it is nonnegligible, it is absolutely small. Absolutely small objects can be described by quantum mechanics, but not by classical mechanics.

**size, relative**—Size is determined by comparing one object to another. An object is big or small only in relation to another object. In classical mechanics it is assumed that size is relative. Classical mechanics cannot describe objects that are small in an absolute sense.

**spatial probability distribution**—The probability of finding a particle, such as an electron, in different regions of space. The spatial probability distribution can be calculated from the quantum mechanical wavefunction for a particle.

**spectroscopy**—The experimental measurement of the amount and colors of light that are absorbed or emitted by a system of atoms or molecules.

**Superposition Principle**—“Whenever a system is in one state, it can always be considered to be partly in each of two or more states.” This quantum mechanical principle says that a system in a particular quantum state can also be described by the superposition (addition) of two or more other states. In practice, this generally means that a particular wavefunction can be expressed as the sum of two or more other wavefunctions. For example, the wavefunctions for molecules can be formed by the superposition of atomic wavefunctions. A photon wave packet can be formed by the superposition of momentum eigenstates

**triple bond**—A chemical bond that holds two atoms together by sharing three pairs of electrons. A triple bond is shorter and stronger (harder to pull the atoms apart) than a double or a single bond.

**Uncertainty Principle**—The statement that the momentum and position of a particle cannot be known exactly simultaneously. If the momentum of a particle is known exactly, then the position is completely uncertain, that is, there can be no information on the position. If the position is known exactly, there can be no information on the magnitude of the momentum. In general, the principle states that the position and momentum can only be known within a certain degree of uncertainty. This is intrinsic to nature and not a consequence of experimental error.

**vector**—A directed line segment usually represented by an arrow. A vector is a quantity with both magnitude and direction. A car going 60 miles per hour has a speed, which is not a vector. A car going 60 miles per hour north has a velocity, which is a vector because it has a magnitude (60 miles per hour) and a direction (north).

**velocity**—A vector describing both the speed and the direction in which an object is moving.

**wave packet**—A superposition of waves that combine to make a particle more or less located in a region of space. The superposition of waves has regions of constructive and destructive interference. The probability of finding the particle is large where there is constructive interference. The superposition of waves more or less localizes a particle in some region of space. The location cannot be perfectly defined because of the Heisenberg Uncertainty Principle.

**wavefunction**—A solution to the Schrödinger Equation for a particular state of a system, such as an atom or molecule. A wavefunction is a probability amplitude wave. It gives information on finding a particle in a particular region of space. For example, the wavefunctions for the hydrogen atom give the probabilities of finding the electron at different distances and directions from the nucleus.

**wavefunction collapse**—A state of a system is frequently a superposition of wavefunctions. Each wavefunction has associated with it a definite value of an observable, for example, the energy. Because a superposition is composed of many wavefunctions, it has associated with it many values of an observable. When a measurement is made, the system goes from being in a superposition of wavefunction to being in a single wavefunction with one value of the observable (e.g., the energy). It is said that the measurement causes the wavefunction to collapse from a superposition of states into a single state with one value of the observable. It is not possible to say beforehand which state the superposition will collapse into. Therefore, it is not possible to say ahead of time which value of the observable will be measured.

**wavelength**—The repeat distance in a wave, that is, the distance from one peak in the wave to another.