The development of quantum mechanics began in 1900 with Planck's study of the light emitted by heated solids, so we start by discussing the nature of light.
In 1803, Thomas Young gave convincing evidence for the wave nature of light by observing diffraction and interference when light went through two adjacent pinholes. (Diffraction is the bending of a wave around an obstacle. Interference is the combining of two waves of the same frequency to give a wave whose disturbance at each point in space is the algebraic or vector sum of the disturbances at that point resulting from each interfering wave. See any first-year physics text.)
In 1864, James Clerk Maxwell published four equations, known as Maxwell's equations, which unified the laws of electricity and magnetism. Maxwell's equations predicted that an accelerated electric charge would radiate energy in the form of electromagnetic waves consisting of oscillating electric and magnetic fields. The speed predicted by Maxwell's equations for these waves turned out to be the same as the experimentally measured speed of light. Maxwell concluded that light is an electromagnetic wave.
In 1888, Heinrich Hertz detected radio waves produced by accelerated electric charges in a spark, as predicted by Maxwell's equations. This convinced physicists that light is indeed an electromagnetic wave.
All electromagnetic waves travel at speed $c=2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}$ in vacuum. The frequency $\nu$ and wavelength $\lambda$ of an electromagnetic wave are related by
\( \begin{equation<em>} \lambda \nu=c \tag{1.1} \end{equation</em>} \)
(Equations that are enclosed in a box should be memorized. The Appendix gives the Greek alphabet.) Various conventional labels are applied to electromagnetic waves depending on their frequency. In order of increasing frequency are radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. We shall use the term light to denote any kind of electromagnetic radiation. Wavelengths of visible and ultraviolet radiation were formerly given in angstroms ( $\AA$ ) and are now given in nanometers ( nm ):
\( \begin{equation<em>} 1 \mathrm{~nm}=10^{-9} \mathrm{~m}, \quad 1 \AA=10^{-10} \mathrm{~m}=0.1 \mathrm{~nm} \tag{1.2} \end{equation</em>} \)
In the 1890 s, physicists measured the intensity of light at various frequencies emitted by a heated blackbody at a fixed temperature, and did these measurements at several temperatures. A blackbody is an object that absorbs all light falling on it. A good approximation to a blackbody is a cavity with a tiny hole. In 1896, the physicist Wien proposed the following equation for the dependence of blackbody radiation on light frequency and blackbody temperature: $I=a \nu^{3} / e^{b \nu / T}$, where $a$ and $b$ are empirical constants, and $I d \nu$ is the energy with frequency in the range $\nu$ to $\nu+d \nu$ radiated per unit time and per unit surface area by a blackbody, with $d \nu$ being an infinitesimal frequency range. Wien's formula gave a good fit to the blackbody radiation data available in 1896, but his theoretical arguments for the formula were considered unsatisfactory.
In 1899-1900, measurements of blackbody radiation were extended to lower frequencies than previously measured, and the low-frequency data showed significant deviations from Wien's formula. These deviations led the physicist Max Planck to propose in October 1900 the following formula: $I=a \nu^{3} /\left(e^{b \nu / T}-1\right)$, which was found to give an excellent fit to the data at all frequencies.
Having proposed this formula, Planck sought a theoretical justification for it. In December 1900, he presented a theoretical derivation of his equation to the German Physical Society. Planck assumed the radiation emitters and absorbers in the blackbody to be harmonically oscillating electric charges ("resonators") in equilibrium with electromagnetic radiation in a cavity. He assumed that the total energy of those resonators whose frequency is $\nu$ consisted of $N$ indivisible "energy elements," each of magnitude $h \nu$, where $N$ is an integer and $h$ (Planck's constant) was a new constant in physics. Planck distributed these energy elements among the resonators. In effect, this restricted the energy of each resonator to be a whole-number multiple of $h v$ (although Planck did not explicitly say this). Thus the energy of each resonator was quantized, meaning that only certain discrete values were allowed for a resonator energy. Planck's theory showed that $a=2 \pi h / c^{2}$ and $b=h / k$, where $k$ is Boltzmann's constant. By fitting the experimental blackbody curves, Planck found $h=6.6 \times 10^{-34} \mathrm{~J} \cdot \mathrm{~s}$.
Planck's work is usually considered to mark the beginning of quantum mechanics. However, historians of physics have debated whether Planck in 1900 viewed energy quantization as a description of physical reality or as merely a mathematical approximation that allowed him to obtain the correct blackbody radiation formula. [See O. Darrigol, Centaurus, 43, 219 (2001); C. A. Gearhart, Phys. Perspect., 4, 170 (2002) (available online at employees.csbsju.edu/cgearhart/Planck/PQH.pdf; S. G. Brush, Am. J. Phys., 70, 119 (2002) (www.punsterproductions.com/~sciencehistory/cautious.htm).] The physics historian Kragh noted that "If a revolution occurred in physics in December 1900, nobody seemed to notice it. Planck was no exception, and the importance ascribed to his work is largely a historical reconstruction" (H. Kragh, Physics World, Dec. 2000, p. 31).
The concept of energy quantization is in direct contradiction to all previous ideas of physics. According to Newtonian mechanics, the energy of a material body can vary continuously. However, only with the hypothesis of quantized energy does one obtain the correct blackbody-radiation curves.
The second application of energy quantization was to the photoelectric effect. In the photoelectric effect, light shining on a metal causes emission of electrons. The energy of a wave is proportional to its intensity and is not related to its frequency, so the electromagnetic-wave picture of light leads one to expect that the kinetic energy of an emitted photoelectron would increase as the light intensity increases but would not change as the light frequency changes. Instead, one observes that the kinetic energy of an emitted electron is independent of the light's intensity but increases as the light's frequency increases.
In 1905, Einstein showed that these observations could be explained by regarding light as composed of particlelike entities (called photons), with each photon having an energy
\( \begin{equation<em>} E_{\text {photon }}=h \nu \tag{1.3} \end{equation</em>} \)
When an electron in the metal absorbs a photon, part of the absorbed photon energy is used to overcome the forces holding the electron in the metal; the remainder appears as kinetic energy of the electron after it has left the metal. Conservation of energy gives $h \nu=\Phi+T$, where $\Phi$ is the minimum energy needed by an electron to escape the metal (the metal's work function), and $T$ is the maximum kinetic energy of an emitted electron. An increase in the light's frequency $\nu$ increases the photon energy and hence increases the kinetic energy of the emitted electron. An increase in light intensity at fixed frequency increases the rate at which photons strike the metal and hence increases the rate of emission of electrons, but does not change the kinetic energy of each emitted electron. (According to Kragh, a strong "case can be made that it was Einstein who first recognized the essence of quantum theory"; Kragh, Physics World, Dec. 2000, p. 31.)
The photoelectric effect shows that light can exhibit particlelike behavior in addition to the wavelike behavior it shows in diffraction experiments.
In 1907, Einstein applied energy quantization to the vibrations of atoms in a solid element, assuming that each atom's vibrational energy in each direction $(x, y, z)$ is restricted to be an integer times $h \nu_{\text {vib }}$, where the vibrational frequency $\nu_{\text {vib }}$ is characteristic of the element. Using statistical mechanics, Einstein derived an expression for the constantvolume heat capacity $C_{V}$ of the solid. Einstein's equation agreed fairly well with known $C_{V}$-versus-temperature data for diamond.
Now let us consider the structure of matter. In the late nineteenth century, investigations of electric discharge tubes and natural radioactivity showed that atoms and molecules are composed of charged particles. Electrons have a negative charge. The proton has a positive charge equal in magnitude but opposite in sign to the electron charge and is 1836 times as heavy as the electron. The third constituent of atoms, the neutron (discovered in 1932), is uncharged and slightly heavier than the proton.
Starting in 1909, Rutherford, Geiger, and Marsden repeatedly passed a beam of alpha particles through a thin metal foil and observed the deflections of the particles by allowing them to fall on a fluorescent screen. Alpha particles are positively charged helium nuclei obtained from natural radioactive decay. Most of the alpha particles passed through the foil essentially undeflected, but, surprisingly, a few underwent large deflections, some being deflected backward. To get large deflections, one needs a very close approach between the charges, so that the Coulombic repulsive force is great. If the positive charge were spread throughout the atom (as J. J. Thomson had proposed in 1904), once the high-energy alpha particle penetrated the atom, the repulsive force would fall off, becoming zero at the center of the atom, according to classical electrostatics. Hence Rutherford concluded that such large deflections could occur only if the positive charge were concentrated in a tiny, heavy nucleus.
An atom contains a tiny ( $10^{-13}$ to $10^{-12} \mathrm{~cm}$ radius), heavy nucleus consisting of neutrons and $Z$ protons, where $Z$ is the atomic number. Outside the nucleus there are $Z$ electrons. The charged particles interact according to Coulomb's law. (The nucleons are held together in the nucleus by strong, short-range nuclear forces, which will not concern us.) The radius of an atom is about one angstrom, as shown, for example, by results from the kinetic theory of gases. Molecules have more than one nucleus.
The chemical properties of atoms and molecules are determined by their electronic structure, and so the question arises as to the nature of the motions and energies of the electrons. Since the nucleus is much more massive than the electron, we expect the motion of the nucleus to be slight compared with the electrons' motions.
In 1911, Rutherford proposed his planetary model of the atom in which the electrons revolved about the nucleus in various orbits, just as the planets revolve about the sun. However, there is a fundamental difficulty with this model. According to classical electromagnetic theory, an accelerated charged particle radiates energy in the form of electromagnetic (light) waves. An electron circling the nucleus at constant speed is being accelerated, since the direction of its velocity vector is continually changing. Hence the electrons in the Rutherford model should continually lose energy by radiation and therefore would spiral toward the nucleus. Thus, according to classical (nineteenth-century) physics, the Rutherford atom is unstable and would collapse.
A possible way out of this difficulty was proposed by Niels Bohr in 1913, when he applied the concept of quantization of energy to the hydrogen atom. Bohr assumed that the energy of the electron in a hydrogen atom was quantized, with the electron constrained to move only on one of a number of allowed circles. When an electron makes a transition from one Bohr orbit to another, a photon of light whose frequency $v$ satisfies
\( \begin{equation<em>} E_{\text {upper }}-E_{\text {lower }}=h \nu \tag{1.4} \end{equation</em>} \)
is absorbed or emitted, where $E_{\text {upper }}$ and $E_{\text {lower }}$ are the energies of the upper and lower states (conservation of energy). With the assumption that an electron making a transition from a free (ionized) state to one of the bound orbits emits a photon whose frequency is an integral multiple of one-half the classical frequency of revolution of the electron in the bound orbit, Bohr used classical mechanics to derive a formula for the hydrogenatom energy levels. Using (1.4), he got agreement with the observed hydrogen spectrum. However, attempts to fit the helium spectrum using the Bohr theory failed. Moreover, the theory could not account for chemical bonds in molecules.
The failure of the Bohr model arises from the use of classical mechanics to describe the electronic motions in atoms. The evidence of atomic spectra, which show discrete frequencies, indicates that only certain energies of motion are allowed; the electronic energy is quantized. However, classical mechanics allows a continuous range of energies. Quantization does occur in wave motion-for example, the fundamental and overtone frequencies of a violin string. Hence Louis de Broglie suggested in 1923 that the motion of electrons might have a wave aspect; that an electron of mass $m$ and speed $v$ would have a wavelength
\( \begin{equation<em>} \lambda=\frac{h}{m v}=\frac{h}{p} \tag{1.5} \end{equation</em>} \)
associated with it, where $p$ is the linear momentum. De Broglie arrived at Eq. (1.5) by reasoning in analogy with photons. The energy of a photon can be expressed, according to Einstein's special theory of relativity, as $E=p c$, where $c$ is the speed of light and $p$ is the photon's momentum. Using $E_{\text {photon }}=h \nu$, we get $p c=h \nu=h c / \lambda$ and $\lambda=h / p$ for a photon traveling at speed $c$. Equation (1.5) is the corresponding equation for an electron.
In 1927, Davisson and Germer experimentally confirmed de Broglie's hypothesis by reflecting electrons from metals and observing diffraction effects. In 1932, Stern observed the same effects with helium atoms and hydrogen molecules, thus verifying that the wave effects are not peculiar to electrons, but result from some general law of motion for microscopic particles. Diffraction and interference have been observed with molecules as large as $\mathrm{C}{48} \mathrm{H}{26} \mathrm{~F}{24} \mathrm{~N}{8} \mathrm{O}{8}$ passing through a diffraction grating [T. Juffmann et al., Nat. Nanotechnol., 7, 297 (2012).]. A movie of the buildup of an interference pattern involving $\mathrm{C}{32} \mathrm{H}{18} \mathrm{~N}{8}$ molecules can be seen at
Thus electrons behave in some respects like particles and in other respects like waves. We are faced with the apparently contradictory "wave-particle duality" of matter (and of light). How can an electron be both a particle, which is a localized entity, and a wave, which is nonlocalized? The answer is that an electron is neither a wave nor a particle, but something else. An accurate pictorial description of an electron's behavior is impossible using the wave or particle concept of classical physics. The concepts of classical physics have been developed from experience in the macroscopic world and do not properly describe the microscopic world. Evolution has shaped the human brain to allow it to understand and deal effectively with macroscopic phenomena. The human nervous system was not developed to deal with phenomena at the atomic and molecular level, so it is not surprising if we cannot fully understand such phenomena.
Although both photons and electrons show an apparent duality, they are not the same kinds of entities. Photons travel at speed $c$ in vacuum and have zero rest mass; electrons always have $v<c$ and a nonzero rest mass. Photons must always be treated relativistically, but electrons whose speed is much less than $c$ can be treated nonrelativistically.