What is Quantum Physics?
The behaviour of matter and radiation on the atomic scale often seems peculiar, and the consequences of quantum theory are accordingly difficult to understand and to believe. Its concepts frequently conflict with common-sense notions derived from observations of the everyday world. There is no reason, however, why the behaviour of the atomic world should conform to that of the familiar, large-scale world. It is important to realize that quantum physics is a branch of physics and that the business of physics is to describe and account for the way the world—on both the large and the small scale—actually is and not how one imagines it or would like it to be.
The study of quantum physics is rewarding for several reasons. First, it illustrates the essential methodology of physics. Second, it has been enormously successful in giving correct results in practically every situation to which it has been applied. There is, however, an intriguing paradox. In spite of the overwhelming practical success of quantum mechanics, the foundations of the subject contain unresolved problems—in particular, problems concerning the nature of measurement. An essential feature of quantum mechanics is that it is generally impossible, even in principle, to measure a system without disturbing it; the detailed nature of this disturbance and the exact point at which it occurs are obscure and controversial. Thus, quantum mechanics attracted some of the ablest scientists of the 20th century, and they erected what is perhaps the finest intellectual edifice of the period.
Early Developments of Quantum Physics
Planck's Radiation Law
By the end of the 19th century, physicists almost universally accepted the wave theory of light. However, though the ideas of classical physics explain interference and diffraction phenomena relating to the propagation of light, they do not account for the absorption and emission of light. All bodies radiate electromagnetic energy as heat; in fact, a body emits radiation at all wavelengths. The energy radiated at different wavelengths is a maximum at a wavelength that depends on the temperature of the body; the hotter the body, the shorter the wavelength for maximum radiation. Attempts to calculate the energy distribution for the radiation from a blackbody using classical ideas were unsuccessful. One formula, proposed by Wilhelm Wien of Germany, did not agree with observations at long wavelengths, and another, proposed by Lord Rayleigh (John William Strutt) of England, disagreed with those at short wavelengths.
In 1900 the German theoretical physicist Max Planck made a bold suggestion. He assumed that the radiation energy is emitted, not continuously, but rather in discrete packets called quanta. The energy E of the quantum is related to the frequency ν by E = hν. The quantity h, now known as Planck’s constant, is a universal constant with the approximate value of 6.62607 × 10−34 joule∙second. Planck showed that the calculated energy spectrum then agreed with observation over the entire wavelength range.
Einstein's Photoelectric Effect
In 1905 Einstein extended Planck’s hypothesis to explain the photoelectric effect, which is the emission of electrons by a metal surface when it is irradiated by light or more-energetic photons. The kinetic energy of the emitted electrons depends on the frequency ν of the radiation, not on its intensity; for a given metal, there is a threshold frequency ν0 below which no electrons are emitted. Furthermore, emission takes place as soon as the light shines on the surface; there is no detectable delay. Einstein showed that these results can be explained by two assumptions: (1) that light is composed of corpuscles or photons, the energy of which is given by Planck’s relationship, and (2) that an atom in the metal can absorb either a whole photon or nothing. Part of the energy of the absorbed photon frees an electron, which requires a fixed energy W, known as the work function of the metal; the rest is converted into the kinetic energy meu2/2 of the emitted electron (me is the mass of the electron and u is its velocity). Thus, the energy relation is
If ν is less than ν0, where hν0 = W, no electrons are emitted. Not all the experimental results mentioned above were known in 1905, but all Einstein’s predictions have been verified since.
Bohr's Theory of Atom
A major contribution to the subject was made by Niels Bohr of Denmark, who applied the quantum hypothesis to atomic spectra in 1913. The spectra of light emitted by gaseous atoms had been studied extensively since the mid-19th century. It was found that radiation from gaseous atoms at low pressure consists of a set of discrete wavelengths. This is quite unlike the radiation from a solid, which is distributed over a continuous range of wavelengths. The set of discrete wavelengths from gaseous atoms is known as a line spectrum, because the radiation (light) emitted consists of a series of sharp lines. The wavelengths of the lines are characteristic of the element and may form extremely complex patterns. The simplest spectra are those of atomic hydrogen and the alkali atoms (e.g., lithium, sodium, and potassium). For hydrogen, the wavelengths λ are given by the empirical formula
where m and n are positive integers with n > m and R∞, known as the Rydberg Constant, has the value 1.097373157 × 107 per metre. For a given value of m, the lines for varying n form a series. The lines for m = 1, the Lyman Series, lie in the ultraviolet part of the spectrum; those for m = 2, the Balmer Series, lie in the visible spectrum; and those for m = 3, the Paschen series, lie in the infrared.
Bohr started with a model suggested by the New Zealand-born British physicist Ernest Rutherford. The model was based on the experiments of Hans Geiger and Ernest Marsden, who in 1909 bombarded gold atoms with massive, fast-moving alpha particles; when some of these particles were deflected backward, Rutherford concluded that the atom has a massive, charged nucleus. In Rutherford's model, the atom resembles a miniature solar system with the nucleus acting as the Sun and the electrons as the circulating planets. Bohr made three assumptions. First, he postulated that, in contrast to classical mechanics, where an infinite number of orbits is possible, an electron can be in only one of a discrete set of orbits, which he termed stationary state. Second, he postulated that the only orbits allowed are those for which the angular momentum of the electron is a whole number n times ℏ (ℏ = h/2π). Third, Bohr assumed that Newton's Law Of Motion, so successful in calculating the paths of the planets around the Sun, also applied to electrons orbiting the nucleus. The force on the electron (the analogue of the gravitational force between the Sun and a planet) is the electrostatic attraction between the positively charged nucleus and the negatively charged electron. With these simple assumptions, he showed that the energy of the orbit has the form
where E0 is a constant that may be expressed by a combination of the known constants e, me, and ℏ. While in a stationary state, the atom does not give off energy as light; however, when an electron makes a transition from a state with energy En to one with lower energy Em, a quantumof energy is radiated with frequency ν, given by the equation
Inserting the expression for En into this equation and using the relation λν = c, where c is the speed of light, Bohr derived the formula for the wavelengths of the lines in the hydrogen spectrum, with the correct value of the Rydberg constant.
Bohr’s theory was a brilliant step forward. Its two most important features have survived in present-day quantum mechanics. They are (1) the existence of stationary, nonradiating states and (2) the relationship of radiation frequency to the energy difference between the initial and final states in a transition. Prior to Bohr, physicists had thought that the radiation frequency would be the same as the electron’s frequency of rotation in an orbit.
Scattering of X-Rays
Soon scientists were faced with the fact that another form of radiation, X-rays, also exhibits both wave and particle properties. Max Von Laue of Germany had shown in 1912 that crystals can be used as three-dimensional diffraction gratings for X-rays; his technique constituted the fundamental evidence for the wavelike nature of X-rays. The atoms of a crystal, which are arranged in a regular lattice, scatter the X-rays. For certain directions of scattering, all the crests of the X-rays coincide. (The scattered X-rays are said to be in phase and to give constructive interference.) For these directions, the scattered X-ray beam is very intense. Clearly, this phenomenon demonstrates wave behaviour. In fact, given the interatomic distances in the crystal and the directions of constructive interference, the wavelength of the waves can be calculated.
In 1922 the American physicist Arthur Holly Crompton showed that X-rays scatter from electrons as if they are particles. Compton performed a series of experiments on the scattering of monochromatic, high-energy X-rays by graphite. He found that part of the scattered radiation had the same wavelength λ0 as the incident X-rays but that there was an additional component with a longer wavelength λ. To interpret his results, Compton regarded the X-ray photon as a particle that collides and bounces off an electron in the graphite target as though the photon and the electron were a pair of (dissimilar) billiard balls. Application of the laws of conservation of energy and momentum to the collision leads to a specific relation between the amount of energy transferred to the electron and the angle of scattering. For X-rays scattered through an angle θ, the wavelengths λ and λ0 are related by the equation
The experimental correctness of Compton’s formula is direct evidence for the corpuscular behaviour of radiation.
Applications Of Quantum Physics
Decay Of the Kaon
The kaon (also called the K0 meson), discovered in 1947, is produced in high-energy collisions between nuclei and other particles. It has zero electric charge, and its mass is about one-half the mass of the proton. It is unstable and, once formed, rapidly decays into either 2 or 3 pi-mesons. The average lifetime of the kaon is about 10−10 second.
In spite of the fact that the kaon is uncharged, quantum theory predicts the existence of an antiparticle with the same mass, decay products, and average lifetime; the antiparticle is denoted by 0. During the early 1950s, several physicists questioned the justification for postulating the existence of two particles with such similar properties. In 1955, however, Murray Gell-Man and Abraham Pais made an interesting prediction about the decay of the kaon. Their reasoning provides an excellent illustration of the quantum mechanical axiom that the wave functiom Ψ can be a superposition of states; in this case, there are two states, the K0 and 0 mesons themselves.
A K0 meson may be represented formally by writing the wave function as Ψ = K0; similarly Ψ = 0 represents a 0 meson. From the two states, K0 and 0, the following two new states are constructed:
From these two equations it follows that
The reason for defining the two states K1 and K2 is that, according to quantum theory, when the K0 decays, it does not do so as an isolated particle; instead, it combines with its antiparticle to form the states K1 and K2. The state K1 (called the K-short [K0S]) decays into two pi-mesons with a very short lifetime (about 9 × 10−11 second), while K2 (called the K-long [K0L]) decays into three pi-mesons with a longer lifetime (about 5 × 10−8 second).
The physical consequences of these results may be demonstrated in the following experiment. K0 particles are produced in a nuclear reaction at the point A (Figure 1). They move to the right in the figure and start to decay. At point A, the wave function is Ψ = K0, which, from equation (16), can be expressed as the sum of K1 and K2. As the particles move to the right, the K1 state begins to decay rapidly. If the particles reach point B in about 10−8 second, nearly all the K1 component has decayed, although hardly any of the K2 component has done so. Thus, at point B, the beam has changed from one of pure K0 to one of almost pure K2, which equation (15) shows is an equal mixture of K0 and 0. In other words, 0 particles appear in the beam simply because K1 and K2 decay at different rates. At point B, the beam enters a block of absorbing material. Both the K0 and 0 are absorbed by the nuclei in the block, but the 0 are absorbed more strongly. As a result, even though the beam is an equal mixture of K0 and 0 when it enters the absorber, it is almost pure K0 when it exits at point C. The beam thus begins and ends as K0.
(Figure :1)
Gell-Mann and Pais predicted all this, and experiments subsequently verified it. The experimental observations are that the decay products are primarily two pi-mesons with a short decay time near A, three pi-mesons with longer decay time near B, and two pi-mesons again near C. (This account exaggerates the changes in the K1 and K2 components between A and B and in the K0 and 0 components between B and C; the argument, however, is unchanged.) The phenomenon of generating the 0 and regenerating the K1 decay is purely quantum. It rests on the quantum axiom of the superposition of states and has no classical counterpart.
Cesium Clock
The cesium clock is the most accurate type of clock yet developed. This device makes use of transitions between the spin states of the cesium nucleus and produces a frequency which is so regular that it has been adopted for establishing the time standard.
Like electrons, many atomic nuclei have spin. The spin of these nuclei produces a set of small effects in the spectra, known as hyperfine structure. (The effects are small because, though the angular momentum of a spinning nucleus is of the same magnitude as that of an electron, its magnetic moment, which governs the energies of the atomic levels, is relatively small.) The nucleus of the cesium atom has spin quantum number 7/2. The total angular momentum of the lowest energy states of the cesium atom is obtained by combining the spin angular momentum of the nucleus with that of the single valence electron in the atom. (Only the valence electron contributes to the angular momentum because the angular momenta of all the other electrons total zero. Another simplifying feature is that the ground states have zero orbital momenta, so only spin angular momenta need to be considered.) When nuclear spin is taken into account, the total angular momentum of the atom is characterized by a quantum number, conventionally denoted by F, which for cesium is 4 or 3. These values come from the spin value 7/2 for the nucleus and 1/2 for the electron. If the nucleus and the electron are visualized as tiny spinning tops, the value F = 4 (7/2 + 1/2) corresponds to the tops spinning in the same sense, and F = 3 (7/2 − 1/2) corresponds to spins in opposite senses. The energy difference ΔE of the states with the two F values is a precise quantity. If electromagnetic radiation of frequency ν0, where
is applied to a system of cesium atoms, transitions will occur between the two states. An apparatus that can detect the occurrence of transitions thus provides an extremely precise frequency standard. This is the principle of the cesium clock.
The apparatus is shown schematically in Figure 2. A beam of cesium atoms emerges from an oven at a temperature of about 100 °C. The atoms pass through an inhomogeneous magnet A, which deflects the atoms in state F = 4 downward and those in state F = 3 by an equal amount upward. The atoms pass through slit S and continue into a second inhomogeneous magnet B. Magnet B is arranged so that it deflects atoms with an unchanged state in the same direction that magnet A deflected them. The atoms follow the paths indicated by the broken lines in the figure and are lost to the beam. However, if an alternating electromagnetic field of frequency ν0 is applied to the beam as it traverses the centre region C, transitions between states will occur. Some atoms in state F = 4 will change to F = 3, and vice versa. For such atoms, the deflections in magnet B are reversed. The atoms follow the whole lines in the diagram and strike a tungsten wire, which gives electric signals in proportion to the number of cesium atoms striking the wire. As the frequency ν of the alternating field is varied, the signal has a sharp maximum for ν = ν0. The length of the apparatus from the oven to the tungsten detector is about one metre.
Each atomic state is characterized not only by the quantum number F but also by a second quantum number mF. For F = 4, mF can take integral values from 4 to −4. In the absence of a magnetic field, these states have the same energy. A magnetic field, however, causes a small change in energy proportional to the magnitude of the field and to the mF value. Similarly, a magnetic field changes the energy for the F = 3 states according to the mF value which, in this case, may vary from 3 to −3. The energy changes are indicated in Figure 3. In the cesium clock, a weak constant magnetic field is superposed on the alternating electromagnetic field in region C. The theory shows that the alternating field can bring about a transition only between pairs of states with mF values that are the same or that differ by unity. However, as can be seen from the figure, the only transitions occurring at the frequency ν0 are those between the two states with mF = 0. The apparatus is so sensitive that it can discriminate easily between such transitions and all the others.
If the frequency of the oscillator drifts slightly so that it does not quite equal ν0, the detector output drops. The change in signal strength produces a signal to the oscillator to bring the frequency back to the correct value. This feedback system keeps the oscillator frequency automatically locked to ν0.The cesium clock is exceedingly stable. The frequency of the oscillator remains constant to about one part in 1013. For this reason, the device is used to redefine the second. This base unit of time in the SI system is defined as equal to 9,192,631,770 cycles of the radiation corresponding to the transition between the levels F = 4, mF = 0 and F = 3, mF = 0 of the ground state of the cesium-133 atom. Prior to 1967, the second was defined in terms of the motion of Earth. The latter, however, is not nearly as stable as the cesium clock. Specifically, the fractional variation of Earth’s rotation period is a few hundred times larger than that of the frequency of the cesium clock.
A quantum Voltage standard
Quantum theory has been used to establish a voltage standard, and this standard has proven to be extraordinarily accurate and consistent from laboratory to laboratory.
If two layers of superconducting material are separated by a thin insulating barrier, a supercurrent (i.e., a current of paired electrons) can pass from one superconducto to the other. This is another example of the tunneling process described earlier. Several effects based on this phenomenon were predicted in 1962 by the British physicist Brian D. Jossephson. Demonstrated experimentally soon afterwards, they are now referred to as the Jossephson Effects.
If a DC (direct-current) voltage V is applied across the two superconductors, the energy of an electron pair changes by an amount of 2eV as it crosses the junction. As a result, the supercurrent oscillates with frequency ν given by the Planck relationship (E = hν). Thus,
This oscillatory behaviour of the supercurrent is known as the AC (alternating-current) Josephson Effect. Measurement of V and ν permits a direct verification of the Planck relationship. Although the oscillating supercurrent has been detected directly, it is extremely weak. A more sensitive method of investigating equation (19) is to study effects resulting from the interaction of microwave radiation with the supercurrent.
Several carefully conducted experiments have verified equation (19) to such a high degree of precision that it has been used to determine the value of 2e/h. This value can in fact be determined more precisely by the AC Josephson effect than by any other method. The result is so reliable that laboratories now employ the AC Josephson effect to set a voltage standard. The numerical relationship between V and ν is
In this way, measuring a frequency, which can be done with great precision, gives the value of the voltage. Before the Josephson method was used, the voltage standard in metrological laboratories devoted to the maintenance of physical units was based on high-stability Weston Cadmium Cells. These cells, however, tend to drift and so caused inconsistencies between standards in different laboratories. The Josephson method has provided a standard giving agreement to within a few parts in 108 for measurements made at different times and in different laboratories.
The experiments described in the preceding two sections are only two examples of high-precision measurements in physics. The values of the fundamental constants, such as c, h, e, and me, are determined from a wide variety of experiments based on quantum phenomena. The results are so consistent that the values of the constants are thought to be known in most cases to better than one part in 106. Physicists may not know what they are doing when they make a measurement, but they do it extremely well.
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